Upload
others
View
4
Download
1
Embed Size (px)
Citation preview
MATH 1300
Fundamentals of
Mathematics
University of Houston Department of Mathematics
MATH 1300, Fundamentals of Mathematics
© 2011 University of Houston Department of Mathematics
MATH 1300 Fundamentals of Mathematics i
MATH 1300 – Fundamentals of Mathematics
Table of Contents
University of Houston Department of Mathematics
CHAPTER 1: INTRODUCTORY INFORMATION AND REVIEW ................................... 1
Section 1.1: Numbers ................................................................................................................ 1
Types of Numbers................................................................................................................... 1
Order on a Number Line ....................................................................................................... 16
Exercise Set 1.1 ..................................................................................................................... 23
Section 1.2: Integers ................................................................................................................ 26
Operations with Integers ....................................................................................................... 26
Exercise Set 1.2 ..................................................................................................................... 35
Section 1.3: Fractions .............................................................................................................. 36
Greatest Common Divisor and Least Common Multiple ..................................................... 36
Addition and Subtraction of Fractions.................................................................................. 43
Multiplication and Division of Fractions.............................................................................. 50
Exercise Set 1.3 ..................................................................................................................... 56
Section 1.4: Exponents and Radicals ...................................................................................... 58
Evaluating Exponential Expressions..................................................................................... 58
Square Roots ......................................................................................................................... 66
Exercise Set 1.4 ..................................................................................................................... 72
Section 1.5: Order of Operations............................................................................................. 75
Evaluating Expressions Using the Order of Operations ....................................................... 75
Exercise Set 1.5 ..................................................................................................................... 79
Section 1.6: Solving Linear Equations .................................................................................... 82
Linear Equations ................................................................................................................... 82
Exercise Set 1.6 ..................................................................................................................... 85
Section 1.7: Interval Notation and Linear Inequalities............................................................ 86
Linear Inequalities ................................................................................................................ 86
Exercise Set 1.7 ..................................................................................................................... 94
University of Houston Department of Mathematics ii
Section 1.8: Absolute Value and Equations ............................................................................ 96
Absolute Value...................................................................................................................... 96
Exercise Set 1.8 ................................................................................................................... 103
CHAPTER 2: POINTS, LINES, AND FUNCTIONS .......................................................... 104
Section 2.1: An Introduction to the Coordinate Plane........................................................... 104
Points in the Coordinate Plane............................................................................................ 104
Exercise Set 2.1 ................................................................................................................... 117
Section 2.2: The Distance and Midpoint Formulas ............................................................... 120
The Distance Formula......................................................................................................... 120
The Midpoint Formula........................................................................................................ 129
Exercise Set 2.2 ................................................................................................................... 134
Section 2.3: Slope and Intercepts of Lines ............................................................................ 136
The Slope of a Line............................................................................................................. 136
Intercepts of Lines............................................................................................................... 142
Exercise Set 2.3 ................................................................................................................... 149
Section 2.4: Equations of Lines............................................................................................. 152
Writing Equations of Lines................................................................................................. 152
Exercise Set 2.4 ................................................................................................................... 160
Section 2.5: Parallel and Perpendicular Lines....................................................................... 162
Pairs of Lines - Parallel and Perpendicular Lines............................................................... 162
Exercise Set 2.5 ................................................................................................................... 168
Section 2.6: An Introduction to Functions ............................................................................ 170
Definition of a Function...................................................................................................... 170
Domain of a Function ......................................................................................................... 177
Exercise Set 2.6 ................................................................................................................... 181
Section 2.7: Functions and Graphs........................................................................................ 185
Graphing a Function ........................................................................................................... 185
Exercise Set 2.7 ................................................................................................................... 200
CHAPTER 3: POLYNOMIALS ............................................................................................ 203
Section 3.1: An Introduction to Polynomial Functions......................................................... 203
Polynomials and Polynomial Functions.............................................................................. 203
Exercise Set 3.1 ................................................................................................................... 213
MATH 1300 Fundamentals of Mathematics iii
Section 3.2: Adding, Subtracting, and Multiplying Polynomials.......................................... 216
Operations with Polynomials.............................................................................................. 216
Exercise Set 3.2 ................................................................................................................... 223
Section 3.3: Dividing Polynomials........................................................................................ 225
Polynomial Long Division and Synthetic Division ............................................................ 225
Exercise Set 3.3 ................................................................................................................... 238
Section 3.4: Quadratic Functions .......................................................................................... 240
The Definition and Graph of a Quadratic Function ............................................................ 240
Exercise Set 3.4 ................................................................................................................... 249
CHAPTER 4: FACTORING .................................................................................................. 250
Section 4.1: Greatest Common Factor and Factoring by Grouping.................................... 250
GCF and Grouping.............................................................................................................. 250
Exercise Set 4.1 ................................................................................................................... 258
Section 4.2: Factoring Special Binomials and Trinomials .................................................... 260
Special Factor Patterns........................................................................................................ 260
Exercise Set 4.2 ................................................................................................................... 268
Section 4.3: Factoring Polynomials....................................................................................... 270
Techniques for Factoring Trinomials.................................................................................. 270
Exercise Set 4.3 ................................................................................................................... 281
Section 4.4: Using Factoring to Solve Equations.................................................................. 283
Solving Quadratic Equations by Factoring ......................................................................... 283
Solving Other Polynomial Equations by Factoring ............................................................ 290
Exercise Set 4.4 ................................................................................................................... 295
CHAPTER 5: RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS ........... 297
Section 5.1: Simplifying Rational Expressions..................................................................... 297
Rational Expressions........................................................................................................... 297
Exercise Set 5.1 ................................................................................................................... 302
Section 5.2: Multiplying and Dividing Rational Expressions................................................ 304
Multiplication and Division ................................................................................................ 304
Exercise Set 5.2 ................................................................................................................... 309
University of Houston Department of Mathematics iv
Section 5.3: Adding and Subtracting Rational Expressions.................................................. 311
Addition and Subtraction .................................................................................................... 311
Exercise Set 5.3 ................................................................................................................... 319
Section 5.4: Complex Fractions ............................................................................................ 321
Simplifying Complex Fractions.......................................................................................... 321
Exercise Set 5.4 ................................................................................................................... 328
Section 5.5: Solving Rational Equations............................................................................... 331
Rational Equations .............................................................................................................. 331
Exercise Set 5.5 ................................................................................................................... 340
Section 5.6: Rational Functions............................................................................................. 342
Working with Rational Functions....................................................................................... 342
Exercise Set 5.6 ................................................................................................................... 354
ODD-NUMBERED ANSWERS TO EXERCISE SETS ....................................................... 357
MATH 1300 Fundamentals of Mathematics v
MATH 1300 – Fundamentals of Mathematics
Online Resources
University of Houston Department of Mathematics
Math 1300 Online:
All materials found in this textbook can also be found online at:
http://online.math.uh.edu/Math1300/
The Math 1300 online site also contains flash lectures which match each additional example in
the text.
Additional Resources:
Math 1300 is designed to prepare students for Math 1310 (College Algebra). There is some
overlap in course material between these two courses, and the online resources for Math 1310
may prove to be helpful.
The Math 1310 materials can be found online at:
http://online.math.uh.edu/Math1310/
In addition to the textbook material, the online site for Math 1310 contains flash lectures
pertaining to most of the topics in the College Algebra course. These lectures simulate the
classroom experience, with audio of course instructors as they present the material on prepared
lesson notes. The lectures are useful in furthering understanding of the course material and can
only be viewed online.
The next page contains a list of some of the overlapping topics between the courses. This should
help students to locate the flash lectures and other textbook materials in Math 1310 that may be
useful for Math 1300. (Keep in mind that Math 1300 and Math 1310 do not cover identical
topics. It may be necessary to search through the Math 1310 sections to find specific types of
examples covered in Math 1300, and some topics in one course may not be covered at all in the
other.)
University of Houston Department of Mathematics vi
Overlapping Topics between Math 1300 and Math 1310
(to use Math 1310 online resources as a reference for Math 1300)
Math 1300 Math 1310
Section 1.6: Linear Equations Section 2.1: Linear Equations
Section 1.7: Interval Notation and Linear
Inequalities
Section 2.6: Linear Inequalities
Section 1.8: Absolute Value and Equations Section 2.8: Absolute Value
Section 2.1: An Introduction to the Coordinate
Plane
Section 1.1: Points, Regions, Distance
and Midpoints
Section 2.2: The Distance and Midpoint Formulas Section 1.1: Points, Regions, Distance
and Midpoints
Section 2.3: Slope and Intercepts of Lines Section 1.2: Lines
Section 2.4: Equations of Lines Section 1.2: Lines
Section 2.5: Parallel and Perpendicular Lines Section 1.2: Lines
Section 2.6: An Introduction to Functions Section 3.1: Basic Ideas
Section 2.7: Functions and Graphs Section 3.2: Functions and Graphs
Section 3.1: An Introduction to Polynomial
Functions
Section 4.1: Polynomial Functions
Section 3.3: Dividing Polynomials Section 4.2: Dividing Polynomials
Section 3.4: Quadratic Functions Section 3.5: Maximum and Minimum
Values
Section 4.1: Greatest Common Factor and
Factoring by Grouping
Section 2.3: Quadratic Equations
Section 2.5: Other Equations
Section 4.2: Factoring Special Binomials and
Trinomials
Section 2.3: Quadratic Equations
Section 4.3: Factoring Polynomials Section 2.3: Quadratic Equations
Section 2.5: Other Equations
Section 4.4: Using Factoring to Solve Equations Section 2.3: Quadratic Equations
Section 2.5: Other Equations
Section 5.6: Rational Functions Section 4.4: Rational Functions
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 1
Chapter 1 Introductory Information and Review
Section 1.1: Numbers
Types of Numbers
Order on a Number Line
Types of Numbers
Natural Numbers:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 2
Example:
Solution:
Even/Odd Natural Numbers:
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 3
Whole Numbers:
Example:
Solution:
Integers:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 4
Example:
Solution:
Even/Odd Integers:
Example:
Solution:
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 5
Rational Numbers:
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 6
Irrational Numbers:
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 7
Real Numbers:
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 8
Note About Division Involving Zero:
Additional Example 1:
Solution:
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 9
Additional Example 2:
Solution:
Natural Numbers:
Whole Numbers:
Integers:
Prime/Composite Numbers:
Positive/Negative Numbers:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 10
Even/Odd Numbers:
Rational Numbers:
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 11
Additional Example 3:
Solution:
Natural Numbers:
Whole Numbers:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 12
Integers:
Prime/Composite Numbers:
Positive/Negative Numbers:
Even/Odd Numbers:
Rational Numbers:
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 13
Additional Example 4:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 14
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 15
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 16
Order on a Number Line
The Real Number Line:
Example:
Solution:
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 17
Inequality Symbols:
The following table describes additional inequality symbols.
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 18
Example:
Solution:
Example:
Solution:
Additional Example 1:
Solution:
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 19
Additional Example 2:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 20
Additional Example 3:
Solution:
SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics 21
Additional Example 4:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 22
Exercise Set 1.1: Numbers
MATH 1300 Fundamentals of Mathematics 23
State whether each of the following numbers is prime,
composite, or neither. If composite, then list all the
factors of the number.
1. (a) 8 (b) 5 (c) 1
(d) 7 (e) 12
2. (a) 11 (b) 6 (c) 15
(d) 0 (e) 2
Answer the following.
3. In (a)-(e), use long division to change the
following fractions to decimals.
(a) 1
9 (b) 2
9 (c) 3
9
(d) 4
9 (e) 5
9 Note: 3 1
9 3
Notice the pattern above and use it as a
shortcut in (f)-(m) to write the following
fractions as decimals without performing
long division.
(f) 6
9 (g) 7
9 (h) 8
9
(i) 9
9 (j) 10
9 (k) 14
9
(l) 25
9 (m) 29
9 Note: 6 2
9 3
4. Use the patterns from the problem above to
change each of the following decimals to either a
proper fraction or a mixed number.
(a) 0.4 (b) 0.7 (c) 2.3
(d) 1.2 (e) 4.5 (f) 7.6
State whether each of the following numbers is
rational or irrational. If rational, then write the
number as a ratio of two integers. (If the number is
already written as a ratio of two integers, simply
rewrite the number.)
5. (a) 0.7 (b) 5 (c) 3
7
(d) 5 (e) 16 (f) 0.3
(g) 12 (h) 2.3
3.5 (i) e
(j) 4 (k) 0.04004000400004...
6. (a) (b) 0.6 (c) 8
(d) 1.3
4.7 (e)
4
5 (f) 9
(g) 3.1 (h) 10 (i) 0
(j) 7
9 (k) 0.03003000300003…
Circle all of the words that can be used to describe
each of the numbers below.
7. 9
Even Odd Positive Negative
Prime Composite Natural Whole
Integer Rational Irrational Real
Undefined
8. 0.7
Even Odd Positive Negative
Prime Composite Natural Whole
Integer Rational Irrational Real
Undefined
9. 2
Even Odd Positive Negative
Prime Composite Natural Whole
Integer Rational Irrational Real
Undefined
10. 4
7
Even Odd Positive Negative
Prime Composite Natural Whole
Integer Rational Irrational Real
Undefined
Answer the following.
11. Which elements of the set
15
48, 2.1, 0.4, 0, 7, , , 5, 12 belong
to each category listed below?
(a) Even (b) Odd
(c) Positive (d) Negative
(e) Prime (f) Composite
(g) Natural (h) Whole
(i) Integer (j) Real
(k) Rational (l) Irrational
(m) Undefined
Exercise Set 1.1: Numbers
University of Houston Department of Mathematics 24
12. Which elements of the set
3 2
4 56.25, 4 , 3, 5, 1, , 1, 2, 10
belong to each category listed below?
(a) Even (b) Odd
(c) Positive (d) Negative
(e) Prime (f) Composite
(g) Natural (h) Whole
(i) Integer (j) Real
(k) Rational (l) Irrational
(m) Undefined
Fill in each of the following tables. Use “Y” for yes if
the row name applies to the number or “N” for no if it
does not.
13.
250
1 35
10 55 13.3
Undefined
Natural
Whole
Integer
Rational
Irrational
Prime
Composite
Real
14.
2.36 0
05 2
2
27
9 3
Undefined
Natural
Whole
Integer
Rational
Irrational
Prime
Composite
Real
Answer the following. If no such number exists, state
“Does not exist.”
15. Find a number that is both prime and even.
16. Find a rational number that is a composite
number.
17. Find a rational number that is not a whole
number.
18. Find a prime number that is negative.
19. Find a real number that is not a rational number.
20. Find a whole number that is not a natural
number.
21. Find a negative integer that is not a rational
number.
22. Find an integer that is not a whole number.
23. Find a prime number that is an irrational number.
24. Find a number that is both irrational and odd.
Answer True or False. If False, justify your answer.j
25. All natural numbers are integers.
26. No negative numbers are odd.
27. No irrational numbers are even.
28. Every even number is a composite number.
29. All whole numbers are natural numbers.
30. Zero is neither even nor odd.
31. All whole numbers are integers.
32. All integers are rational numbers.
33. All nonterminating decimals are irrational
numbers.
34. Every terminating decimal is a rational number.
Answer the following.
35. List the prime numbers less than 10.
36. List the prime numbers between 20 and 30.
37. List the composite numbers between 7 and 19.
38. List the composite numbers between 31 and 41.
39. List the even numbers between 13 and 97 .
40. List the odd numbers between 29 and 123 .
Exercise Set 1.1: Numbers
MATH 1300 Fundamentals of Mathematics 25
Fill in the appropriate symbol from the set , , .
41. 7 ______ 7
42. 3 ______ 3
43. 7 ______ 7
44. 3 ______ 3
45. 81 ______ 9
46. 5 ______ 25
47. 5.32 ______53
10
48. 7
100______ 0.07
49. 1
3 ______
1
4
50. 1
6 ______
1
5
51. 1
3 ______
1
4
52. 1
6 ______
1
5
53. 15 ______ 4
54. 7 ______ 49
55. 3 ______ 9
56. 29 ______ 5
Answer the following.
57. Find the additive inverse of the following
numbers. If undefined, write “undefined.”
(a) 3 (b) 4 (c) 1
(d) 23
(e) 37
2
58. Find the multiplicative inverse of the following
numbers. If undefined, write “undefined.”
(a) 3 (b) 4 (c) 1
(d) 23
(e) 37
2
59. Find the multiplicative inverse of the following
numbers. If undefined, write “undefined.”
(a) 2 (b) 59
(c) 0
(d) 35
1 (e) 1
60. Find the additive inverse of the following
numbers. If undefined, write “undefined.”
(a) 2 (b) 59
(c) 0
(d) 35
1 (e) 1
61. Place the correct number in each of the following
blanks:
(a) The sum of a number and its additive
inverse is _____. (Fill in the correct
number.)
(b) The product of a number and its
multiplicative inverse is _____. (Fill in the
correct number.)
62. Another name for the multiplicative inverse is
the ____________________.
Order the numbers in each set from least to greatest
and plot them on a number line.
(Hint: Use the approximations 2 1.41 and
3 1.73 .)
63. 0 9
1, 2, 0.4, , , 0.495 4
64. 2
3 ,1 , 0.65 , , 1.5 , 0.643
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 26
Section 1.2: Integers
Operations with Integers
Operations with Integers
Absolute Value:
SECTION 1.2 Integers
MATH 1300 Fundamentals of Mathematics 27
Addition of Integers:
Example:
Solution:
Subtraction of Integers:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 28
Example:
Solution:
Multiplication of Integers:
Example:
Solution:
SECTION 1.2 Integers
MATH 1300 Fundamentals of Mathematics 29
Division of Integers:
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 30
Additional Example 1:
Solution:
SECTION 1.2 Integers
MATH 1300 Fundamentals of Mathematics 31
Additional Example 2:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 32
Additional Example 3:
SECTION 1.2 Integers
MATH 1300 Fundamentals of Mathematics 33
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 34
Additional Example 4:
Solution:
Exercise Set 1.2: Integers
MATH 1300 Fundamentals of Mathematics 35
Evaluate the following.
1. (a) 3 7 (b) 3 ( 7) (c) 3 7
(d) 3 ( 7) (e) 3 0
2. (a) 8 5 (b) 8 5 (c) 8 ( 5)
(d) 8 ( 5) (e) 0 ( 5)
3. (a) 0 4 (b) 4 0 (c) 0 ( 4)
(d) 4 0
4. (a) 6 0 (b) 0 ( 6) (c) 0 6
(d) 6 0
5. (a) 10 2 (b) 10 ( 2) (c) 10 2
(d) 2 ( 10) (e) 2 ( 10) (f) 2 10
(g) 2 10 (h) 10 ( 2)
6. (a) 7 ( 9) (b) 7 9 (c) 7 9
(d) 9 ( 7) (e) 9 ( 7) (f) 9 7
(g) 7 ( 9) (f) 9 7
Fill in the appropriate symbol from the set , , .
7. (a) 1(4) ____ 0 (b) 7( 2) ____ 0
(c) 5( 1)( 2) ____ 0 (d) 3( 1)(0) ____ 0
8. (a) 3( 2) ____ 0 (b) 7( 1) ____ 0
(c) 5(0)( 2) ____ 0 (d) 2( 2)( 2) ___ 0
Evaluate the following. If undefined, write
“Undefined.”
9. (a) 6(0) (b) 6
0 (c)
0
6
(d) 6( 1) (e) 6(1) (f) 6( 1)
(g) 6( 1) (h) 6
1
(i)
6
1
(j) 6
0
(k) 6( 1)( 1) (l)
0
6
10. (a) 1(7) (b) 7
1
(c) 7( 1)
(d) 0( 7) (e) 1( 7) (f) 0
7
(g) 7
1
(h)
0
7 (i)
7
0
(j) 7( 1)( 1) (k) 7(0)( 1) (l) 7
0
11. (a) 10( 2) (b) 10
2
(c) 10(2)
(d) 10
2 (e)
10
2
(f)
10
2
12. (a) 6
3
(b) 6( 3) (c)
6
3
(d) 6(3) (e) 6( 3) (f) 6
3
13. (a) 2( 3)( 4) (b) ( 2)( 3)( 4)
(c) 1( 2)( 3)( 4)
(d) 1(2)( 3)( 4)
14. (a) 3( 2)(5) (b) 3( 2)(5)
(c) 3( 2)( 1)(5)
(d) 3( 2)( 2)( 5)
15. (a) 8 2 (b) 8 ( 2) (c) 8( 2)
(d) 8
2
(e) 8 ( 2) (f) ( 8)(0)
(g) 8( 1) (h) 8 1 (i) 8
1
(j) 0 8 (k) 2 ( 8) (l) 0
8
(m) 2
8
(n)
2
0 (o) 2 8
16. (a) 12
3 (b) 12( 3) (c) 12 3
(d) 3 12 (e) 0( 3) (f) 0 ( 3)
(g) ( 3)(12) (h) 12
1 (i)
3
0
(j) 3
12
(k) 1 ( 3) (l) 1(12)
(m) 0
3 (n) 3 ( 1) (o) 3(1)
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 36
Section 1.3: Fractions
Greatest Common Divisor and Least Common Multiple
Addition and Subtraction of Fractions
Multiplication and Division of Fractions
Greatest Common Divisor and Least Common Multiple
Greatest Common Divisor:
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 37
A Method for Finding the GCD:
Least Common Multiple:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 38
A Method for Finding the LCM:
Example:
Solution:
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 39
The LCM is
Additional Example 1:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 40
The LCM is 2 2 2 3 5 120 .
Additional Example 2:
Solution:
The LCM is 2 3 3 5 7 630 .
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 41
Additional Example 3:
Solution:
The LCM is 2 2 3 3 2 72 .
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 42
Additional Example 4:
Solution:
The LCM is 2 3 3 2 5 180 .
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 43
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions with Like Denominators:
a b a b
c c c
and
a b a b
c c c
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 44
Addition and Subtraction of Fractions with Unlike
Denominators:
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 45
Example:
Solution:
Additional Example 1:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 46
Solution:
Additional Example 2:
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 47
Solution:
Additional Example 3:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 48
Solution:
(b) We must rewrite the given fractions so that they have a common denominator.
Find the LCM of the denominators 14 and 21 to find the least common denominator.
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 49
Additional Example 4:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 50
Multiplication and Division of Fractions
Multiplication of Fractions:
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 51
Example:
Solution:
Division of Fractions:
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 52
Additional Example 1:
Solution:
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 53
Additional Example 2:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 54
Additional Example 3:
Solution:
SECTION 1.3 Fractions
MATH 1300 Fundamentals of Mathematics 55
Additional Example 4:
Solution:
Exercise Set 1.3: Fractions
University of Houston Department of Mathematics 56
For each of the following groups of numbers,
(a) Find their GCD (greatest common divisor).
(b) Find their LCM (least common multiple).
1. 6 and 8
2. 4 and 5
3. 7 and 10
4. 12 and 15
5. 14 and 28
6. 6 and 22
7. 8 and 20
8. 9 and 18
9. 18 and 30
10. 60 and 210
11. 16, 20, and 24
12. 15, 21, and 27
Change each of the following improper fractions to a
mixed number.
13. (a) 97
(b) 23
5 (c)
19
3
14. (a) 103
(b) 17
6 (c)
49
9
15. (a) 274
(b) 32
11 (c)
73
10
16. (a) 1513
(b) 43
8 (c)
57
7
Change each of the following mixed numbers to an
improper fraction.
17. (a) 16
5 (b) 49
7 (c) 23
8
18. (a) 12
3 (b) 78
10 (c) 35
6
19. (a) 57
2 (b) 23
5 (c) 14
12
20. (a) 19
4 (b) 45
11 (c) 37
9
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as a mixed number.)
21. (a) 2 17 7 (b)
8 4 3
11 11 11
22. (a) 3 1
5 5 (b)
4 5 2
9 9 9
23. (a) 4 15 5
8 2 (b) 7 23
3 3
24. (a) 3 21
5 5 (b) 6 2
11 117 5
25. (a) 3 14 4
5 2 (b) 3 45 5
6 7
26. (a) 5 37 7
9 2 (b) 511
4
27. (a) 23
7 (b) 3 910 10
7 3
28. (a) 7 1112 12
6 2 (b) 516 6
8 2
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as a mixed number.)
29. (a) 1 1
4 2 (b)
1 1
3 7
30. (a) 1 1
8 10 (b)
1 1
6 5
31. (a) 1 1 1
4 5 6 (b)
2 3
7 5
32. (a) 1 1 1
2 7 5 (b)
4 3
11 7
33. (a) 1 1
35 10 (b)
3 5
4 6
Exercise Set 1.3: Fractions
MATH 1300 Fundamentals of Mathematics 57
34. (a) 1 1
6 24 (b)
8 7
15 12
35. (a) 3 17 6
4 5 (b) 7 110 2
7 5
36. (a) 5 17 4
10 3 (b) 3112 8
6 4
37. (a) 3 45 7
7 8 (b) 4 29 3
5 1
38. (a) 514 6
7 3 (b) 7 138 24
2 9
39. (a) 7215 12
5 2 (b) 7 516 6
9 2
40. (a) 9 510 8
7 6 (b) 5 314 4
11
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as an improper
fraction.)
41. (a) 2 3
9 4 (b)
4 8
15 9
42. (a) 7 9
16 10 (b)
11 17
14 35
43. (a) 13
5 (b) 23
7
44. (a) 25
9 (b) 27
6
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as an improper
fraction.)
45. (a) 1
53 (b)
521
6 (c)
516
4
46. (a) 3
87 (b)
124
18 (c)
1125
10
47. (a) 1 25
7 11 (b)
10 9
21 8
(c)
3 16
20 15
48. (a) 36 1
25 8
(b) 8 7
19 3 (c)
1 42
14 5
49. (a) 1
520
(b) 8
43 (c)
75
10
50. (a) 3
611
(b) 8
205
(c)
422
9
51. (a) 12 18
35 7 (b)
35
59
(c)
15 5
16 24
52. (a)
14
516
(b) 36 9
5 50 (c)
49 35
24 32
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as a mixed number.)
53. (a) 1045 77
8 (b) 7 98 10
1
54. (a) 329 4
2 (b) 7 416 5
3
55. (a) 1 1732 5 (b) 3 3
5 116 2
56. (a) 1 17 4
3 5 (b) 3 115 1225
57. (a) 5 18 4
5 2 (b) 1719 18
11 1
58. (a) 545 7
4 1 (b) 5 111 22
2 2
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 58
Section 1.4: Exponents and Radicals
Evaluating Exponential Expressions
Square Roots
Evaluating Exponential Expressions
Two Rules for Exponential Expressions:
Example:
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics 59
Solution:
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 60
Additional Properties for Exponential Expressions:
Two Definitions:
Quotient Rule for Exponential Expressions:
Exponential Expressions with Bases of Products:
Exponential Expressions with Bases of Fractions:
Example:
Evaluate each of the following:
(a) 32 (b)
9
6
5
5 (c)
32
5
Solution:
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics 61
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 62
Additional Example 1:
Solution:
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics 63
Additional Example 2:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 64
Additional Example 3:
Solution:
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics 65
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 66
Square Roots
Definitions:
Two Rules for Square Roots:
Writing Radical Expressions in Simplest Radical Form:
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics 67
Example:
Solution:
Example:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 68
Solution:
Exponential Form:
Additional Example 1:
Solution:
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics 69
Additional Example 2:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 70
Solution:
Additional Example 3:
Solution:
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics 71
Exercise Set 1.4: Exponents and Radicals
University of Houston Department of Mathematics 72
Write each of the following products instead as a base
and exponent. (For example, 26 6 6 )
1. (a) 7 7 7 (b) 10 10
(c) 8 8 8 8 8 8 (d) 3 3 3 3 3 3 3
2. (a) 9 9 9 (b) 4 4 4 4 4
(c) 5 5 5 5 (d) 17 17
Fill in the appropriate symbol from the set , , .
3. 27 ______ 0
4. 4
9 ______ 0
5. 6
8 ______ 0
6. 68 ______ 0
7. 210 ______ 2
10
8. 310 ______ 3
10
Evaluate the following.
9. (a) 13 (b)
23 (c) 33
(d) 13 (e)
23 (f) 33
(g) 1
3 (h) 2
3 (i) 3
3
(j) 03 (k)
03 (l) 0
3
(m) 43 (n)
43 (o) 4
3
10. (a) 05 (b)
05 (c)
05
(d) 15 (e)
15 (f)
15
(g) 25 (h)
25 (i)
25
(j) 35 (k)
35 (l)
35
(m) 45 (n)
45 (o)
45
11. (a) 2
0.5 (b)
21
5
(c)
21
9
12. (a) 2
0.03 (b)
41
3
(c)
21
12
Write each of the following products instead as a base
and exponent. (Do not evaluate; simply write the base
and exponent.) No answers should contain negative
exponents.
13. (a) 2 65 5 (b)
2 65 5
14. (a) 8 53 3 (b)
8 53 3
15. (a) 9
2
6
6 (b)
9
2
6
6
16. (a) 9
5
7
7 (b)
9
5
7
7
17. (a) 7 3
8
4 4
4
(b)
11 3
8 5
4 4
4 4
18. (a) 12
5 4
8
8 8 (b)
4 9
4 1
8 8
8 8
19. (a) 6
37 (b) 3
425
20. (a) 4
23 (b) 4
532
Rewrite each expression so that it contains positive
exponent(s) rather than negative exponent(s), and then
evaluate the expression.
21. (a) 15
(b) 25
(c) 35
22. (a) 13
(b) 23
(c) 33
23. (a) 32 (b) 52
24. (a) 27
(b) 410
25. (a)
11
5
(b)
12
3
26. (a)
11
7
(b)
16
5
27. (a) 25 (b)
25
28. (a) 2
8
(b) 28
Exercise Set 1.4: Exponents and Radicals
MATH 1300 Fundamentals of Mathematics 73
Evaluate the following.
29. (a) 3
8
2
2
(b)
2
6
2
2
30. (a) 1
2
5
5
(b) 1
3
5
5
31. (a) 2
032 (b)
21
32
32. (a) 2
213
(b) 0
123
Simplify the following. No answers should contain
negative exponents.
33. (a) 3
3 4 23x y z (b) 3
3 4 23x y z
34. (a) 2
5 3 46x y z (b) 2
5 3 46x y z
35.
13 4 6
7
x x x
x
36.
2 3 4
14 1
x x x
x x
37.
3 2
31 2
k m
k m
38.
44 3 7
3 5 9
a b c
a b c
39. 4 3
1 0 9
2
4
a b
a b
40. 7 0
1 2 4
5
3
d e
d e
41.
0 0
0
a b
a b
42.
0 0
0
c d
c d
43.
23 6
3 2
3
2
a b
a b
44.
32 2
2
5
6
a b
a b
Write each of the following expressions in simplest
radical form or as a rational number (if appropriate).
If it is already in simplest radical form, say so.
45. (a) 1236 (b) 7 (c) 18
46. (a) 20 (b) 49 (c) 1232
47. (a) 1250 (b) 14 (c)
81
16
48. (a) 1219 (b)
16
49 (c) 55
49. (a) 28 (b) 72 (c) 1227
50. (a) 1245 (b) 48 (c) 500
51. (a) 54 (b) 1280 (c) 60
52. (a) 120 (b) 180 (c) 1284
53. (a) 1
5 (b)
123
4
(c) 2
7
54. (a) 1
3 (b)
5
9 (c)
122
5
55. (a) 7
4 (b)
1
10 (c)
3
11
56. (a) 1
6 (b)
11
9 (c)
5
2
Exercise Set 1.4: Exponents and Radicals
University of Houston Department of Mathematics 74
57. (a) 53 (b) 4 5 7x y z
58. (a) 72 (b) 2 9 5a b c
Evaluate the following.
59. (a) 2
5 (b) 4
6 (c) 6
2
60. (a) 2
7 (b) 4
3 (c) 6
10
We can evaluate radicals other than square roots.
With square roots, we know, for example, that
49 7 , since 2
7 49 , and 49 is not a real
number. (There is no real number that when squared
gives a value of 49 , since 27 and
27 give a value
of 49, not 49 . The answer is a complex number,
which will not be addressed in this course.) In a
similar fashion, we can compute the following:
Cube Roots 3 125 5 , since
35 125 .
3 125 5 , since 3
5 125 .
Fourth Roots 4 10,000 10 , since
410 10,000 .
4 10,000 is not a real number.
Fifth Roots 5 32 2 , since
52 32 .
5 32 2 , since 5
2 32 .
Sixth Roots
1 1664 2 , since
61
264 .
1664
is not a real number.
Evaluate the following. If the answer is not a real
number, state “Not a real number.”
61. (a) 64 (b) 64 (c) 64
62. (a) 25 (b) 25 (c) 25
63. (a) 3 8 (b) 3 8 (c) 3 8
64. (a) 4 81 (b) 4 81 (c) 4 81
65. (a) 6 1,000,000 (b) 6 1,000,000
(c) 6 1,000,000
66. (a) 5 32 (b) 5 32 (c) 5 32
67. (a) 1416
(b) 1416
(c) 1416
68. (a) 1327
(b) 1327
(c) 1327
69. (a) 15100,000
(b) 15100,000
(c) 15100,000
70. (a) 6 1 (b) 6 1 (c) 6 1
SECTION 1.5 Order of Operations
MATH 1300 Fundamentals of Mathematics 75
Section 1.5: Order of Operations
Evaluating Expressions Using the Order of Operations
Evaluating Expressions Using the Order of Operations
Rules for the Order of Operations:
1) Operations that are within parentheses and other grouping symbols are performed
first. These operations are performed in the order established in the following steps.
If grouping symbols are nested, evaluate the expression within the innermost
grouping symbol first and work outward.
2) Exponential expressions and roots are evaluated first.
3) Multiplication and division are performed next, moving left to right and performing
these operations in the order that they occur.
4) Addition and subtraction are performed last, moving left to right and performing
these operations in the order that they occur.
Upon removing all of the grouping symbols, repeat the steps 2 through 4 until the
final result is obtained.
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 76
Example:
Solution:
Example:
Solution:
Additional Example 1:
SECTION 1.5 Order of Operations
MATH 1300 Fundamentals of Mathematics 77
Solution:
Additional Example 2:
Solution:
Additional Example 3:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 78
Additional Example 4:
Solution:
Additional Example 5:
Solution:
Exercise Set 1.5: Order of Operations
MATH 1300 Fundamentals of Mathematics 79
Answer the following.
1. In the abbreviation PEMDAS used for order of
operations,
(a) State what each letter stands for:
P: ____________________
E: ____________________
M: ____________________
D: ____________________
A: ____________________
S: ____________________
(b) If choosing between multiplication and
division, which operation should come first?
(Circle the correct answer.)
Multiplication
Division
Whichever appears first
(c) If choosing between addition and
subtraction, which operation should come
first? (Circle the correct answer.)
Addition
Subtraction
Whichever appears first
2. When performing order of operations, which of
the following are to be viewed as if they were
enclosed in parentheses? (Circle all that apply.)
Absolute value bars
Radical symbols
Fraction bars
Evaluate the following.
3. (a) 3 4 5 (b) (3 4) 5
(c) 3 4 5 (d) (3 4) 5
(e) 3 4 5 (f) 3 (4 5)
4. (a) 10 6 7 (b) (10 6) 7
(c) 10 6(7) (d) 10(6 7)
(e) 7 10 6 (f) 7 (10 6)
5. (a) 3 7 (b) 7 3
(c) 3 7 (d) 7 3
6. (a) 2 5 (b) 2 5
(c) 2 5 (d) 2 5
7. (a) 2 7 5 (b) 2 (7 5)
(c) 2 ( 7) 5 (d) 2 7( 5)
(e) 2(7 ( 5)) (f) 2(7) 5 7
8. (a) 6 2 ( 4) (b) 6 2 ( 4)
(c) 6 2( 4) (d) ( 6 2)( 4)
(e) 2 ( 6) 4 (f) 2 4( 6 2)
9. (a) 2 1 1
5 3 4 (b)
2 1 1
5 3 4
(c) 2 1 1 1
5 3 4 4
(d)
2 1 1
5 3 4
10. (a) 3 5
12 6
(b)
3 51
2 6
(c) 3 5
12 6
(d)
3 51
2 6
11. (a) 2
5 4 7 (b) 2
1 7
(c) 5 1 4 7 (d) 2
7 4 1 5
(e) 2 25 1 (f)
25 1
12. (a) 22 3 (b)
232 3
(c) 2 3(1 4) (d) 3
( 2 3) 1 4
(e) 2 22 3 (f)
22 3
13. (a) 20 2(10) (b) 20 2 10
(c) 20 10 ( 2) 10 5
14. (a) 24 4( 2) (b) (24 4) 2
(c) 24( 2) 4 2( 2)
15. (a) 210 5 2 (b)
210 5 2
(c) 22 10 2 5 5
16. (a) (3 9) 3 4 (b) 3 (9 3) 4
(c) 33 9 3 4
17. (a) 1
1
63
(b) 1
1
63
(c) 1
1
63
18. (a) 1
2
35
(b) 1
2
35
(c) 1
2
35
Exercise Set 1.5: Order of Operations
University of Houston Department of Mathematics 80
19. 1 17 4 5
20. 1 18 3 7
21. 2 47 5 2 3
22. 3 23 2 3 4
23. 1 1 3
2 3 4
24. 3 3 10
5 10 3
25. 25
5 3 3
26. 16
3 2 16
27. 2 3 4 1
28. 2 3 4 1
29. 2 3 4 1
30. 2 3 4 1
31. 2
2 3 4 1
32. 2
2 3 4 1
33. 3 7 7 3
12 2 3 3
34.
3 52 4 1 1
5 12 6 3
35. 281 2 4 3 2
36. 2 364 5 4 2
37. 2 24 121 5 4 3
38. 2 2144 5 2 6 12 3
39. 249 3 2
3 49
40. 23 49 2
3 49
41. 29 16 1
9 16
42. 29 16 1
9 16
43.
222 3 5
2 8 2 4
44.
22 3 5
2 8 2 4
45.
2 2 3 2 4
2 2
5 3 3 7 2 4 1
4 2 2 1 81 2 3
46.
23 2
2
5 2 25 2 2 2 3 3
81 16 2 1 3 1 1 4 2
Exercise Set 1.5: Order of Operations
MATH 1300 Fundamentals of Mathematics 81
Evaluate the following expressions for the given values
of the variables.
47. r
Pk
for 5, 1, and 7P r k .
48. x y
y z for 4, 3, and 8x y z .
49. 2
2
8b b c
c
for 4b and 2c .
50. 2 4
2
b b ac
a
for 1, 3, and 18a b c .
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 82
Section 1.6: Solving Linear Equations
Linear Equations
Linear Equations
Rules for Solving Equations:
Linear Equations:
Example:
SECTION 1.6 Solving Linear Equations
MATH 1300 Fundamentals of Mathematics 83
Solution:
Example:
Solution:
Additional Example 1:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 84
Additional Example 2:
Solution:
Additional Example 3:
Solution:
Exercise Set 1.6: Solving Linear Equations
MATH 1300 Fundamentals of Mathematics 85
Solve the following equations algebraically.
1. 5 12x
2. 8 9x
3. 4 7x
4. 2 8x
5. 6 30x
6. 4 28x
7. 6 10x
8. 8 26x
9. 1373 x
10. 6115 x
11. 7432 xx
12. 6425 xx
13. 3)8(59)2(3 xx
14. 3)4(25)3(4 xx
15. )37(4)52(3 xx
16. )51(648327 xx
17. 75
x
18. 103
x
19. 3
92
x
20. 4
127
x
21. 5
36
x
22. 8
49
x
23. 7152 x
24. 2743 x
25. 1)7(52
35 xx
26. 3)12(1261
94 xx
27. xxx
37
5
3
22
28. 12
1
6
5
8
7
xxx
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 86
Section 1.7: Interval Notation and Linear Inequalities
Linear Inequalities
Linear Inequalities
Rules for Solving Inequalities:
SECTION 1.7 Interval Notation and Linear Inequalities
MATH 1300 Fundamentals of Mathematics 87
Interval Notation:
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 88
Example:
Solution:
Example:
SECTION 1.7 Interval Notation and Linear Inequalities
MATH 1300 Fundamentals of Mathematics 89
Solution:
Additional Example 1:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 90
Additional Example 2:
Solution:
SECTION 1.7 Interval Notation and Linear Inequalities
MATH 1300 Fundamentals of Mathematics 91
Additional Example 3:
Solution:
Additional Example 4:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 92
Additional Example 5:
Solution:
Additional Example 6:
Solution:
SECTION 1.7 Interval Notation and Linear Inequalities
MATH 1300 Fundamentals of Mathematics 93
Additional Example 7:
Solution:
Exercise Set 1.7: Interval Notation and Linear Inequalities
University of Houston Department of Mathematics 94
For each of the following inequalities:
(a) Write the inequality algebraically.
(b) Graph the inequality on the real number line.
(c) Write the inequality in interval notation.
1. x is greater than 5.
2. x is less than 4.
3. x is less than or equal to 3.
4. x is greater than or equal to 7.
5. x is not equal to 2.
6. x is not equal to 5 .
7. x is less than 1.
8. x is greater than 6 .
9. x is greater than or equal to 4 .
10. x is less than or equal to 2 .
11. x is not equal to 8 .
12. x is not equal to 3.
13. x is not equal to 2 and x is not equal to 7.
14. x is not equal to 4 and x is not equal to 0.
Write each of the following inequalities in interval
notation.
15. 3x
16. 5x
17. 2x
18. 7x
19. 53 x
20. 27 x
21. 7x
22. 9x
Write each of the following inequalities in interval
notation.
23.
24.
25.
26.
27.
28.
Given the set 31,3,4,2 S , use substitution to
determine which of the elements of S satisfy each of
the following inequalities.
29. 1052 x
30. 1424 x
31. 712 x
32. 013 x
33. 1012 x
34. 5
21
x
For each of the following inequalities:
(a) Solve the inequality.
(b) Graph the solution on the real number line.
(c) Write the solution in interval notation.
35. 102 x
36. 243 x
Exercise Set 1.7: Interval Notation and Linear Inequalities
MATH 1300 Fundamentals of Mathematics 95
37. 305 x
38. 404 x
39. 1152 x
40. 1743 x
41. 2038 x
42. 010 x
43. 47114 xx
44. 7395 xx
45. 62710 xx
46. xx 5648
47. 1485 xx
48. 9810 xx
49. )7(2)54(3 xx
50. )20()23(4 xx
51. )5(21
31
65 xx
52. xx 1031
21
52
53. 82310 x
54. 13329 x
55. 17734 x
56. 34519 x
57. 54
15103
32 x
58. 35
625
43 x
Which of the following inequalities can never be true?
59. (a) 95 x
(b) 59 x
(c) 73 x
(d) 35 x
60. (a) 53 x
(b) 18 x
(c) 82 x
(d) 107 x
Answer the following.
61. You go on a business trip and rent a car for $75
per week plus 23 cents per mile. Your employer
will pay a maximum of $100 per week for the
rental. (Assume that the car rental company
rounds to the nearest mile when computing the
mileage cost.)
(a) Write an inequality that models this
situation.
(b) What is the maximum number of miles
that you can drive and still be
reimbursed in full?
62. Joseph rents a catering hall to put on a dinner
theatre. He pays $225 to rent the space, and pays
an additional $7 per plate for each dinner served.
He then sells tickets for $15 each.
(a) Joseph wants to make a profit. Write an
inequality that models this situation.
(b) How many tickets must he sell to make
a profit?
63. A phone company has two long distance plans as
follows:
Plan 1: $4.95/month plus 5 cents/minute
Plan 2: $2.75/month plus 7 cents/minute
How many minutes would you need to talk each
month in order for Plan 1 to be more cost-
effective than Plan 2?
64. Craig’s goal in math class is to obtain a “B” for
the semester. His semester average is based on
four equally weighted tests. So far, he has
obtained scores of 84, 89, and 90. What range of
scores could he receive on the fourth exam and
still obtain a “B” for the semester? (Note: The
minimum cutoff for a “B” is 80 percent, and an
average of 90 or above will be considered an
“A”.)
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 96
Section 1.8: Absolute Value and Equations
Absolute Value
Absolute Value
Equations of the Form |x| = C:
Special Cases for |x| = C:
Example:
SECTION 1.8 Absolute Value and Equations
MATH 1300 Fundamentals of Mathematics 97
Solution:
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 98
Example:
Solution:
Example:
Solution:
SECTION 1.8 Absolute Value and Equations
MATH 1300 Fundamentals of Mathematics 99
Example:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 100
Additional Example 1:
Solution:
Additional Example 2:
Solution:
SECTION 1.8 Absolute Value and Equations
MATH 1300 Fundamentals of Mathematics 101
Additional Example 3:
Solution:
Additional Example 4:
Solution:
CHAPTER 1 Introductory Information and Review
University of Houston Department of Mathematics 102
Additional Example 5:
Solution:
Exercise Set 1.8: Absolute Value and Equations
MATH 1300 Fundamentals of Mathematics 103
Solve the following equations.
1. 7x
2. 5x
3. 9x
4. 10x
5. 122 x
6. 303 x
7. 54 x
8. 27 x
9. 4 5x
10. 7 2x
11. 843 x
12. 345 x
13. 3 4 8x
14. 5 4 3x
15. 1732 x
16. 31
65
21 x
17. 10734 x
18. 2825 x
19. 115123 x
20. 46922 x
21. 1131421 x
22. 875 x
23. 1523 xx
24. 674 xx
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 104
Chapter 2 Points, Lines, and Functions
Section 2.1: An Introduction to the Coordinate Plane
Points in the Coordinate Plane
Points in the Coordinate Plane
The Rectangular Coordinate System:
SECTION 2.1 An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 105
Plotting Points in the Coordinate Plane:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 106
Example:
Solution:
SECTION 2.1 An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 107
Graphing Horizontal and Vertical Lines:
Example:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 108
Graphing Other Lines:
Example:
Solution:
SECTION 2.1 An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 109
Additional Example 1:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 110
Additional Example 2:
SECTION 2.1 An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 111
Solution:
Additional Example 3:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 112
Solution:
Additional Example 4:
Solution:
SECTION 2.1 An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 113
(c) Draw a line through the points.
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 114
Additional Example 5:
Solution:
SECTION 2.1 An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 115
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 116
Exercise Set 2.1: An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 117
Plot the following points in a coordinate plane.
1. A(3, 4)
2. B(2, -5)
3. C(-3, -1)
4. D(-4, -6)
5. E(-5, 0)
6. F(0, -2)
Write the coordinates of each of the points shown in
the figure below. Then identify the quadrant or axis
in which the point is located.
7. G
8. H
9. I
10. J
11. K
12. L
Plot each of the following sets of points in a coordinate
plane. Then identify the quadrant or axis in which
each point is located.
13. (a) A(2, 5)
(b) B(-2, -5)
(c) C(2, -5)
(d) D(-2, 5)
14. (a) A(4, -3)
(b) B(-4, -3)
(c) C(-4, 3)
(d) D(4, 3)
15. (a) A(0, -2)
(b) B(-2, 0)
(c) C(2, 0)
(d) D(0, 2)
16. (a) A(-3, 0)
(b) B(3, 0)
(c) (0, -3)
(d) D(0, 3)
17. If the point (a, b) is in Quadrant I, identify the
quadrant of each of the following points:
(a) (-a, -b) (b) (-a, b) (c) (a, a)
18. If the point (a, b) is in Quadrant I, identify the
quadrant of each of the following points:
(a) (-b, a) (b) (b, b) (c) (-b, -a)
19. If the point (a, b) is in Quadrant II, then 0a <
and 0b > . Identify the quadrant of each of the
following points:
(a) (-a, -b) (b) (b, a) (c) (a, -b)
20. If the point (a, b) is in Quadrant III, then 0a <
and 0b < . Identify the quadrant of each of the
following points:
(a) (-a, b) (b) (b, a) (c) (-a, -b)
21. If the point (a, b) is in Quadrant IV, identify the
quadrant of each of the following points:
(a) (b, -b) (b) (-a, -a) (c) (b, a)
22. If the point (a, b) is in Quadrant II, identify the
quadrant of each of the following points:
(a) (-a, b) (b) (b, b) (c) (a, -a)
23. If the point (a, b) is in Quadrant III, identify the
axis on which each of the following points lies:
(a) (a, 0) (b) (0, b) (c) (-b, 0)
24. If the point (a, b) is in Quadrant IV, identify the
axis on which each of the following points lies:
(a) (0, -b) (b) (-a, 0) (c) (b, 0)
Answer True or False.
25. The point (0, 5) is on the x-axis.
26. The point (-4, 0) is in Quadrant II.
27. The point (1, -3) is in Quadrant IV.
28. The point (-2, -5) is in Quadrant III.
29. The point (0, 0) is in Quadrant I.
30. The point (-6, 1) is in Quadrant IV.
31. If the point (a, b) is in Quadrant IV, then 0b < .
32. If the point (a, b) is in Quadrant II, then 0a > .
33. If the point (a, b) is in Quadrant I, then the point
(b, a) is also in Quadrant I.
−4 −2 2 4 6
−4
−2
2
4
x
y
H
G
I
J
K
L
Exercise Set 2.1: An Introduction to the Coordinate Plane
University of Houston Department of Mathematics 118
34. If the point (a, b) is in Quadrant I, then the point
(a, -b) is in Quadrant II.
35. If the point (a, b) is in Quadrant II, then the point
(-a, -b) is in Quadrant III .
36. If the point (a, b) is in Quadrant IV, then the
point (-b, a) is in Quadrant I.
37. If the point (a, b) is in Quadrant III, then 0b > .
38. If the point (a, b) is on the y-axis, then 0a > .
39. If the point (a, b) is on the y-axis, then 0b > .
40. If the point (a, b) is on the y-axis, then 0a = .
41. If the point (a, b) is on the y-axis, then the point
(b, a) is on the x-axis.
42. If the point (a, b) is on the x-axis, then the point
(a, 3) lies in Quadrant I .
Answer the following.
43. Given the following points:
A(3, 5), B(3, 1), C(3, 0), D(3, -2)
(a) Plot the above points on a coordinate plane.
(b) What do the above points have in common?
(c) Draw a line through the above points.
(d) What is the equation of the line drawn in
part (c)?
44. Given the following points:
A(-3, 4), B(0, 4), C(1, 4), D(3, 4)
(a) Plot the above points on a coordinate plane.
(b) What do the above points have in common?
(c) Draw a line through the above points.
(d) What is the equation of the line drawn in
part (c)?
45. (a) List four points that are on the x-axis.
(b) Analyze the coordinates of the points you
have listed. What do they have in common?
(c) Give the equation of the x-axis.
46. (a) List four points that are on the y-axis.
(b) Analyze the coordinates of the points you
have listed. What do they have in common?
(c) Give the equation of the y-axis.
47. Graph the line 2x = .
48. Graph the line 5y = − .
49. Graph the line 4y = .
50. Graph the line 3x = − .
51. On the same set of axes, graph the lines 1x = −
and 3y = .
52. On the same set of axes, graph the lines 5x =
and 2y = − .
53. On the same set of axes, graph the lines 72
x =
and 0y = .
54. On the same set of axes, graph the lines 0x =
and 52
y = − .
Graph the following lines by first completing the table
and then plotting the points on a coordinate plane.
55. 3 2y x= +
56. 2 5y x= − +
x y
-2
-1
0
1
2
x y
-2
-1
0
1
2
Exercise Set 2.1: An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 119
57. 4 7y x= − +
58. 5 1y x= −
Answer the following.
59. Graph the line segment with endpoints (-7, 0)
and (0, 7).
60. Graph the line segment with endpoints (3, 5) and
and (-5, -3).
61. Graph the line segment with endpoints (1, -4)
and (-1, 4)
62. Graph the line segment with endpoints (-2, 6)
and (6, 2).
x y
0
14
-5
2
32
−
x y
2
-1
35
-6
0
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 120
Section 2.2: The Distance and Midpoint Formulas
The Distance Formula
The Midpoint Formula
The Distance Formula
Finding the Distance Between Two Points:
Example:
SECTION 2.2 The Distance and Midpoint Formulas
MATH 1300 Fundamentals of Mathematics 121
Solution:
Additional Example 1:
Solution:
Additional Example 2:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 122
Additional Example 3:
Solution:
Additional Example 4:
Solution:
SECTION 2.2 The Distance and Midpoint Formulas
MATH 1300 Fundamentals of Mathematics 123
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 124
Additional Example 5:
Solution:
SECTION 2.2 The Distance and Midpoint Formulas
MATH 1300 Fundamentals of Mathematics 125
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 126
Additional Example 6:
Solution:
SECTION 2.2 The Distance and Midpoint Formulas
MATH 1300 Fundamentals of Mathematics 127
Additional Example 7:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 128
Use the Pythagorean Theorem to determine c.
SECTION 2.2 The Distance and Midpoint Formulas
MATH 1300 Fundamentals of Mathematics 129
The Midpoint Formula
Finding the Midpoint of a Line Segment:
Example:
Solution:
Additional Example 1:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 130
Solution:
Additional Example 2:
Solution:
Additional Example 3:
SECTION 2.2 The Distance and Midpoint Formulas
MATH 1300 Fundamentals of Mathematics 131
Solution:
Additional Example 4:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 132
Additional Example 5:
Solution:
SECTION 2.2 The Distance and Midpoint Formulas
MATH 1300 Fundamentals of Mathematics 133
Exercise Set 2.2: The Distance and Midpoint Formulas
University of Houston Department of Mathematics 134
Use the Pythagorean Theorem to find the missing side
of each of the following triangles.
Pythagorean Theorem: In a right triangle, if a and b
are the measures of the legs, and c is the measure of the hypotenuse, then a2 + b2 = c2.
1.
2.
3.
4.
Answer the following.
5. Given the following points:
(1, 2)A and (4, 7)B
(a) Plot the above points on a coordinate plane.
(b) Draw segment AB. This will be the
hypotenuse of triangle ABC.
(c) Find a point C such that triangle ABC is a
right triangle. Draw triangle ABC.
(d) Use the Pythagorean theorem to find the
distance between A and B (the length of the
hypotenuse of the triangle).
6. Given the following points:
( 3, 1)A and (1, 5)B
(a) Plot the above points on a coordinate plane.
(b) Draw segment AB. This will be the
hypotenuse of triangle ABC.
(c) Find a point C such that triangle ABC is a
right triangle. Draw triangle ABC.
(d) Use the Pythagorean theorem to find the
distance between A and B (the length of the
hypotenuse of the triangle).
Use the distance formula to find the distance between
the two given points. (You can also use the method from
the previous two problems to double-check your answer.)
7. (3, 6) and (5, 9)
8. (4, 7) and (2, 3)
9. ( 5, 0) and ( 2, 6)
10. (9, 4) and (2, 3)
11. (4, 0) and (0, 7)
12. ( 4, 8) and ( 10, 1)
13. 12
5, and 56
3,
14. 23
, 1 and 34
, 0
Find the midpoint of the line segment joining points A
and B.
15. (7, 6)A and (3, 8)B
16. (5, 9)A and (1, 3)B
17. ( 7, 0)A and ( 4, 8)B
18. (7, 5)A and (4, 3)B
19. (3, 0)A and (0, 9)B
20. ( 6, 7)A and ( 10, 6)B
21. 13
, 5A and 35
, 7B
22. 12
3,A and 56
8,B
c 5
12
a 7
5
2 6
b
c 6
8
c a
b
Exercise Set 2.2: The Distance and Midpoint Formulas
MATH 1300 Fundamentals of Mathematics 135
Answer the following.
23. (a) Graph the line segment with endpoints
( 2, 6)A and (5, 4)B .
(b) Find the distance from A to B.
(c) Find the midpoint of AB .
24. (a) Graph the line segment with endpoints
(4, 0)A and ( 2, 5)B .
(b) Find the distance from A to B.
(c) Find the midpoint of AB .
25. If (4, 7)M is the midpoint of the line segment
joining points A and B, and A has coordinates
(2, 3) , find the coordinates of B.
26. If (5, 3)M is the midpoint of the line segment
joining points A and B, and A has coordinates
(1, 6) , find the coordinates of B.
27. If (3, 5)M is the midpoint of the line segment
joining points A and B, and B has coordinates
( 1, 2) ,
(a) Find the coordinates of A.
(b) Find the length of AB .
28. If ( 2, 1)M is the midpoint of the line segment
joining points A and B, and B has coordinates
( 5, 3) ,
(a) Find the coordinates of A.
(b) Find the length of AB .
29. Determine which of the following points is
closer to the origin: (5, 6)A or ( 3, 7)B ?
30. Determine which of the following points is
closer to the point (4, 1) : ( 2, 3)A or
(6, 6)B ?
31. A circle has a diameter with endpoints
( 5, 9)A and (3, 5)B .
(a) Find the coordinates of the center of the
circle.
(b) Find the length of the radius of the circle.
32. A circle has a diameter with endpoints (2, 7)A
and (8, 1)B .
(a) Find the coordinates of the center of the
circle.
(b) Find the length of the radius of the circle.
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 136
Section 2.3: Slope and Intercepts of Lines
The Slope of a Line
Intercepts of Lines
The Slope of a Line
Finding the Slope of a Line:
SECTION 2.3 Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 137
Example:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 138
SECTION 2.3 Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 139
Additional Example 1:
Solution:
Additional Example 2:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 140
Additional Example 3:
Solution:
Additional Example 4:
Solution:
SECTION 2.3 Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 141
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 142
Intercepts of Lines
Finding Intercepts of Lines:
SECTION 2.3 Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 143
Horizontal Lines:
Vertical Lines:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 144
Example:
Solution:
SECTION 2.3 Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 145
Example:
Solution:
Additional Example 1:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 146
Additional Example 2:
Solution:
SECTION 2.3 Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 147
Additional Example 3:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 148
Solution:
Exercise Set 2.3: Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 149
State whether the slope of each of the following lines is
positive, negative, zero, or undefined.
1. p
2. q
3. r
4. s
5. t
6. w
Find the slope of the line that passes through the
following points. If undefined, state „Undefined.‟
7. )7,3( and )0,0(
8. (8, 0) and (3, 6)
9. )10,4( and )5,2(
10. )9,5( and )3,7(
11. (6, 4) and (2, 4)
12. (5, 1) and (5, 8)
13. )7,6( and )3,2(
14. ( 2, 6) and ( 5, 10)
15. ( 3, 8) and ( 3, 4)
16. )7,1( and )7,8(
17. (2, 8) and (0, 3)
18. (1, 4) and ( 7, 2)
19. 12
, 1 and 2 13 6
,
20. 34
2, and 515 8
,
21. 2 47 9
, and 5 16 2
,
22. 3 75 10
, and 714 8
,
Find the slope of each of the following lines. If
undefined, state „Undefined.‟
23. c
24. d
25. e
26. f
For each of the following:
(a) Complete the given table.
(b) Plot the points on a coordinate plane and
graph the line.
(c) Use two points from the table to find the slope
of the line.
27. 4 1y x
28. 3 2y x
x y
0
2
3
0
12
x y
2
2
4
43
3
x
y
pq
r
s
t
w
x
y
c
d
e
f
Exercise Set 2.3: Slope and Intercepts of Lines
University of Houston Department of Mathematics 150
29. 23
4y x
30. 35
6y x
Answer the following.
31. Examine the relationship in numbers 27-30
between each of the equations and the
corresponding slope that you found for each line.
Do you see any pattern? Can you determine the
slope of the line from simply looking at its
equation?
32. Based on the pattern found in the previous
problem, state the slope of the following lines
without graphing the line or performing any
calculations:
(a) 2 9y x
(b) 7 5y x
(c) 45
2y x
(d) 37
4y x
For each of the following graphs:
(a) State the x-intercept.
(b) State the y-intercept.
(c) State the coordinates of the x-intercept.
(d) State the coordinates of the y-intercept.
(e) Find the slope of the line.
33.
34.
For each of the following equations:
(a) Find the x- and y-intercepts of the line.
(b) State the coordinates of the intercepts.
(c) Plot the x- and y-intercepts on a coordinate
plane.
(d) Graph the line, based on the intercepts.
35. 2 8y x
36. 3 6y x
37. 54 xy
38. 73 xy
39. 2025 yx
x y
4
5
9
8
32
x y
5
0
7
8
0
x
y
x
y
Exercise Set 2.3: Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 151
40. 2 3 18x y
41. 3 5 30x y
42. 3 24 4x y
43. 2 3 10x y
44. 4 6 9x y
45. 5 3 21 0x y
46. 4 7 8 0x y
47. 2 2 7x y
48. 3 15x
49. 4 12y
50. 4 4 15x y
51. 6 24x
52. 2 14y
For each of the following:
(a) Complete the given table.
(b) Plot the points on a coordinate plane and
graph the line.
(c) Find the x- and y-intercepts of the line.
(d) Find the slope of the line.
53. 2 8y x
54. 3y x
Answer the following.
55. Examine the relationship in numbers 53 and 54
between each of the equations and the
corresponding y-intercept that you found for
each line. Do you see any pattern? Can you
determine the y-intercept of the line from simply
looking at its equation?
56. Based on the pattern found in the previous
problem, state the y-intercept of the following
lines without graphing the line or performing any
calculations:
(a) 2 9y x
(b) 7 5y x
(c) 45
2y x
(d) 37
4y x
x y
0
0
2
6
0.5
x y
0
0
3
1.5
2
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 152
Section 2.4: Equations of Lines
Writing Equations of Lines
Writing Equations of Lines
Different Forms for Equations of Lines:
Example:
Solution:
SECTION 2.4 Equations of Lines
MATH 1300 Fundamentals of Mathematics 153
Example:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 154
Example:
Solution:
SECTION 2.4 Equations of Lines
MATH 1300 Fundamentals of Mathematics 155
Example:
Solution:
Additional Example 1:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 156
Additional Example 2:
Solution:
To sketch the graph, begin by using
the y-intercept to plot the point 0,1 .
SECTION 2.4 Equations of Lines
MATH 1300 Fundamentals of Mathematics 157
Additional Example 3:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 158
Additional Example 4:
Solution:
SECTION 2.4 Equations of Lines
MATH 1300 Fundamentals of Mathematics 159
Exercise Set 2.4: Equations of Lines
University of Houston Department of Mathematics 160
Write an equation in slope-intercept form for each of
the following lines.
1.
2.
3.
4.
For each of the following equations,
(a) Write the equation in slope-intercept form. (b) Identify the slope and the y-intercept of the
line.
(c) Graph the line.
5. 52 yx
6. 4 0y x
7. 5 1x y
8. 63 yx
9. 04 yx
10. 3 9x y
11. 5 4 12x y
12. 1052 yx
13. 5 2 30 0y x
14. 3 2 8 0x y
15. 5 14 2
1x y
16. 2 13 2
1x y
Each set of conditions below describes the properties
of a particular line. Using these conditions,
(a) Graph the line.
(b) Write an equation for the line in point-slope
form.
(c) Write an equation for the line in slope-
intercept form. (Do this algebraically, and
then check to see if your result matches your
graph.)
17. Slope 3
2; passes through 6, 4
18. Slope 2
5 ; passes through 4, 3
19. Passes through 8, 2 and 4, 7
20. Passes through 4, 7 and 1, 3
x
y
x
y
x
y
x
y
Exercise Set 2.4: Equations of Lines
MATH 1300 Fundamentals of Mathematics 161
Write an equation in slope-intercept form for the line
that satisfies the given conditions.
21. Slope 4
7 ; y-intercept 3
22. Slope 4 ; y-intercept 5
23. Slope 4
5; passes through 5, 3
24. Slope 3
4 ; passes through 12, 5
25. Slope 9
2 ; passes through 3, 2
26. Slope 5
1; passes through 4, 2
27. Passes through 10, 2 and 5, 7
28. Passes through 6, 1 and 9, 4
29. Passes through 4, 5 and 1, 2
30. Passes through 7, 0 and 3, 5
31. x-intercept 7 ; y-intercept 5
32. x-intercept 2 ; y-intercept 6
33. Slope 2
3 ; x-intercept 4
34. Slope 5
1; x-intercept 6
Answer the following, assuming that each situation
can be modeled by a linear equation.
35. If a company can make 21 computers for
$23,000, and can make 40 computers for
$38,200, write an equation that represents the
cost C of x computers.
36. A certain electrician charges a $40 traveling fee,
and then charges $55 per hour of labor. Write an
equation that represents the cost C of a job that
takes x hours.
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 162
Section 2.5: Parallel and Perpendicular Lines
Pairs of Lines – Parallel and Perpendicular Lines
Pairs of Lines - Parallel and Perpendicular Lines
Parallel Lines:
Perpendicular Lines:
Two lines with slopes 1m and
2m perpendicular if and only if 1 2 1m m .
SECTION 2.5 Parallel and Perpendicular Lines
MATH 1300 Fundamentals of Mathematics 163
Example:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 164
Example:
Solution:
SECTION 2.5 Parallel and Perpendicular Lines
MATH 1300 Fundamentals of Mathematics 165
Additional Example 1:
Solution:
Additional Example 2:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 166
Additional Example 3:
Solution:
Additional Example 4:
Solution:
SECTION 2.5 Parallel and Perpendicular Lines
MATH 1300 Fundamentals of Mathematics 167
Exercise Set 2.5: Parallel and Perpendicular Lines
University of Houston Department of Mathematics 168
State whether the following pairs of lines are parallel,
perpendicular, or neither.
1. 3 5y x
3 7y x
2. 25
1y x
52
3y x
3. 73
5y x
37
4y x
4. 23
7y x
23
5y x
5. 2 5y x
2 5y x
6. 5 7y x
15
3y x
7. 2 5 7x y
5 2 6x y
8. 3 4 8x y
3 4 8x y
9. 2 3 5x y
4 6 11x y
10. 5 0x y
2x y
11. The line passing through (2, 5) and (7, 9)
The line passing through ( 2, 6) and (2, 1)
12. The line passing through ( 4, 7) and (0, 5)
The line passing through ( 3, 8) and ( 5, 9)
13. The line passing through ( 6, 0) and (4, 10)
The line passing through (3, 7) and (7, 11)
14. The line passing through ( 1, 7) and (2, 5)
The line passing through ( 6, 6) and ( 2, 5)
15. 4y
14
y
16. 3x
3y
17. 2y
0x
18. 5x
5x
19. The line passing through (4, 5) and ( 1, 5)
The line passing through (2, 3) and (0, 3)
20. The line passing through (2, 6) and (2, 8)
The line passing through ( 3, 4) and (5, 4)
Each set of conditions below describes a particular
line. Using these conditions, write an equation for each
line in the following two forms:
(a) Point-slope form
(b) Slope-intercept form
21. Passes through (4, 7) ; parallel to the line
2 5y x
22. Passes through (4, 7) ; perpendicular to the line
2 5y x
23. Passes through ( 12, 5) ; perpendicular to the
line 6 1y x
24. Passes through ( 12, 5) ; parallel to the line
6 1y x
25. Passes through (3, 7) ; parallel to the line
54
2y x
26. Passes through (3, 7) ; perpendicular to the
line 54
2y x
27. Passes through ( 1, 6) ; perpendicular to the line
2 3 7x y
28. Passes through ( 1, 6) ; parallel to the line
2 3 7x y
Exercise Set 2.5: Parallel and Perpendicular Lines
MATH 1300 Fundamentals of Mathematics 169
Write an equation for the line that satisfies the given
conditions. With the exception of vertical lines, write
all equations in slope-intercept form.
29. Passes through (1, 4) ; parallel to the x-axis
30. Passes through (1, 4) ; parallel to the y-axis
31. Passes through (2, 6) ; parallel to the line
4x
32. Passes through (2, 6) ; parallel to the line
4y
33. Passes through ( 2, 3) ; and is
(a) parallel to the line 23
5y x
(b) perpendicular to the line 23
5y x
34. Passes through (20, 2) ; and is
(a) parallel to the line 35 xy
(b) perpendicular to the line 35 xy
35. Passes through (2, 3) ; parallel to the line
625 yx
36. Passes through ( 1, 5) ; parallel to the line
834 yx
37. Passes through (2, 3) ; perpendicular to the line
625 yx
38. Passes through ( 1, 5) ; perpendicular to the
line 834 yx
39. Passes through (4, 6) ; parallel to the line
containing (3, 5) and (2, 1)
40. Passes through (8, 3) ; parallel to the line
containing ( 2, 3) and ( 4, 6)
41. Perpendicular to the line containing ( 3, 5) and
(7, 1) ; passes through the midpoint of the line
segment connecting these points
42. Perpendicular to the line containing (4, 2) and
(10, 4) ; passes through the midpoint of the line
segment connecting these points
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 170
Section 2.6: An Introduction to Functions
Definition of a Function
Domain of a Function
Definition of a Function
Definition:
SECTION 2.6 An Introduction to Functions
MATH 1300 Fundamentals of Mathematics 171
Defining a Function by an Equation in the Variables x and y:
The Function Notation:
Example:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 172
Example:
Solution:
SECTION 2.6 An Introduction to Functions
MATH 1300 Fundamentals of Mathematics 173
Additional Example 1:
Solution:
Additional Example 2:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 174
Additional Example 3:
Solution:
Additional Example 4:
SECTION 2.6 An Introduction to Functions
MATH 1300 Fundamentals of Mathematics 175
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 176
Additional Example 5:
Solution:
SECTION 2.6 An Introduction to Functions
MATH 1300 Fundamentals of Mathematics 177
Domain of a Function
Finding the Domain of a Function:
Example:
Solution:
Example:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 178
Additional Example 1:
Solution:
SECTION 2.6 An Introduction to Functions
MATH 1300 Fundamentals of Mathematics 179
Additional Example 2:
Solution:
Additional Example 3:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 180
Exercise Set 2.6: An Introduction to Functions
MATH 1300 Fundamentals of Mathematics 181
For each of the examples below, determine whether
the mapping makes sense within the context of the
given situation, and then state whether or not the
mapping represents a function.
1. Erik conducts a science experiment and maps the
temperature outside his kitchen window at
various times during the morning.
2. Dr. Kim counts the number of people in
attendance at various times during his lecture this
afternoon.
State whether or not each of the following mappings
represents a function.
3.
4.
5.
6.
Express each of the following rules in function
notation. (For example, “Subtract 3, then square”
would be written as 2( ) ( 3)f x x .)
7. (a) Divide by 7, then add 4
(b) Add 4, then divide by 7
8. (a) Multiply by 2, then square
(b) Square, then multiply by 2
9. (a) Take the square root, then subtract 6 squared
(b) Take the square root, subtract 6, then square
10. (a) Add 4, square, then subtract 2
(b) Subtract 2, square, then add 4
Complete the table for each of the following functions.
11. 3( ) 5f x x
12. 2( ) ( 4) 1g x x
Find the domain of each of the following functions.
Write the domain first as an inequality, and then
express it in interval notation.
13. 1
( )f xx
14. 4
( )f xx
x ( )f x
2
1
0
1
2
x ( )f x
3
1
1
4
6
A B
7 9 -3
0 5 4
A B
0
8
4
2
A B
-2
9
-6
1
A B
9
-6
8 4 -7
9
10
57
62
65
Time Temp. (oF)
Time
1
2
3
85
87
# of People
Exercise Set 2.6: An Introduction to Functions
University of Houston Department of Mathematics 182
15. 3
5)(
xxf
16. 7
( )8
f xx
17. 6
( )4
xf x
x
18. 4
( )6
xh x
x
19. 8
( )2 5
f tt
20. 2
( )3 4
h tt
21. 4 1
( )4 9
xg x
x
22. 5 7
( )3 7
xf x
x
23. 2
1( )
9
xg x
x
24. 2
2( )
25
xh x
x
25. 242)( 2 xxxf
26. xxf 27)(
27. ( ) 3 5g x x
28. 2( ) 16h x x
29. ttf )(
30. 3)( xxh
31. 5)( xxf
32. 7)( xxg
33. 3( ) 5f x x
34. 3( ) 7g x x
35. ( ) 2 9h x x
36. ( ) 3 2h t t
37. ( ) 1 5g x x
38. ( ) 4f x x
39. ( ) 8 5 2f x x
40. ( ) 2 7 4f x x
41. 2
( )6
xH x
x
42. 3
( )x
G xx
43. 3( ) 1f t t
44. 3( ) 2 9g x x
45. 31
( )5
th t
t
46. 32 9
( )4 7
xf x
x
47. 5( )h x x
48. 4( )h x x
49. 6( ) 3 5g x x
50. 5( ) 2 7g x x
51. ( )f x x
52. ( ) 2g x x
53. ( ) 2 6H x x
54. ( ) 3 5f x x
Exercise Set 2.6: An Introduction to Functions
MATH 1300 Fundamentals of Mathematics 183
55. 2
( )7
f xx
56. 5
( )f xx
57. 3
( )4
xf x
x
58. 9
( )1
xf x
x
Evaluate the following.
59. If 45)( xxf ,
(a) Find (3)f
(b) Find x when ( ) 3f x
(c) Find 12
f
(d) Find x when 12
( )f x
(e) Find 0f
(f) Find x when ( ) 0f x
60. If ( ) 3 1f x x ,
(a) Find ( 5)f
(b) Find x when ( ) 5f x
(c) Find 34
f
(d) Find x when 34
( )f x
(e) Find 0f
(f) Find x when ( ) 0f x
61. If ( ) 3h x x ,
(a) Find (1)h
(b) Find x when ( ) 1h x
(c) Find 2h
(d) Find x when ( ) 2h x
(e) Find 7h
(f) Find x when ( ) 7h x
62. If ( ) 7g x x ,
(a) Find (0)g
(b) Find x when ( ) 0g x
(c) Find 2g
(d) Find x when ( ) 2g x
(e) Find 3g
(f) Find x when ( ) 3g x
63. If ( ) 2h x x , find
(a) (7)h
(b) (25)h
(c) 14
h
64. If ( ) 2h x x , find
(a) (7)h
(b) (25)h
(c) 14
h
65. If ( ) 3f x x , find
(a) (16)f
(b) (12)f
(c) 9f
66. If ( ) 3f x x , find
(a) (16)f
(b) (12)f
(c) 9f
67. If 2( ) 5 6g x x x ,
(a) Find (3)g
(b) Find 4g
(c) Find 12
g
(d) Find 0g
68. If 2( ) 2 15h t t t ,
(a) Find (0)h
(b) Find (6)h
(c) Find 5h
(d) Find 23
h
Exercise Set 2.6: An Introduction to Functions
University of Houston Department of Mathematics 184
69. If 3
2)(
x
xxf ,
(a) Find ( 7)f
(b) Find (0)f
(c) Find 5f
(d) Find 3f
(e) Find 2f
70. If 5 2
( )4
xg x
x
,
(a) Find (2)g
(b) Find ( 4)g
(c) Find 52
g
(d) Find 3g
(e) Find (0)g
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics 185
Section 2.7: Functions and Graphs
Graphing a Function
Graphing a Function
The Graph of a Function:
The Vertical Line Test:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 186
Example:
Solution:
\
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics 187
Example:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 188
Example:
Solution:
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics 189
Additional Example 1:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 190
Additional Example 2:
The graph of y f x is shown below.
(a) Find the domain of f.
(b) Find the range of f.
(c) Find the following function values: 3 ; 1 ; 0 ; 1f f f f .
(d) For what value(s) of x is 2f x ?
Solution:
Part (a):
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics 191
Part (b):
Part (c):
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 192
Part (d):
Additional Example 3:
Solution:
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics 193
Additional Example 4:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 194
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics 195
Additional Example 5:
Solution:
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 196
Additional Example 6:
Solution:
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics 197
CHAPTER 2 Points, Lines, and Functions
University of Houston Department of Mathematics 198
Additional Example 7:
Solution:
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics 199
Exercise Set 2.7: Functions and Graphs
University of Houston Department of Mathematics 200
x
yDetermine whether or not each of the following graphs
represents a function.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
For each set of points,
(a) Graph the set of points.
(b) Determine whether or not the set of points
represents a function. Justify your answer.
11. (1, 5), (2, 4), ( 3, 4), (2, 1), (3, 6)
12. ( 3, 2), (1, 2), (0, 3), (2,1), ( 2,1)
13. (2, 0), (4, 1), (6, 0), (3, 1), (5, 2)
14. ( 1, 4), ( 2, 3), (4,1), (4, 2), ( 2, 3)
Answer the following.
15. Analyze the coordinates in each of the sets
above. Describe a method of determining
whether or not the set of points represents a
function without graphing the points.
16. Determine whether or not each set of points
represents a function without graphing the
points. Justify each answer.
(a) ( 7, 3), (3, 7), (1, 5), (5,1), ( 2,1)
(b) (6, 3), ( 4, 3), (2, 3), ( 3, 3), (5, 3)
(c) (3, 6), (3, 4), (3, 2), (3, 3), (3, 5)
(d) ( 2, 5), ( 5, 2), (2, 5), (5, 2), (5, 2)
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
Exercise Set 2.7: Functions and Graphs
MATH 1300 Fundamentals of Mathematics 201
Answer the following.
17. The graph of )(xfy is shown below.
(a) Find the domain of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
(c) Find the following function values: )6();4();0();2( ffff
(d) For what value(s) of x is ( ) 9f x ?
18. The graph of )(xgy is shown below.
(a) Find the domain of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
(c) Find the following function values:
( 2); (0); (1); (3); (6)g g g g g
(d) For what value(s) of x is ( ) 2g x ?
19. The graph of )(xgy is shown below.
(a) Find the domain of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
(c) Find the following function values:
( 2); (0); (2); (4); (6)g g g g g
(d) Which is greater, ( 2)g or (3)g ?
20. The graph of )(xfy is shown below.
(a) Find the domain of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
(c) Find the following function values: )4();1();1();2();3( fffff
(d) Which is smaller, )0(f or )3(f ?
x
y
g
x
yf
x
y
g
x
y
f
Exercise Set 2.7: Functions and Graphs
University of Houston Department of Mathematics 202
For each of the following functions:
(a) State the domain of the function. Write your
answer in interval notation.
(b) Choose x-values corresponding to the domain
of the function, calculate the corresponding y-
values, plot the points, and draw the graph of
the function.
21. 6)(23 xxf
22. 4)(32 xxf
23. 31,53)( xxxh
24. 23,2)( xxxh
25. 3)( xxg
26. 4)( xxg
27. 3)( xxf
28. xxf 5)(
29. xxxF 4)( 2
30. 1)3()( 2 xxG
For each of the following equations,
(a) Solve for y.
(b) Determine whether the equation defines y as a
function of x. (Do not graph.)
31. 3 5 8y x
32. 2692 yx
33. 22 3 7y x
34. 2 1 5y x
35. 23 yx
36. 32 yx
37. xy 2
38. 43 yx
39. 2 5 7 0y x
40. 3 4 8 0x y
SECTION 3.1 An Introduction to Polynomial Functions
MATH 1300 Fundamentals of Mathematics 203
Chapter 3 Polynomials
Section 3.1: An Introduction to Polynomial Functions
Polynomials and Polynomial Functions
Polynomials and Polynomial Functions
Polynomials:
A polynomial in a single variable x is the sum of a finite number of terms of the form nax , where a is a constant and the exponent n is a whole number. Recall that the set
of whole numbers is 0,1, 2, ...
Examples of polynomials in x are 33 , 5 8 ,x x x and 24 7 1x x . They can be
classified according to the number of terms:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 204
Degree of a Polynomial:
Example:
Solution:
SECTION 3.1 An Introduction to Polynomial Functions
MATH 1300 Fundamentals of Mathematics 205
Polynomial Functions:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 206
Evaluating Polynomial Functions:
Example:
Solution:
SECTION 3.1 An Introduction to Polynomial Functions
MATH 1300 Fundamentals of Mathematics 207
Graphs of Polynomial Functions:
Example:
Solution:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 208
Additional Example 1:
SECTION 3.1 An Introduction to Polynomial Functions
MATH 1300 Fundamentals of Mathematics 209
Solution:
Additional Example 2:
Solution:
Leading term:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 210
Degree:
Leading Coefficient:
Constant term:
Additional Example 3:
Solution:
SECTION 3.1 An Introduction to Polynomial Functions
MATH 1300 Fundamentals of Mathematics 211
Additional Example 4:
Solution:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 212
Additional Example 5:
Solution:
Exercise Set 3.1: An Introduction to Polynomial Functions
MATH 1300 Fundamentals of Mathematics 213
Answer the following.
(a) State whether or not each of the following
expressions is a polynomial. (Yes or No.)
(b) If the answer to part (a) is yes, then state the
degree of the polynomial.
(c) If the answer to part (a) is yes, then classify
the polynomial as a monomial, binomial,
trinomial, or none of these. (Polynomials of
four or more terms are not generally given
specific names.)
1. 34 3x
2. 5 3 86 3x x
x
3. 3 5x
4. 3 22 4 7 4x x x
5. 3 2
2
5 6 7
4 5
x x
x x
6. 8
7. 3 25 8x x x
8. 27 52 3
9x x
9. 3 2
7 5 32
xx x
10. 1 4 13 7 2x x
11. 11 1
34 29 2 4x x x
12. 2 3 1x x
13. 632
x
14. 3 26 8x x
x
15. 43 7x
16. 2 110 3 5x x
17. 10
18. 7 4x
19. 1 5 1 3 1 25 2 3 4x x x
20. 2 4 93 5 6 3x x x
21. 3 4 2 23 2a b a b
22. 5 2 4 94 3x y x y
23. 5 3
2
34x y
xy
24. 2 9 3 4 22 15 4
3x y z xy x y z
25. 3 7 4 3 2325 7
4xyz y x y z
26. 7 3 5 6 2 42 3a a b b a b
Answer True or False.
27. (a) 37 2x x is a trinomial.
(b) 37 2x x is a third degree polynomial.
(c) 37 2x x is a binomial.
(d) 37 2x x is a first degree polynomial
28. (a) 25 3 2x x is a trinomial.
(b) 25 3 2x x is a third degree polynomial.
(c) 25 3 2x x is a binomial.
(d) 25 3 2x x is a second degree polynomial.
29. (a) 36x is a monomial.
(b) 36x is a third degree polynomial.
(c) 36x is a first degree polynomial.
(d) 36x is a trinomial.
30. (a) 2 34 7x x x is a second degree
polynomial.
(b) 2 34 7x x x is a binomial.
(c) 2 34 7x x x is a third degree polynomial.
(d) 2 34 7x x x is a trinomial.
31. (a) 7 4 6 83 2 3x x y y is a tenth degree
polynomial.
(b) 7 4 6 83 2 3x x y y is a binomial.
(c) 7 4 6 83 2 3x x y y is an eighth degree
polynomial.
(d) 7 4 6 83 2 3x x y y is a trinomial.
32. (a) 4 53a b is a fifth degree polynomial.
(b) 4 53a b is a trinomial.
(c) 4 53a b is a ninth degree polynomial.
(d) 4 53a b is a monomial.
Exercise Set 3.1: An Introduction to Polynomial Functions
University of Houston Department of Mathematics 214
Each of the graphs below represents a polynomial
function. Use the graph to determine the x- and y-
intercept(s). The equation of each graph is given for
informational purposes, but the intercepts can be
determined entirely from the graph.
33. 3 2( ) 5 2 8f x x x x
34. 2( ) 8 12f x x x
35. 5 4 3( ) 2 16 36 54f x x x x x
36. 4 3 27 5 3113 3 3 3
( ) 10f x x x x x
x
y
x
y
x
y
x
y
Exercise Set 3.1: An Introduction to Polynomial Functions
MATH 1300 Fundamentals of Mathematics 215
For each of the following polynomial functions,
(a) Classify the function as linear, quadratic, or
cubic.
(b) Find the x- and y-intercept(s) of the function.
(Do this algebraically without drawing the
graph.)
(c) Find ( 4), ( 1)f f and (6)f .
37. 2( ) 64f x x
38. ( ) 3 8f x x
39. 3( ) 32 4f x x
40. 2( ) 50 2f x x
41. ( ) 12 5f x x
42. 3( ) 2 54f x x
Follow the directions above for numbers 43 and 44,
but in part (b), find the y-intercept only. (Do not
find the x-intercept(s).)
43. 2( ) 3 28f x x x
44. 2( ) 18 9 2f x x x
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 216
Section 3.2: Adding, Subtracting, and Multiplying
Polynomials
Operations with Polynomials
Operations with Polynomials
Like Terms:
Addition of Polynomials:
Example:
SECTION 3.2 Adding, Subtracting, and Multiplying Polynomials
MATH 1300 Fundamentals of Mathematics 217
Solution:
Subtraction of Polynomials:
Example:
Solution:
Multiplication of Polynomials:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 218
Example:
Solution:
Special Case - Multiplying Two Binomials:
Example:
SECTION 3.2 Adding, Subtracting, and Multiplying Polynomials
MATH 1300 Fundamentals of Mathematics 219
Solution:
Special Products:
Example:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 220
Solution:
Additional Example 1:
Solution:
Additional Example 2:
SECTION 3.2 Adding, Subtracting, and Multiplying Polynomials
MATH 1300 Fundamentals of Mathematics 221
Solution:
Additional Example 3:
(a) 26 2 7x x
(b) 25 3 6x x x
Solution:
Additional Example 4:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 222
Solution:
Exercise Set 3.2: Adding, Subtracting, and Multiplying Polynomials
MATH 1300 Fundamentals of Mathematics 223
Multiply. Write your final answer with the terms in
descending order, from highest to lowest degree.
1. 5 8x x
2. 2 10x x
3. 1 3x x
4. 9 2x x
5. 3 12x x
6. 15 10x x
7. 4 20x x
8. 7 8x x
9. 4 4x x
10. 9 9x x
11. 7 7x x
12. 6 6x x
13. 2 1 3x x
14. 4 3 2x x
15. 5 1 4 7x x
16. 3 1 2 5x x
17. 3 2 4x x
18. 6 1 3 4x x
19. 2 25 7x x
20. 2 23 4 2 1x x
21. 4 3 2 37 2 5x x x x
22. 5 2 42 5 3x x x
Perform the indicated operations. Write your final
answer with the terms in descending order, from
highest to lowest degree.
23. (a) 5 2x x
(b) 5 2x x
(c) 5 2x x
24. (a) 3 37 4x x
(b) 3 37 4x x
(c) 3 37 4x x
25. (a) 3 4 32 5 6x x x
(b) 3 4 32 5 6x x x
(c) 3 4 32 5 6x x x
26. (a) 7 4 45 3x x x
(b) 7 4 45 3x x x
(c) 7 4 45 3x x x
27. (a) 2 33 2 9x x x
(b) 2 33 2 9x x x
(c) 2 33 2 9x x x
28. (a) 37 5x x x
(b) 37 5x x x
(c) 37 5x x x
29. (a) 2 210 5 7x x x
(b) 2 210 5 7x x x
(c) 2 210 5 7x x x
30. (a) 38 2 5x x
(b) 38 2 5x x
(c) 38 2 5x x
31. (a) 3 + 7x x
(b) 3 7x x
(c) 3 7x x
Exercise Set 3.2: Adding, Subtracting, and Multiplying Polynomials
University of Houston Department of Mathematics 224
32. (a) 2 8x x
(b) 2 8x x
(c) 2 8x x
33. (a) 2 3 2 7x x x
(b) 2 3 2 7x x x
(c) 2 3 2 7x x x
34. (a) 2 3 3 26 2 7x x x x
(b) 2 3 3 26 2 7x x x x
(c) 2 3 3 26 2 7x x x x
35. (a) 25 5 4x x x
(b) 25 5 4x x x
(c) 25 5 4x x x
36. (a) 2 27 2 4 9x x x
(b) 2 27 2 4 9x x x
(c) 2 27 2 4 9x x x
37. (a) 22 3 4 12x x x
(b) 22 3 4 12x x x
(c) 22 3 4 12x x x
38. (a) 23 1 4 2 6x x x
(b) 23 1 4 2 6x x x
(c) 23 1 4 2 6x x x
39. (a) 3 2 32 5 8 2x x x x x
(b) 3 2 32 5 8 2x x x x x
(c) 3 22 5 8 2x x x x
40. (a) 4 3 25 3 4 2 3x x x x
(b) 4 3 25 3 4 2 3x x x x
(c) 4 3 25 3 4 2 3x x x x
41. (a) 2 23 2 1 2 5 3x x x x
(b) 2 23 2 1 2 5 3x x x x
(c) 2 23 2 1 2 5 3x x x x
42. (a) 2 25 7 2 4 3x x x x
(b) 2 25 7 2 4 3x x x x
(c) 2 25 7 2 4 3x x x x
43. (a) 4 3 5 32 5 3 2x x x x x x
(b) 4 3 5 32 5 3 2x x x x x x
(c) 4 3 5 32 5 3 2x x x x x x
44. (a) 7 4 5 44 2 6 3 5x x x x x x
(b) 7 4 5 44 2 6 3 5x x x x x x
(c) 7 4 5 44 2 6 3 5x x x x x x
45. (a) 3 2 34 2 3 5 4x x x x
(b) 3 2 34 2 3 5 4x x x x
(c) 3 2 34 2 3 5 4x x x x
46. (a) 2 4 35 2 8 2 4x x x x x
(b) 2 4 35 2 8 2 4x x x x x
(c) 2 4 35 2 8 2 4x x x x x
47. (a) 2
1 7x x
(b) 2
1 7x x
(c) 2
1 7x x
48. (a) 2
2 3x x
(b) 2
2 3x x
(c) 2
2 3x x
SECTION 3.3 Dividing Polynomials
MATH 1300 Fundamentals of Mathematics 225
Section 3.3: Dividing Polynomials
Polynomial Long Division and Synthetic Division
Polynomial Long Division and Synthetic Division
Long Division of Polynomials:
Example:
Solution:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 226
SECTION 3.3 Dividing Polynomials
MATH 1300 Fundamentals of Mathematics 227
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 228
SECTION 3.3 Dividing Polynomials
MATH 1300 Fundamentals of Mathematics 229
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 230
Synthetic Division of Polynomials:
Example:
Solution:
SECTION 3.3 Dividing Polynomials
MATH 1300 Fundamentals of Mathematics 231
A Comparison of Long Division and Synthetic Division
Let us now analyze the previous two examples, both of which solved the same problem using
long division and then synthetic division.
Long Division Synthetic Division
4 3 22 05 8 3 5x x x x x
Constant: 5
Change the sign of the constant term when
performing synthetic division.
4 3 22 05 8 3 5x x x x x
Notice the coefficients of the dividend: 2, 0, 8, 3, 5
Write the coefficients of the dividend (without
changing any signs). Do not forget the
‘placeholder’ for 30x .
Notice that the coefficients in each column of
the subtraction problems under the division
sign (at the left) are similar to the numbers in
each column of the synthetic division problem
(above). Remember that at the left, the signs
are changed when the expressions are
subtracted.
3 2
4 3 2
4 3
3 2
3 2
2
2
2 10 42 213
5 2 0 8 3 5
2
10 8
10
4
10
2 3
42
213 5
213
1
50
210
106
060
5
x x x
x x x x x
x x
x x
x x
x x
x x
x
x
10 50 210 1
5 | 2 0 8 3 5
2 10 42 213 1060
065
2 0 8| 55 3
| 2 0 3 55 8
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 232
Notice that the numbers in the answer line of
the synthetic division problem are the same as
the coefficients of the quotient plus the final
remainder in the long division problem.
In the long division problem, there is one
column for each power of x, and the
arithmetic in each column is done with the
coefficients.
Synthetic division is a shortcut for doing the
arithmetic with the coefficients without having
to write down all the variables. Remember that
this synthetic division procedure ONLY works
when the divisor is of the form D x x c .
The Remainder Theorem:
3 2
4 3 2
4 3
3 2
3 2
2
2
5 2 0 8 3 5
2 10
10 8
10 50
42 3
42
2 10 42 213
1060
210
213 5
213 1065
x x x
x x x x x
x x
x x
x x
x x
x x
x
x
5 | 2 0 8 3 5
10 50 210 1
2 10 42 213 1 6
065
0 0
SECTION 3.3 Dividing Polynomials
MATH 1300 Fundamentals of Mathematics 233
Additional Example 1:
Solution:
Additional Example 2:
Solution:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 234
Additional Example 3:
Solution:
SECTION 3.3 Dividing Polynomials
MATH 1300 Fundamentals of Mathematics 235
Additional Example 4:
Solution:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 236
SECTION 3.3 Dividing Polynomials
MATH 1300 Fundamentals of Mathematics 237
Additional Example 5:
Solution:
Exercise Set 3.3: Dividing Polynomials
University of Houston Department of Mathematics 238
Use long division to find the quotient and the
remainder.
1. 2 6 11
2
x x
x
2. 2 5 12
3
x x
x
3. 2 7 2
1
x x
x
4. 2 6 5
4
x x
x
5. 3 22 19 12
3
x x x
x
6. 5
33222 23
x
xxx
7. 12
12656 23
x
xxx
8. 3 212 13 22 14
3 4
x x x
x
9. 3 2
2
2 13 28 21
3 1
x x x
x x
10. 4 3 2
2
7 4 42 12
7 2
x x x x
x x
11. 64
144433222
2345
x
xxxx
12. 362
4282201024
23468
xx
xxxxx
13. 5
1532
34
x
xx
14. xx
xxx
2
72432
35
Use synthetic division to find the quotient and the
remainder.
15. 2 8 4
10
x x
x
16. 3
642
x
xx
17. 5
286133 23
x
xxx
18. 4
312 23
x
xxx
19. 1
43 24
x
xx
20. 1
8732 45
x
xxx
21. 5
101827113 234
x
xxxx
22. 2
1251832 234
x
xxxx
23. 2
83
x
x
24. 3
814
x
x
25. 21
3 574
x
xx
26. 31
234 9106
x
xxx
Evaluate P(c) using the following two methods:
(a) Substitute c into the function.
(b) Use synthetic division along with the
Remainder Theorem.
27. 2;254)( 23 cxxxxP
28. 1;3875)( 23 cxxxxP
Exercise Set 3.3: Dividing Polynomials
MATH 1300 Fundamentals of Mathematics 239
29. 1;12487)( 23 cxxxxP
30. 3;14672)( 34 cxxxxP
Evaluate P(c) using synthetic division along with the
Remainder Theorem. (Notice that substitution without
a calculator would be quite tedious in these examples,
so synthetic division is particularly useful.)
31. 5;321703883)( 23567 cxxxxxxP
32. 2;11235103)( 2456 cxxxxxxP
33. 43234 ;12254)( cxxxxP
34. 273456 ;135932196)( cxxxxxxP
When the remainder is zero, the dividend can be
written as a product of two factors (the divisor and the
quotient), as shown below.
30
65 , so 30 5 6 .
2
62
3
x xx
x
, so 2
6 3 2x x x x
In the following examples, use either long division or
synthetic division to find the quotient, and then write
the dividend as a product of two factors.
35. 2 11 24
8
x x
x
36. 2 3 40
5
x x
x
37. 2 7 18
2
x x
x
38. 2 10 21
3
x x
x
39. 24 25 21
7
x x
x
40. 23 22 24
6
x x
x
41. 22 7 5
1
x x
x
42. 25 4 12
2
x x
x
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 240
Section 3.4: Quadratic Functions
The Definition and Graph of a Quadratic Function
The Definition and Graph of a Quadratic Function
Definition:
Graph:
SECTION 3.4 Quadratic Functions
MATH 1300 Fundamentals of Mathematics 241
Example:
Solution:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 242
Additional Example 1:
Solution:
SECTION 3.4 Quadratic Functions
MATH 1300 Fundamentals of Mathematics 243
Additional Example 2:
Solution:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 244
Additional Example 3:
Solution:
(b) Since 1 0a , the parabola opens upward.
SECTION 3.4 Quadratic Functions
MATH 1300 Fundamentals of Mathematics 245
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 246
Additional Example 4:
SECTION 3.4 Quadratic Functions
MATH 1300 Fundamentals of Mathematics 247
Solution:
CHAPTER 3 Polynomials
University of Houston Department of Mathematics 248
Exercise Set 3.4: Quadratic Functions
MATH 1300 Fundamentals of Mathematics 249
Each of the quadratic functions below is written in the
form 2
( )f x ax bx c . The graph of a quadratic
function is a parabola with vertex (h, k), where
2ba
h and 2ba
k f .
(a) Give the coordinates of the vertex of the
parabola.
(b) Does the parabola open upward (with the
vertex being the lowest point on the graph) or
downward (with the vertex being the highest
point on the graph)?
(c) Find the y-intercept of the parabola.
(d) Find the axis of symmetry. (Be sure to write
your answer as an equation of a line.)
(e) Draw a sketch of the parabola which includes
the features from parts (a) through (c). (Do
not worry about the accuracy of the x-
intercepts on your graph; we will learn about
these in a later section.)
1. 2( ) 6 7f x x x
2. 2( ) 8 21f x x x
3. 2( ) 2f x x x
4. 2( ) 6 9f x x x
5. 2( ) 2 8 11f x x x
6. 2( ) 10f x x x
7. 2( ) 14 49f x x x
8. 2( ) 16f x x
9. 2( ) 8 9f x x x
10. 2( ) 3 18 15f x x x
11. 2( ) 2 5f x x
12. 2( ) 4 7f x x x
13. 2( ) 4 40 115f x x x
14. 2( ) 5 10 8f x x x
15. 2( ) 2 8 14f x x x
16. 2( ) 4 24 27f x x x
17. 2( ) 5 3f x x x
18. 2( ) 7 1f x x x
19. 2( ) 2 3 4f x x x
20. 2( ) 7 3f x x x
For each of the following quadratic functions,
(a) Multiply the factors to obtain a function of the
form 2
( )f x ax bx c .
(b) Find the coordinates of the vertex (h, k) of the
parabola, using the formulas 2ba
h and
2ba
k f .
(c) Match the function to its appropriate graph
from the choices shown below:
I. II.
III. IV.
21. ( ) 3 5f x x x
22. ( ) 2 8f x x x
23. ( ) 2 1 5f x x x
24. ( ) 3 2 4f x x x
xy
x
y
x
y
x
y
CHAPTER 4 Factoring
University of Houston Department of Mathematics 250
Chapter 4 Factoring
Section 4.1: Greatest Common Factor and Factoring by
Grouping
GCF and Grouping
GCF and Grouping
Finding the Greatest Common Factor:
Example:
Solution:
SECTION 4.1 Greatest Common Factor and Factoring by Grouping
MATH 1300 Fundamentals of Mathematics 251
Factoring Out the Greatest Common Factor:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 252
Example:
Solution:
Factoring by Grouping:
SECTION 4.1 Greatest Common Factor and Factoring by Grouping
MATH 1300 Fundamentals of Mathematics 253
Additional Example 1:
Solution:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 254
Additional Example 2:
Solution:
The GCF is the product of the factors that are shared by all three monomials.
SECTION 4.1 Greatest Common Factor and Factoring by Grouping
MATH 1300 Fundamentals of Mathematics 255
Additional Example 3:
Solution:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 256
Additional Example 4:
Solution:
SECTION 4.1 Greatest Common Factor and Factoring by Grouping
MATH 1300 Fundamentals of Mathematics 257
Additional Example 5:
Solution:
Exercise Set 4.1: Greatest Common Factor and Factoring by Grouping
University of Houston Department of Mathematics 258
Find the GCF (Greatest Common Factor) of the
following monomials.
1. 3 2 2 5 418 , 24 , 12x y x y xy
2. 3 5 7 3 4 920 , 32 , 8x y x y x y
3. 7 4 4 5 7 8, 7 , 14a b a b a b
4. 6 10 4 7 412 , 15 , 21c d c d c d
5. 3 12 10 5 6 7 516 , 32 , 100a b c a bc a c
6. 5 2 7 3 7 330 , 90 , 45a b a c b c
7. 6 9 8 3 5 7 5 6 410 , 18 , 7x y z x y z x y z
8. 7 5 3 4 8 6 5 29 , 50 , 20x y z x y x y z
Find the GCF of the terms of the polynomial and
factor it out. If the first term that appears in the
polynomial has a negative coefficient, then factor out
the negative of the GCF.
9. 5 10a
10. 4 12x
11. 3 15b
12. 4 24y
13. 9 24x y
14. 10 25a c
15. 6 8x xy
16. 8 12ab bc
17. 3 26 2a b ab
18. 5 73 6ac a c
19. 2 215 20r t rt
20. 4 3 3 630 2u v u w
21. 3 24 2 8x x x
22. 5 3 218 36 45x x x
23. 3 2 4 5 8 35 3 7x y x y x y
24. 3 6 4 5 220 8 12a b ab a b
25. 7 4 9 2 5 3 9 635 28 21a b c a b c a b c
26. 3 7 8 2 5 4 2 6 736 12 48x y z x y z x y z
27. 3 2 5 4 7 3 810 21 49a b c a c b c
28. 7 4 6 4 2 6 34 35 9x y z y z x y z
Factor the following expressions.
29. (a) 5xy y
(b) 4 5 4x x x
30. (a) 3xy y
(b) 6 3 6x x x
31. (a) 3b ab
(b) 3 5 5c a c
32. (a) ap cp
(b) 2 2a b c b
33. 3 ( 5) 4 ( 5)a a b a
34. 4 ( 7) 3 ( 7)x x y x
35. 2 ( 8) ( 8)x x x
36. 3 ( 2) ( 2)b b b
37. ( 3)( 5) ( 2)( 5)x x x x
38. ( 4)( 1) ( 4)( 6)x x x x
39. ( 2)(4 3) ( 8)( 2)a a a a
40. (3 1)(2 6) (3 1)( 2)a a a a
Factor by grouping.
41. 2 2b c ab ac
42. 3 3x y xz yz
43. 5 5y z xy xz
44. 4 4a b ca cb
45. 2 3 3x x xy y
46. 3 5 15xy x y
47. ac ad bc bd
48. 2p pr tp tr
49. 4 4xy x y
50. 2 2b ab a
Exercise Set 4.1: Greatest Common Factor and Factoring by Grouping
MATH 1300 Fundamentals of Mathematics 259
51. 2y xy y x
52. 2px p x p
53. 12 8 3 2b ab a
54. 18 24 15 20xy x y
55. 26 2 3t tx t x
56. 215 5 6 2x xy x y
57. 12 3 8 2ac ad bc bd
58. 24 15 8 5xy xz y z
Factor by grouping. (Hint: Use groups of three.)
59. ad ae af bd be bf
60. 4 3 3 12xy xz x y z
61. 23 12 15 2 8 10x xz x xy yz y
62. 212 8 20 3 2 5ab ac ad b bc bd
Each of the following expressions contains like terms.
Do not combine the like terms; instead, simply factor
by grouping. (This method will be helpful in the next
section when factoring trinomials.)
63. 2 3 2 6x x x
64. 2 5 7 35x x x
65. 2 4 3 12x x x
66. 2 3 6 18x x x
67. 26 10 9 15x x x
68. 221 3 14 2x x x
69. 29 21 6 14x x x
70. 225 5 20 4x x x
71. 24 14 14 49x x x
72. 29 15 15 25x x x
CHAPTER 4 Factoring
University of Houston Department of Mathematics 260
Section 4.2: Factoring Special Binomials and Trinomials
Special Factor Patterns
Special Factor Patterns
Factoring the Difference of Two Squares:
Example:
Solution:
SECTION 4.2 Factoring Special Binomials and Trinomials
MATH 1300 Fundamentals of Mathematics 261
Note:
Factoring the Difference of Two Cubes:
Example:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 262
Solution:
Factoring the Sum of Two Cubes:
Example:
SECTION 4.2 Factoring Special Binomials and Trinomials
MATH 1300 Fundamentals of Mathematics 263
Solution:
Factoring Perfect Square Trinomials:
Example:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 264
Solution:
Additional Example 1:
Solution:
SECTION 4.2 Factoring Special Binomials and Trinomials
MATH 1300 Fundamentals of Mathematics 265
Additional Example 2:
Solution:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 266
(c) The monomials 310yz and 10y share a common factor of 10 .y The first step in
factoring the given binomial to factor out the GCF of 10y .
Additional Example 3:
Solution:
Additional Example 4:
SECTION 4.2 Factoring Special Binomials and Trinomials
MATH 1300 Fundamentals of Mathematics 267
Solution:
Exercise Set 4.2: Factoring Special Binomials and Trinomials
University of Houston Department of Mathematics 268
Multiply the following.
1. (a) 4 4x x
(b) 2
4x
(c) 2
4x
2. (a) 9 9x x
(b) 2
9x
(c) 2
9x
Answer True or False.
3. 22 49 7x x
4. 2 64 8 8x x x
5. 2 26 12 36x x x
6. 2 24 16x x
7. 210 10 100x x x
8. 22 24 144 12x x x
9. 22 81 9x x
10. 2 25 25x x
Factor the following polynomials. If the polynomial
can not be factored any further within the real
number system, then write the original polynomial as
your answer.
11. (a) 2 9x
(b) 2 9x
(c) 2 6 9x x
(d) 2 6 9x x
12. (a) 2 25x
(b) 2 25x
(c) 2 10 25x x
(d) 2 10 25x x
13. 2 49x
14. 2 36x
15. 2 144x
16. 2 81a
17. 2 1p
18. 21 p
19. 2 100x
20. 2 4x
21. 225 c
22. 2144 d
23. 24 9b
24. 225 49x
25. 216 1x
26. 236 1x
27. 2 249 100x y
28. 2 264 25a b
29. 2 225 16c d
30. 2 24 9z w
31. 2
49
x
32. 2
116
x
33. 2 2
2 2
x a
y b
34. 2 2
2 2
p r
q t
35. 2 216
25 9
x y
36. 2 2
2
100
81
a b
c
37. 2 20 100x x
38. 2 8 16x x
39. 2 2 1x x
40. 2 14 49x x
41. 2 18 81x x
42. 2 24 144x x
43. 24 12 9x x
Exercise Set 4.2: Factoring Special Binomials and Trinomials
MATH 1300 Fundamentals of Mathematics 269
44. 29 30 25x x
45. 225 40 16x x
46. 236 12 1x x
47. 2 22x bx b
48. 2 22x cx c
49. 2 2 24 20 25b c bcd d
50. 2 2 29 24 16x xyz y z
When the remainder is zero, the dividend can be
written as a product of two factors (the divisor and the
quotient), as shown below.
30
65 , so 30 5 6 .
2
62
3
x xx
x
, so 2
6 3 2x x x x
In the following examples, use either long division or
synthetic division to find the quotient, and then write
the dividend as a product of two factors.
51. 3 8
2
x
x
52. 3 27
3
x
x
Factor the following polynomials.
53. 3 64x
54. 3 1m
55. 3 27p
56. 3 125x
57. 3 3x y
58. 3 3c d
59. 3 3125 8a b
60. 3 364 27x y
CHAPTER 4 Factoring
University of Houston Department of Mathematics 270
Section 4.3: Factoring Polynomials
Techniques for Factoring Trinomials
Techniques for Factoring Trinomials
Factorability Test for Trinomials:
Example:
Solution:
SECTION 4.3 Factoring Polynomials
MATH 1300 Fundamentals of Mathematics 271
Factoring Trinomials with Leading Coefficient 1:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 272
Example:
Solution:
SECTION 4.3 Factoring Polynomials
MATH 1300 Fundamentals of Mathematics 273
Factoring Trinomials with Leading Coefficient Different from 1:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 274
SECTION 4.3 Factoring Polynomials
MATH 1300 Fundamentals of Mathematics 275
CHAPTER 4 Factoring
University of Houston Department of Mathematics 276
Example:
Solution:
Additional Example 1:
(a) 22 3 8x x
(b) 242 25 3x x
SECTION 4.3 Factoring Polynomials
MATH 1300 Fundamentals of Mathematics 277
Solution:
Additional Example 2:
Solution:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 278
Additional Example 3:
Solution:
SECTION 4.3 Factoring Polynomials
MATH 1300 Fundamentals of Mathematics 279
Additional Example 4:
Solution:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 280
Additional Example 5:
Solution:
Exercise Set 4.3: Factoring Polynomials
MATH 1300 Fundamentals of Mathematics 281
At times, it can be difficult to tell whether or not a
quadratic of the form 2
ax bx c can be factored
into the form dx e fx g , where a, b, c, d, e, f,
and g are integers. If 2
4b ac is a perfect square, then
the quadratic can be factored in the above manner.
For each of the following problems,
(a) Compute 2
4b ac .
(b) Use the information from part (a) to
determine whether or not the quadratic can
be written as factors with integer coefficients.
(Do not factor; simply answer Yes or No.)
1. 2 5 3x x
2. 2 7 10x x
3. 2 6 16x x
4. 2 6 4x x
5. 29 x
6. 27x x
7. 22 7 4x x
8. 26 1x x
9. 22 2 5x x
10. 25 4 1x x
Factor the following polynomials. If the polynomial
can not be rewritten as factors with integer
coefficients, then write the original polynomial as your
answer.
11. 2 4 5x x
12. 2 9 14x x
13. 2 5 6x x
14. 2 6x x
15. 2 7 12x x
16. 2 8 15x x
17. 2 12 20x x
18. 2 7 18x x
19. 2 5 24x x
20. 2 9 36x x
21. 2 16 64x x
22. 2 6 9x x
23. 2 15 56x x
24. 2 6 27x x
25. 2 11 60x x
26. 2 19 48x x
27. 2 17 42x x
28. 2 12 64x x
29. 2 49x
30. 2 36x
31. 2 3x
32. 2 8x
33. 29 25x
34. 216 81x
35. 22 5 3x x
36. 23 16 15x x
37. 28 2 3x x
38. 24 16 15x x
39. 29 9 4x x
40. 25 17 6x x
41. 24 3 10x x
42. 29 21 10x x
43. 212 17 6x x
44. 28 26 7x x
Factor the following. Remember to first factor out the
Greatest Common Factor (GCF) of the terms of the
polynomial, and to factor out a negative if the leading
coefficient is negative.
45. 2 9x x
46. 2 16x x
47. 25 20x x
48. 24 28x x
Exercise Set 4.3: Factoring Polynomials
University of Houston Department of Mathematics 282
49. 22 18x
50. 28 8x
51. 4 25 20x x
52. 33 75x x
53. 22 10 8x x
54. 23 12 63x x
55. 210 10 420x x
56. 24 40 100x x
57. 3 29 22x x x
58. 3 27 6x x x
59. 3 24 4x x x
60. 5 4 310 21x x x
61. 4 3 26 6x x x
62. 3 22 80x x x
63. 5 39 100x x
64. 12 1049 64x x
65. 250 55 15x x
66. 230 24 72x x
Factor the following polynomials. (Hint: Factor first
by grouping, and then continue to factor if possible.)
67. 3 22 25 50x x x
68. 3 23 4 12x x x
69. 3 25 4 20x x x
70. 3 29 18 25 50x x x
71. 3 24 36 9x x x
72. 3 29 27 4 12x x x
SECTION 4.4 Using Factoring to Solve Equations
MATH 1300 Fundamentals of Mathematics 283
Section 4.4: Using Factoring to Solve Equations
Solving Quadratic Equations by Factoring
Solving Other Polynomials Equations by Factoring
Solving Quadratic Equations by Factoring
Zero-Product Property:
Example:
Solution:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 284
Example:
Solution:
The x-Intercepts of the Graph of a Quadratic Function:
Example:
Solution:
SECTION 4.4 Using Factoring to Solve Equations
MATH 1300 Fundamentals of Mathematics 285
Additional Example 1:
Solution:
Additional Example 2:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 286
Solution:
Additional Example 3:
Solution:
SECTION 4.4 Using Factoring to Solve Equations
MATH 1300 Fundamentals of Mathematics 287
Additional Example 4:
Solution:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 288
Additional Example 5:
Solution:
SECTION 4.4 Using Factoring to Solve Equations
MATH 1300 Fundamentals of Mathematics 289
Additional Example 6:
Solution:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 290
(c) Since 1 0a , the parabola opens upward.
Solving Other Polynomial Equations by Factoring
Solving Polynomial Equations by Factoring:
Example:
SECTION 4.4 Using Factoring to Solve Equations
MATH 1300 Fundamentals of Mathematics 291
Solution:
Example:
Solution:
The x-Intercepts of the Graph of a Polynomial Function:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 292
Example:
Solution:
SECTION 4.4 Using Factoring to Solve Equations
MATH 1300 Fundamentals of Mathematics 293
Additional Example 1:
Solution:
Additional Example 2:
Solution:
Additional Example 3:
Solution:
CHAPTER 4 Factoring
University of Houston Department of Mathematics 294
Exercise Set 4.4: Using Factoring to Solve Equations
MATH 1300 Fundamentals of Mathematics 295
Solve the following equations by factoring.
1. 021102 xx
2. 040132 xx
3. 01282 xx
4. 04062 xx
5. 35)2( xx
6. 20)8( xx
7. 72142 xx
8. xx 11602
9. 01572 2 xx
10. 0473 2 xx
11. 12176 2 xx
12. 6710 2 xx
13. 253 2 xx
14. 0568 2 xx
15. 0252 x
16. 0492 x
17. 094 2 x
18. 02536 2 x
Solve the following equations by factoring. To simplify
the process, remember to first factor out the Greatest
Common Factor (GCF) and to factor out a negative if
the leading coefficient is negative.
19. 082 xx
20. 0102 xx
21. 2 5 36 0x x
22. 2 14 48 0x x
23. 0213 2 xx
24. 0305 2 xx
25. 0123 2 x
26. 077 2 x
27. 090155 2 xx
28. 024204 2 xx
29. 03023080 2 xx
30. 0187512 2 xx
31. 3 25 6 0x x x
32. 3 27 18 0x x x
Each of the quadratic functions below is written in the
form 2
( )f x ax bx c . The graph of a quadratic
function is a parabola with vertex (h, k), where
2ba
h and 2ba
k f .
(a) Find the x-intercept(s) of the parabola by
setting ( ) 0f x and solving for x.
(b) Write the coordinates of the x-intercept(s)
found in part (a).
(c) Find the y-intercept of the parabola and write
its coordinates.
(d) Give the coordinates of the vertex (h, k) of the
parabola, using the formulas 2ba
h and
2ba
k f .
(e) Does the parabola open upward (with the
vertex being the lowest point on the graph) or
downward (with the vertex being the highest
point on the graph)?
(f) Find the axis of symmetry. (Be sure to write
your answer as an equation of a line.)
(g) Draw a graph of the parabola that includes
the features from parts (b) through (e).
33. 2( ) 6 8f x x x
34. 2( ) 2 15f x x x
35. 2( ) 8 16f x x x
36. 2( ) 10 16f x x x
37. 2( ) 4 21f x x x
38. 2( ) 10 25f x x x
39. 2( ) 3 12 36f x x x
40. 2( ) 4 8 5f x x x
41. 2( ) 16f x x
Exercise Set 4.4: Using Factoring to Solve Equations
University of Houston Department of Mathematics 296
42. 2( ) 25f x x
43. 2( ) 9 4f x x
44. 2( ) 9 100f x x
Find the x-intercept(s) of the following.
45. 3 2( ) 7 10f x x x x
46. 3 2( ) 2 99f x x x x
47. 3( ) 25f x x x
48. 3( ) 4f x x x
49. 3 2( ) 2 9 18f x x x x
50. 3 2( ) 4 4f x x x x
For each of the following problems:
(a) Model the situation by writing appropriate
equation(s).
(b) Solve the equation(s) and then answer the
question posed in the problem.
51. The length of a rectangular frame is 5 cm longer
than its width. If the area of the frame is 36 cm2,
find the length and width of the frame.
52. The width of a rectangular garden is 8 m shorter
than its length. If the area of the field is 180 m2,
find the length and the width of the garden
53. The height of a triangle is 3 cm shorter than its
base. If the area of the triangle is 90 cm2, find the
base and height of the triangle.
54. Find x if the area of the figure below is 26cm2.
(Note that the figure may not be drawn to scale.)
x cm
x cm
3 cm
8 cm
SECTION 5.1 Simplifying Rational Expressions
MATH 1300 Fundamentals of Mathematics 297
Chapter 5 Rational Expressions, Equations, and Functions
Section 5.1: Simplifying Rational Expressions
Rational Expressions
Rational Expressions
Definition:
Simplifying:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 298
Example:
Solution:
SECTION 5.1 Simplifying Rational Expressions
MATH 1300 Fundamentals of Mathematics 299
Additional Example 1:
Solution:
Additional Example 2:
Solution:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 300
Additional Example 3:
Solution:
Additional Example 4:
SECTION 5.1 Simplifying Rational Expressions
MATH 1300 Fundamentals of Mathematics 301
Solution:
Exercise Set 5.1: Simplifying Rational Expressions
University of Houston Department of Mathematics 302
Simplify the following rational expressions. If the
expression cannot be simplified any further, then
simply rewrite the original expression.
1. 15
25
2. 30
36
3. 48
64
4. 26
39
5. 2 5
5 3
60
48
x y
x y
6. 4 9
7 10
49
56
a b
a b
7.
73
35
5
10
x x y
x x y
8.
26
3
8
12
c c d
c c d
9. x y
y x
10. c d
d c
11.
2
6
a b c d
b a
12.
12
6
x y w z
z w x y
13. 4 8
2
x
x
14. 3
5 15
x
x
15. 2
5
25
x
x
16. 2
3
9
x
x
17. 2 2a b
a b
18. 2 16
4
x
x
19. 2
2
49
9 18
c
c c
20. 2
2
11 10
100
x x
x
21. 2
2
2 15
10 21
x x
x x
22. 2
2
20
30
m m
m m
23. 2
2
5 6
12
x x
x x
24. 2
2
7 12
7 30
x x
x x
25. 2
2
8 12
13 42
x x
x x
26. 2
2
7 10
7 10
x x
x x
27. 2
2
36
12 36
x
x x
28. 2
2
8 16
16
x x
x
29. 2
9 36
4
x
x x
30. 27 14
2
x x
x
31. 2
2
10 30
5 10
x x
x x
Exercise Set 5.1: Simplifying Rational Expressions
MATH 1300 Fundamentals of Mathematics 303
32. 2
3 2
6 8
9 12
x x
x x
33. 2
2
7 6
8 8
x x
x x
34. 2
2
4 20
4 5
x x
x x
35. 2
2
6 24 18
4 8 60
x x
x x
36. 2
2
5 10 40
10 30 20
x x
x x
37. 2
2
4 17 15
5 13 6
x x
x x
38. 2
2
4 8 21
8 24 14
x x
x x
39. 2
2
6 5 4
10 9 2
x x
x x
40. 2
2
15 4 4
5 22 8
x x
x x
41. 2
2
8 30 7
16 1
x x
x
42. 2
2
9 25
6 13 5
x
x x
43. 3 2
3 2
1m m m
m m m n n
44. 2 2
ax ay bx by
ax ay x y
45. 3 2 6
3 5 15
xy x y
yz z y
46. 2 2
5 2 10
4 5 20
ab a b
a b b a
47. 3 8
2
x
x
48. 3
5
125
x
x
49. 3
2
27
3 9
x
x x
50. 3
2
1
1
x
x x
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics
304
Section 5.2: Multiplying and Dividing Rational Expressions
Multiplication and Division
Multiplication and Division
Multiplication of Rational Expressions:
To multiply two fractions, place the product of the numerators over the product
of the denominators.
Example:
Solution:
SECTION 5.2 Multiplying and Dividing Rational Expressions
MATH 1300 Fundamentals of Mathematics 305
Division of Rational Expressions:
Example:
Solution:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics
306
Additional Example 1:
Solution:
Additional Example 2:
SECTION 5.2 Multiplying and Dividing Rational Expressions
MATH 1300 Fundamentals of Mathematics 307
Solution:
Additional Example 3:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics
308
Solution:
Exercise Set 5.2: Multiplying and Dividing Rational Expressions
MATH 1300 Fundamentals of Mathematics 309
Multiply the following rational expressions and
simplify. No answers should contain negative
exponents.
1. 6 14
7 18
2. 8 45
9 32
3. 2
105
4. 3
128
5. 4 7 3
5 8 6 9
ab c d
c d a b
6. 5 6 3
3 8 10 9
x y wz
w z x y
7. 8 2 4 6 6 2
3 5 5 3 7
m n n t p t
p t m m n
8. 3 4 8 2 2 3 4
2 7 7 5
x y a b x y a
ab x y b y
9. 2
5
32
6x
x
10. 4 56
2x
x
11. 5 3
3 10
x x
x x
12. 6 5
1 6
x
x x
13. 5
22
xx
x
14. 3
( 1)1
xx
x
15. 5
(7 )7
xx
x
16. 3
( 2)2
xx
17. 3 2
( 5)5
xx
x
18. 2 1
(3 )3
xx
x
19. 3
( 4)5 20
xx
20. 2
(4 28)7
xx
21. 3 4
(2 8)3 12
xx
x
22. 2 2
(3 3)4 4
xx
x
23. 6 12 4 12
3 3 6
x x
x x
24. 7 6 24
2 8 5 35
x x
x x
25. 6 10 3
5 15 9
x
x x
26. 2
2 4 6
6 9
x x
x x x
27. 2 2
2 2
6 6 5
3 4 2 15
x x x x
x x x x
28. 2 2
2 2
2 12
8 15 9 14
x x x x
x x x x
29. 2 2
2
3 10 2 4
56 24
x x x x
xx x
30. 2 2
2 2
6 30 4 21
6 40 8
x x x x
x x x x
31. 2
2
4 9
3 16
x x
x x
Exercise Set 5.2: Multiplying and Dividing Rational Expressions
University of Houston Department of Mathematics 310
32. 2 225 12 36
6 5
x x x
x x
33. 2 2
2 2
2 9 10 7 12
5 6 2 3 5
x x x x
x x x x
34. 2 2
2 2
2 8 3 14 5
3 16 5 20
x x x x
x x x x
35. 7 2 14
7 3 21 2 2
ax bx ay by ax x a
ax x a ax bx a b
36. 2 22 2
3 3
ac ad bc bd c d
ac ad bc bd ac ad bc bd
Divide the following rational expressions and simplify.
No answers should contain negative exponents.
37. 5 15
8 32
38. 6 4
25 5
39. 25
102
40. 12
67
41. 2
63
42. 4
405
43. 4 3
2 7 5
x x z
y z y
44. 3 7 5 9
4 2
a c b c
b a
45. 5 6
5 2
2 5
a ba d
c d
46. 3 2
4 5
6
x yx z
w z
47. 3 5
1 1
x x
x x
48. 4 3
2 2
x x
x x
49. 27 7 1
21 3
x x
x x
50. 2
7 4
39
x
xx
51. 2 1 1
6 3 18
x x
x x
52. 2
5
24
x x
xx
53. 2
5 10
416 xx
54. 2
2
4 2
5 25
x x
x x
55. 2
2 2
9 3
1 2 1
x x
x x x
56. 2
2
4 9 2 3
510 25
x x
xx x
57. 2 2
2 2
3 10 6
3 28 12
x x x x
x x x x
58. 2 2
2 2
4 4 8 20
6 16 9 8
x x x x
x x x x
59. 2 2
2 2
6 1 3 2 1
6 5 1 3 4 1
x x x x
x x x x
60. 2 2
2 2
10 17 6 6 5 4
5 4 12 3 2 8
x x x x
x x x x
61. 3 3
3 3
am an bm bn am an bm bn
am an bm bn am an bm bn
62. 2
2 2 5 5
5 53 3
cx dx cy dy cx cy dx dy
cx dx c dx x xy y
SECTION 5.3 Adding and Subtracting Rational Expressions
MATH 1300 Fundamentals of Mathematics 311
Section 5.3: Adding and Subtracting Rational Expressions
Addition and Subtraction
Addition and Subtraction
Addition and Subtraction of Rational Expressions with Like
Denominators:
Example:
Perform the following operations. All results should be in simplified form.
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 312
Solution:
Addition and Subtraction of Rational Expressions with Unlike
Denominators:
Example:
SECTION 5.3 Adding and Subtracting Rational Expressions
MATH 1300 Fundamentals of Mathematics 313
Solution:
Additional Example 1:
Perform the following operations. All results should be in simplified form.
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 314
Solution:
Additional Example 2:
Perform the addition. Give the result in simplified form.
Solution:
SECTION 5.3 Adding and Subtracting Rational Expressions
MATH 1300 Fundamentals of Mathematics 315
Additional Example 3:
Perform the subtraction. Give the result in simplified form.
Solution:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 316
Additional Example 4:
Perform the subtraction. Give the result in simplified form.
Solution:
SECTION 5.3 Adding and Subtracting Rational Expressions
MATH 1300 Fundamentals of Mathematics 317
Additional Example 5:
Perform the following operations. Give all results in simplified form.
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 318
Solution:
Exercise Set 5.3: Adding and Subtracting Rational Expressions
MATH 1300 Fundamentals of Mathematics 319
Perform the indicated operations and simplify.
(Whenever possible, write both the numerator and
denominator of the answer in factored form.)
1. 2 3
5 7
x y
2. 4 2
5 3
a b
3. 3 2
4 9a b
4. 7 5
2 3c c
5. 2 2 5
7 2
x y xy
6. 4 7 5 4
3 2
a b a b
7. 8 7
5 5
x x
x x
8. 3 4 6
1 1
x x
x x
9. 3 2 2 6
5 20 5 20
x x
x x
10. 2 3 10 9
4 3 4 3
x x
x x
11. 2 3
1 5x x
12. 5 6
4 7x x
13. 3 8
1x x
14. 5 2
4x x
15. 3 4
1 2x x
16. 1 2
2 2x x
17. 6 2
3 7x x
18. 7 4
9 2x x
19. 5 4
3 3
x
x x
20. 2
5 5
x x
x x
21. 2
35x
22. 5
47x
23. 7
22x
24. 6
43x
25. 4 2
1 3
x
x x
26. 3 1
3 3
x
x x
27. 2 3
2
x
x x
28. 4
3 5
x
x x
29. 1 1
1 2 1
x
x x
30. 2 3 6
1
x
x x
31. 3 1
2 4
x x
x x
Exercise Set 5.3: Adding and Subtracting Rational Expressions
University of Houston Department of Mathematics 320
32. 1 2
3 1
x x
x x
33. 5 2
4 3
x x
x x
34. 4 2
1 1
x x
x x
35. 7 5
8 12 6 6x x
36. 5 2
12 6 10 40x x
37. 3 8 6
1 2x x x
38. 2 4 3
3 2x x x
39. 2
2
4 35
2 8
x
x x
40. 2
2
3 52
3 4
x x
x x
41. 2 2 2
2 5
2 8 2 4
x
x x x x x x
42. 2 2 2
1 1
3 18 6 3
x
x x x x x x
43. 2 2 2
4 2
10 24 12 32 14 48
x
x x x x x x
44. 2 2 2
2 3
7 12 4 3 5 4
x
x x x x x x
SECTION 5.4 Complex Fractions
MATH 1300 Fundamentals of Mathematics 321
Section 5.4: Complex Fractions
Simplifying Complex Fractions
Simplifying Complex Fractions
Definition:
Simplifying:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 322
Example:
Solution:
Method 1:
SECTION 5.4 Complex Fractions
MATH 1300 Fundamentals of Mathematics 323
Method 2:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 324
Additional Example 1:
Solution:
Additional Example 2:
Solution:
SECTION 5.4 Complex Fractions
MATH 1300 Fundamentals of Mathematics 325
Additional Example 3:
Solution:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 326
Additional Example 4:
Solution:
SECTION 5.4 Complex Fractions
MATH 1300 Fundamentals of Mathematics 327
Additional Example 5:
Solution:
Exercise Set 5.4: Complex Fractions
University of Houston Department of Mathematics 328
Simplify the following. No answers should contain
negative exponents.
1.
7
125
8
2.
6
72
3
3. 4
3
4
5
x
y
x
y
4.
5
2
7
12
4
a
b
c
b
5.
2
3
3
4
2
8
5
ab
c d
a c
bd
6.
4
6 5
2
7
3
8
9
4
x y
w z
x w
z
7. 3
2
5
4
8
x yz
x
y
8.
35
2
10
b c
d
bd
9.
2 3
3 41 2
2 5
10.
5 1
6 23 2
5 3
11.
5
61 2
2 3
12.
11
71
7
13.
12
41
22
14.
5
82
43
15.
2
53
10
x
x
16.
7
81
12
x
x
17.
3
a b
bb a
ab
18.
2x y
x y
xy
19.
2
2
4
7 12
8
20
6
x x
x
x x
x
Exercise Set 5.4: Complex Fractions
MATH 1300 Fundamentals of Mathematics 329
20.
5
2
3
2
9
6 16
18
11 24
x
x x
x
x x
21. 3 2
3 2
x x
x x
22. 4 5
3 4
a a
a a
23.
2 1
1 1
2 2
x
x
x
24. 2
5 1
3 3
25
x
x
x
25.
2 3
5 4a b
a b
26.
7 2
3 4
x y
x y
27.
6
19
xx
xx
28.
4
10
7
xx
xx
29.
2
2 3
5 52
25
x x
x
30. 2
6
24 1
2 1
x
x x
x x
31.
1 3
3 43 2
3 1
x x
x x
32.
2 5
1 23 2
2
x x
x x
33.
152
127
xx
xx
34.
149
76
xx
xx
For each of the following expressions,
(a) Rewrite the expression so that it contains
positive exponents rather than negative
exponents.
(b) Simplify the expression.
35. 1
1 1
x
x
36. 1
1
3 x
x
37. 1 1
1 1
x y
x y
38. 1 1
2 2
c d
c d
39. 2 2
1 1
x y
x y
Exercise Set 5.4: Complex Fractions
University of Houston Department of Mathematics 330
40. 1 1
2 2
a b
b a
41. 1 1
3 3
c d
c d
42. 3 3
1 1
x y
x y
43. 3 3
2 2
a b
a b
44. 2 2
3 3
x y
x y
45. 1
11
1 x
46. 2
11
1 x
47. 1
54
5 x
48. 1
23
2 x
SECTION 5.5 Solving Rational Equations
MATH 1300 Fundamentals of Mathematics 331
Section 5.5: Solving Rational Equations
Rational Equations
Rational Equations
Definition of a Rational Equation:
Solving a Rational Equation:
Example:
Solution:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 332
Example:
Solution:
SECTION 5.5 Solving Rational Equations
MATH 1300 Fundamentals of Mathematics 333
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 334
Extraneous Solutions:
Example:
Solution:
SECTION 5.5 Solving Rational Equations
MATH 1300 Fundamentals of Mathematics 335
Additional Example 1:
Solution:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 336
Additional Example 2:
Solution:
SECTION 5.5 Solving Rational Equations
MATH 1300 Fundamentals of Mathematics 337
Additional Example 3:
Solution:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 338
SECTION 5.5 Solving Rational Equations
MATH 1300 Fundamentals of Mathematics 339
Additional Example 4:
Solution:
Exercise Set 5.5: Solving Rational Equations
University of Houston Department of Mathematics 340
Solve the following. Remember to identify any
extraneous solutions.
1. 2
25 3
x x
2. 3 2
14 3
a a
3. 3 2
222 5
c c
4. 5
148 4
x x
5. 5 3
26 10
x x
6. 7 3
58 20
x x
7. 4 7 3
5 5
x x
x x
8. 3 4 8
2 2
x x
x x
9. 5 2 6
1 1
x x
x x
10. 3 4 5 7
6 6
x x
x x
11. 34
7
5
2
xx
12. 24
5
6
7
xx
13. 3
42x
14. 5
37x
15. 2
05x
16. 5
02
x
x
17. 3 1
75
x
x
18. 3
27
x
x
19. 2
71
9x
20. 2
50
4x
21. 2
51
7 12
x
x x
22. 2
111
3 10
x
x x
23. 5 9
27 3t
24. 3 12
31 5x
25. 7 8
19 1
x
x
26. 2 1
19 4
a
a
27. 7 2
37 3
x
x
28. 2 2
23 3
t
t
29. 1 3 13
1 4 12
w
w
30. 4 1 9
9 2 14
x
x
31. 5 3 7
3 4 4
x
x x
32. 3 3
2 7 2
x x
x x
Exercise Set 5.5: Solving Rational Equations
MATH 1300 Fundamentals of Mathematics 341
33. 153
8
3
1
5
4
xx
34. 3
5
63
4
2
7
xx
35. 3 2 1
4 8 3 6 36a a
36. 5 1 7
3 15 2 10 12c c
37. 2
3 1 7
5 3 2 15x x x x
38. 2
2 1 4
1 2 2x x x x
39. 2
4 2 8
3 1 2 3x x x x
40. 2
7 2 10
4 5 9 20x x x x
41. 2
3 4 8
2 2 4x x x
42. 2
3 6 24
4 4 16x x x
43. 2
1 61
x x
44. 2
12 11
xx
45. 2
7 42
x x
46. 2
4 113
xx
47. 6 1
14x x
48. 7 4
15x x
49. 4 1
14 1x x
50. 5 2
14 2x x
51. 7 8
15 8x x
52. 5 6
17 9x x
53. 4 2
15 10
x
x x
54. 2 1
17 3
x
x x
55. 1 4
2 5 3 2 5
x
x x x
56. 2 1 6
3 1 3 1
x
x x x
57. 4 3 3
3 2 1 1
x
x x x
58. 5 2
2 3 2 3
x
x x x
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 342
Section 5.6: Rational Functions
Working with Rational Functions
Working with Rational Functions
Definition of a Rational Function:
Domain of a Rational Function:
Example:
SECTION 5.6 Rational Functions
MATH 1300 Fundamentals of Mathematics 343
Solution:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 344
Graph of a Rational Function:
Example:
Solution:
SECTION 5.6 Rational Functions
MATH 1300 Fundamentals of Mathematics 345
The graph of the function is shown below, labeled with the information from parts (b)-(d).
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 346
Vertical Asymptotes:
SECTION 5.6 Rational Functions
MATH 1300 Fundamentals of Mathematics 347
Finding Vertical Asymptotes
Example:
Solution:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 348
Horizontal Asymptotes:
SECTION 5.6 Rational Functions
MATH 1300 Fundamentals of Mathematics 349
Additional Example 1:
Solution:
0
30
1
f x
x
x
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 350
SECTION 5.6 Rational Functions
MATH 1300 Fundamentals of Mathematics 351
Additional Example 2:
Solution:
Additional Example 3:
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics 352
Solution:
SECTION 5.6 Rational Functions
MATH 1300 Fundamentals of Mathematics 353
Exercise Set 5.6: Rational Functions
University of Houston Department of Mathematics 354
Find the indicated function values. If undefined, state
“Undefined.”
1. If ( )3
xf x
x
, find
(a) (0)f (b) ( 1)f (c) 13
f
2. If 5
( )5
f xx
, find
(a) (0)f (b) ( 5)f (c) 15
f
3. If 3 2
( )7
xf x
x
, find
(a) (0)f (b) ( 3)f (c) 45
f
4. If 2 7
( )6
xf x
x
, find
(a) (0)f (b) (4)f (c) 34
f
5. If 2
2( )
6f x
x x
, find
(a) ( 2)f (b) (0)f (c) (5)f
6. If 2
1( )
2 1
xf x
x x
, find
(a) ( 4)f (b) (0)f (c) (1)f
7. If 2
( )121
xf x
x
, find
(a) ( 3)f (b) (0)f (c) (12)f
8. If 2
1
5 14x x , find
(a) (0)f (b) ( 1)f (c) (7)f
9. If 2
3( )
11 28
xf x
x x
, find
(a) (3)f (b) ( 4)f (c) (0)f
10. If 2
5( )
12
xf x
x x
, find
(a) (0)f (b) ( 2)f (c) ( 5)f
The graph of each of the following functions has a
horizontal asymptote at 1y . (You will learn how to
find horizontal asymptotes in a later mathematics
course.) For each function,
(a) Find the domain of the function and express it
as an inequality.
(b) Write the equation of the vertical
asymptote(s) of the function.
(c) Find the x- and y-intercept(s) of the function,
if they exist. If an intercept does not exist,
state “None.”
(d) Find (1)f and ( 1)f .
(e) Based on the features from (a)-(d), match the
function with its corresponding graph, using
the choices (Graphs I-IV) below.
11. 4
( )3
xf x
x
12. 6
( )2
xf x
x
13. 6
( )3
xf x
x
14. 4
( )2
xf x
x
Graph IV:
Graph I: Graph II:
Graph III:
x
y
x
y
x
y
x
y
Exercise Set 5.6: Rational Functions
MATH 1300 Fundamentals of Mathematics 355
The graph of each of the following functions has a
horizontal asymptote at 0y . (You will learn how to
find horizontal asymptotes in a later mathematics
course.) For each function,
(a) Find the domain of the function and express it
as an inequality.
(b) Write the equation of the vertical
asymptote(s) of the function.
(c) Find the x- and y-intercept(s) of the function,
if they exist. If an intercept does not exist,
state “None.”
(d) Find (1)f and ( 1)f .
(e) Based on the features from (a)-(d), match the
function with its corresponding graph, using
the choices (Graphs I-IV) below.
15. 4
( )2
f xx
16. 8
( )f xx
17. 4
( )f xx
18. 8
( )2
f xx
For each of the following functions,
(a) Find the domain of the function and express it
as an inequality. Then write the domain of the
function in interval notation.
(b) Write the equation of the vertical
asymptote(s) of the function.
(c) Find the x- and y- intercept(s) of the function.
If an intercept does not exist, state “None."
19. If 10
( )5
f xx
20. 12
( )3
f xx
21. 6
( )2
xf x
x
22. ( )8
xf x
x
23. 3
( )x
f xx
24. 4
( )1
xf x
x
25. 2
9( )
9
xf x
x
26. 2
8( )
16
xf x
x
27. 2
24( )
8 12f x
x x
28. 2
2( )
20
xf x
x x
29. 2
5( )
2 1
xf x
x x
30. 2
8( )
5 4
xf x
x x
31. 2
4( )
8f x
x x
32. 2
4( )
6
xf x
x x
Graph IV:
Graph I: Graph II:
Graph III:
x
y
x
y
x
y
x
y
Exercise Set 5.6: Rational Functions
University of Houston Department of Mathematics 356
33. 2 10 25
( )5
x xf x
x
34. 2 7 18
( )5
x xf x
x
35. 2
2( )
25
xf x
x
36. 2
1( )
16
xf x
x
37. 2 5 14
( )5 7
x xf x
x
38. 29 1
( )3 2
xf x
x
39. 2
2
25 36( )
5 4
xf x
x x
40. 2
2
7 6( )
5 24
x xf x
x x
Odd-Numbered Answers to Exercise Set 1.1: Numbers
MATH 1300 Fundamentals of Mathematics 357
1. (a) Composite; 1, 2, 4, 8
(b) Prime
(c) Neither
(d) Neither
(e) Composite; 1, 2, 3, 4, 6, 12
3. (a) 0.1
(b) 0.2
(c) 0.3
(d) 0.4
(e) 0.5
(f) 0.6
(g) 0.7
(h) 0.8
(i) 0.9 1=
(j) 1.1 (since 10 1
9 91= )
(k) 1.5 (since 514
9 91= )
(l) 2.7 (since 25 7
9 92= )
(m) 3.2 (since 29 2
9 93= )
5. (a) Rational; 7
10
(b) Irrational
(c) Rational; 3
7
(d) Rational; 5
1
−
(e) Rational; 4
1
(f) Rational; 1
3
(g) Rational; 12
1
(h) Rational; 23
35
(i) Irrational
(j) Rational; 2
1
−
(k) Irrational
7. Odd, Negative, Integer, Rational, Real
9. Positive, Irrational, Real
11. (a) 8, 0,12−
(b) 5
(c) 15
47, , , 5, 12π
(d) 8, 2.1, 0.4− − −
(e) 5
(f) 12
(g) 5,12
(h) 0, 5, 12
(i) 8, 0, 5, 12−
(j) All numbers in the set:
15
48, 2.1, 0.4, 0, 7, , , 5, 12π− − −
(k) 15
48, 2.1, 0.4, 0, , 5,12− − −
(l) 7, π
(m) None
13.
15. 2 is the only number that is both prime and even.
17. Answers vary. Some possible answers are:
2 4
3 7, , 0.6, 0.37, 0.2, 8− − −
(Note: Any repeating decimal is a rational number.
There are methods for changing repeating decimals to
fractions which will not be learned in this course.)
19. Answers vary. Some possible answers are:
2, 3, 5, 6, 10, , , 0.080080008...eπ− −
21. Does not exist
23. Does not exist
25. True
27. True
29. False. The number 0 is a whole number but not a
natural number.
31. True
33. False. A repeating decimal such as 4
90.4 = is a
nonterminating decimal, but is a rational number.
35. 2, 3, 5, 7
37. 8, 9. 10, 12, 14, 15, 16, 18
39. 4, 6, 8
250
1 35
10 −55 13.3
Undefined Y N N N N
Natural N Y N N N
Whole N Y N N N
Integer N Y N Y N
Rational N Y Y Y Y
Irrational N N N N N
Prime N N N N N
Composite N N N N N
Real N Y Y Y Y
Odd-Numbered Answers to Exercise Set 1.1: Numbers
University of Houston Department of Mathematics 358
−3 −2 −1 0 1
41. <
43. >
45. =
47. >
49. >
51. <
53. <
55. =
57. (a) 3−
(b) 4
(c) 1−
(d) 2
3
(e) 3
72−
59. (a) 1
2−
(b) 9
5
(c) Undefined
(d) 5
8 (Note: 3 8
5 51 = )
(e) 1−
61. (a) 0
(b) 1
63. Numbers ordered from least to greatest:
9 0
, 2, 1, , 0.4, 0.494 5
− − −
The above numbers plotted on a number line:
Odd-Numbered Answers to Exercise Set 1.2: Integers
MATH 1300 Fundamentals of Mathematics 359
1. (a) 10
(b) 10−
(c) 4
(d) 4−
(e) 3−
3. (a) 4−
(b) 4
(c) 4
(d) 4−
5. (a) 12−
(b) 8−
(c) 8
(d) 8
(e) 12
(f) 8−
(g) 12−
(h) 12
7. (a) <
(b) >
(c) >
(d) =
9. (a) 0
(b) Undefined
(c) 0
(d) 6−
(e) 6
(f) 6−
(g) 6
(h) 6
(i) 6−
(j) Undefined
(k) 6−
(l) 0
11. (a) 20
(b) 5
(c) 20−
(d) 5−
(e) 5−
(f) 5
13. (a) 24
(b) 24−
(c) 24
(d) 24−
15. (a) 6
(b) 10−
(c) 16
(d) 4−
(e) 6−
(f) 0
(g) 8
(h) 9−
(i) 8−
(j) 8−
(k) 10
(l) 0
(m) 1
4, or 0.25
(n) Undefined
(o) 6
Odd-Numbered Answers to Exercise Set 1.3: Fractions
University of Houston Department of Mathematics 360
1. (a) GCD: 2 (b) LCM: 24
3. (a) GCD: 1 (b) LCM: 70
5. (a) GCD: 14 (b) LCM: 28
7. (a) GCD: 4 (b) LCM: 40
9. (a) GCD: 6 (b) LCM: 90
11. (a) GCD: 4 (b) LCM: 240
13. (a) 27
1 (b) 35
4 (c) 13
6
15. (a) 34
6− (b) 1011
2− (c) 3
107−
17. (a) 316
(b) 679
(c) 263
19. (a) 197
− (b) 173
− (c) 494
−
21. (a) 37
(b) 9
11
23. (a) 35
6 (b) 13
5−
25. (a) 2 14 2
3 3= (b) 25
14
27. (a) 13
6 (b) 4 2
10 53 3=
29. (a) 34
(b) 421
31. (a) 1760
(b) 3135
33. (a) 5 1
70 14− −= (b)
712
1
35. (a) 2542
9 (b) 2 1
10 52 2=
37. (a) 6
3516 (b)
79
3
39. (a) 33 1160 20
2 2= (b) 1348
12
41. (a) 3536
(b) 2845
−
43. (a) 163
(b) 193
45. (a) 53
(b) 352
(c) 20−
47. (a) 2577
(b) 1528
(c) 425
49. (a) 100 (b) 23
(c) 507
−
51. (a) 2
15 (b)
2725
(c) 92
53. (a) 17
1 (b) 1116
1
55. (a) 12 (b) 15
57. (a) 12
2 (b) 57
5−
Odd-Numbered Answers to Exercise Set 1.4: Exponents and Radicals
MATH 1300 Fundamentals of Mathematics 361
1. (a) 37 (b)
210
(c) 68 (d)
73
3. <
5. >
7. <
9. (a) 3 (b) 9 (c) 27
(d) 3− (e) 9− (f) 27−
(g) 3− (h) 9 (i) 27−
(j) 1 (k) 1− (l) 1
(m) 81 (n) 81− (o) 81
11. (a) 0.25, or 1
4 (b)
1
25 (c)
1
81
13. (a) 85 (b)
45
15. (a) 76 (b)
116
17. (a) 24 (b)
54
19. (a) 187 (b)
245
21. (a) 1
1 1
55= (b)
2
1 1
255= (c)
3
1 1
1255=
23. (a) 3
1 1
82= (b)
5
1 1
322=
25. (a)
( )1
15
15= (b)
( )1
23
1 3
2=
27. (a) 2
1 1
255− = − (b)
( )2
1 1
255=
−
29. (a) 1
32− (b) 16
31. (a) 1 (b) 1
64
33. (a)
9 12
6
27x y
z (b)
6
9 1227
z
x y
35. 20x
37.
4
4
k
m
39. 4 68a b−
41. 2
43. 12 8
4
9a b−
45. (a) 6 (b) 7 (c) 9 2 3 2=
47. (a) 25 2 5 2= (b) 14 (c) 9
4
49. (a) 4 7 2 7= (b) 36 2 6 2=
(c) 9 3 3 3=
51. (a) 9 6 3 6= (b) 16 5 4 5=
(c) 4 15 2 15=
53. (a) 5
5 (b)
3
2 (c)
14
7
55. (a) 7
2 (b)
10
10 (c)
3 11
11
57. (a) 9 3 (b) 2 2 3x y z yz
59. (a) 5 (b) 36 (c) 8
61. (a) 8 (b) Not a real number (c) 8−
63. (a) 2 (b) 2− (c) 2−
65. (a) 10 (b) Not a real number (c) 10−
67. (a) 1
2 (b) Not a real number (c)
1
2−
69. (a) 1
10 (b)
1
10− (c)
1
10−
Odd-Numbered Answers to Exercise Set 1.5: Order of Operations
University of Houston Department of Mathematics 362
1. (a) P: Parentheses
E: Exponents
M: Multiplication
D: Division
A: Addition
S: Subtraction
(b) Whichever appears first
(c) Whichever appears first
3. (a) 23 (b) 35
(c) 17− (d) 5−
(e) 4 (f) 6−
5. (a) 10 (b) 4
(c) 4− (d) 10
7. (a) 4− (b) 14−
(c) 10 (d) 33
(e) 24− (f) 16
9. (a) 1960
(b) 1160
−
(c) 7
30− (d) 1
30−
11. (a) 45 (b) 49−
(c) 8 (d) 57
(e) 26− (f) 36
13. (a) 100 (b) 1 (c) 5,000
15. (a) 40 (b) 16 (c) 33
17. (a) 619
(b) 2 (c) 18
19. 1920
6
21. 281
44
23. 18
1−
25. 514
27. 7−
29. 2 3 5−
31. 25−
33. 8
15−
35. 19
37. 18−
39. 13
−
41. 57
−
43. 2
45. 115
47. 67
4
49. 4
Odd-Numbered Answers to Exercise Set 1.6: Solving Linear Equations
MATH 1300 Fundamentals of Mathematics 363
1. 7x =
3. 3x = −
5. 5x =
7. 53
x =
9. 2−=x
11. 5=x
13. 4
11822 ==x
15. 136=x
17. 35−=x
19. 6x =
21. 185
x =
23. 20=x
25. 10=x
27. 5227=x
Odd-Numbered Answers to Exercise Set 1.7:
Interval Notation and Linear Inequalities
University of Houston Department of Mathematics 364
1 2 3 4 5 6 7 8 9
1. (a) 5x >
(b)
(c) ( )5, ∞
3. (a) 3x ≤
(b)
(c) ( ], 3−∞
5. (a) 2x ≠
(b)
(c) ( ) ( ), 2 2,−∞ ∞∪
7. (a) 1x < −
(b)
(c) ( ), 1−∞ −
9. (a) 4x ≥ −
(b)
(c) [ )4,− ∞
11. (a) 8x ≠ −
(b)
(c) ( ) ( ), 8 8,−∞ − − ∞∪
13. (a) 2x ≠ and 7x ≠
(b)
(c) ( ) ( ) ( ), 2 2, 7 7,−∞ ∞∪ ∪
15. ( )∞,3
17. ( ]2, −∞−
19. ( ]5,3
21. ( ) ( ), 7 7,−∞ − − ∞∪
23. ( )3,1−
25. ( ]4,∞−
27. ( ) ( ), 0 0,−∞ ∞∪
29. { }31,3,2 −
31. { }31,3,2 −
33. { }31,2
35. (a) 5<x
(b)
(c) ( )5,∞−
37. (a) 6−≤x
(b)
(c) ( ]6, −∞−
39. (a) 3−≥x
(b)
(c) [ )∞− ,3
41. (a) 4−<x
(b)
(c) ( )4, −∞−
43. (a) 5−>x
(b)
(c) ( )∞− ,5
45. (a) 8
13≥x
(b)
(c) [ )∞,8
13
47. (a) 31≤x
(b)
(c) ( ]31,∞−
2 3 4 5 6 7 8
−9 −8 −7 −6 −5 −4 −3
−5 −4 −3 −2 −1 0 1
−7 −6 −5 −4 −3 −2 −1
813
−7 −6 −5 −4 −3 −2 −1
31
−11 −10 −9 −8 −7 −6 −5 −4
−6 −5 −4 −3 −2 −1 0
−4 −3 −2 −1 0 1 2
−1 0 1 2 3 4 5 6
0 1 2 3 4 5 6
3 4 5 6 7 8 9 10
Odd-Numbered Answers to Exercise Set 1.7:
Interval Notation and Linear Inequalities
MATH 1300 Fundamentals of Mathematics 365
−5 −4 −3 −2 −1 0 1 2 3 4
−3 −2 −1 0 1 2 3
49. (a) 172>x
(b)
(c) ( )∞,172
51. (a) 2−≥x
(b)
(c) [ )∞− ,2
53. (a) 24 <≤− x
(b)
(c) [ )2,4−
55. (a) 12 i.e.,21 ≤≤−−≥≥ xx
(b)
(c) [ ]1,2−
57. (a) 3
223
20 << x
(b)
(c) ( )3
223
20 ,
59. b (since 9 is not less than 5), and
d (since -5 is not greater than -3)
61. (a) 10023.075 ≤+ x , where x represents the # of
miles driven.
(b) 7.108≤x , so you can drive a maximum of 108
miles and still be reimbursed in full.
63. You would need to talk for more than 110 minutes in
order for Plan 1 to be more cost-effective than Plan 2.
172
2−
322
320
Odd-Numbered Answers to Exercise Set 1.8: Absolute Value and Equations
University of Houston Department of Mathematics 366
1. 7±=x
3. No solution (since the absolute value of any quantity
is always 0≥ , and therefore cannot be negative)
5. 6±=x
7. 9,1 −== xx
9. 1x = ±
11. 34,4 −== xx
13. 4x = ±
15. 9,12 == xx
17. 37
31 , == xx
19. 23
21 , −== xx
21. No solution (since the absolute value of any quantity
is always 0≥ , and therefore cannot be negative)
23. 81
23 , −== xx
Odd-Numbered Answers to Exercise Set 2.1:
An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 367
1, 3, 5:
7. (-4, 2); Quadrant II
9. (0, 3); y-axis
11. (-3, -4); Quadrant III
13. Graph:
(a) Quadrant I
(b) Quadrant III
(c) Quadrant IV
(d) Quadrant II
15. Graph:
(a) y-axis
(b) x-axis
(c) x-axis
(d) y-axis
17. (a) III (b) II (c) I
19. (a) IV (b) IV (c) III
21. (a) II (b) III (c) II
23. (a) x-axis (b) y-axis (c) x-axis
25. False
27. True
29. False
31. True
33. True
35. False
37. False
39. False
41. True
−6 −4 −2 2 4 6
−4
−2
2
4
6
x
y
C
E
A
−6 −4 −2 2 4 6
−6
−4
−2
2
4
6
x
y
CB
AD
−4 −2 2 4
−4
−2
2
4
x
y
CB
A
D
Odd-Numbered Answers to Exercise Set 2.1:
An Introduction to the Coordinate Plane
University of Houston Department of Mathematics 368
43. (a), (c): See graph below.
(b) They all have an x-value of 3
(d) 3x =
45. (a) Answers vary for part (a). One possible set of
points is ( 3, 0), (0, 0), (2, 0), (6, 0)− .
(b) All points on the x-axis have a y-value of zero.
(c) 0y =
47.
49.
51.
53.
−2 2 4
−2
2
4
x
y
−4 −2 2 4
−2
2
4
x
y
−4 −2 2 4
−2
2
4
x
y
−2 2 4
−2
2
4
x
y
−2 2 4 6
−2
2
4
x
y
C
B
A
D
Odd-Numbered Answers to Exercise Set 2.1:
An Introduction to the Coordinate Plane
MATH 1300 Fundamentals of Mathematics 369
55. 3 2y x= +
Graph:
57. 4 7y x= − +
Graph:
59.
61.
x y
-2 -4
-1 -1
0 2
1 5
2 8
x y
0 7
14
6
3 -5
5
41.25==== 2
32
− 13
−6 −4 −2 2 4 6
−4
−2
2
4
6
8
x
y
−8 −6 −4 −2 2 4 6 8 10
−4
−2
2
4
6
8
10
12
x
y
−8 −6 −4 −2 2 4
−2
2
4
6
8
x
y
−4 −2 2 4
−4
−2
2
4
x
y
Odd-Numbered Answers to Exercise Set 2.2:
The Distance and Midpoint Formulas
University of Houston Department of Mathematics 370
1. 13c =
3. 32 16 2 4 2b = = =
5. (a) – (c): See graph.
Note: Point C could also be placed at (1, 7).
(d) 222
)(53 AB=+
34=AB
7. 13
9. 535945 ==
11. 65
13. 373
15. ( )7 ,5
17. ( )112
, 4−
19. ( )3 92 2
, −
21. ( )215
, 1−
23. (a)
(b) 149
(c) ( )32
, 1
25. (6,11)
27. (a) (7, 8)−
(b) 5
29. Point B is closer to the origin, since 6158 <
31. (a) The center of the circle is ( 1, 2)− − .
(b) The length of the radius of the circle is 65 .
−2 2 4 6 8
−2
2
4
6
8
x
y
A
B
C
−4 −2 2 4 6 8
−4
−2
2
4
6
x
yA
B
Odd-Numbered Answers to Exercise Set 2.3:
Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 371
1. Positive
3. Zero
5. Negative
7. 73
9. 52
11. 0
13. 54
−
15. Undefined
17. 52
−
19. 5−
21. 7
141
23. 34
25. Undefined
27. (a) 4 1y x= − +
(b)
(c) Slope: 4−
29. (a) 23
4y x= −
(b)
(c) Slope: 23
31. Summary of slopes from numbers 27-30:
4 1y x= − + Slope: 4−
3 2y x= + Slope: 3
23
4y x= − Slope: 23
35
6y x= − + Slope: 35
−
The slope is the coefficient of the x-term. The
equation of a line is often written in the form
y mx b= + , and m represents the slope of the line.
33. (a) x-intercept: 4
(b) y-intercept: 2
(c) Coordinates of x-intercept: ( )4, 0
(d) Coordinates of y-intercept: ( )0, 2
(e) Slope: 12
−
x y
0 1
2− 9
1 3−
1
4 0
12
− 3
x y
0 4−
5 2
3−−−−
9 2
6−−−− 8−
32
3−−−−
−6 −4 −2 2 4 6 8
−4
−2
2
4
6
8
10
x
y
−6 −4 −2 2 4 6 8 10
−10
−8
−6
−4
−2
2
4
x
y
Odd-Numbered Answers to Exercise Set 2.3:
Slope and Intercepts of Lines
University of Houston Department of Mathematics 372
35. (a) x-intercept: 4
y-intercept: 8
(b) Coordinates of x-intercept: ( )4, 0
Coordinates of y-intercept: ( )0, 8
(c), (d): See graph below.
37. (a) x-intercept: 5 14 4
1=
y-intercept: 5−
(b) Coordinates of x-intercept: ( )14
1 , 0
Coordinates of y-intercept: ( )0, 5−
(c), (d): See graph below.
39. (a) x-intercept: 4
y-intercept: 10
(b) Coordinates of x-intercept: ( )4, 0
Coordinates of y-intercept: ( )0, 10
(c), (d): See graph below.
41. (a) x-intercept: 10
y-intercept: 6−
(b) Coordinates of x-intercept: ( )10, 0
Coordinates of y-intercept: ( )0, 6−
(c), (d): See graph below.
−4 −2 2 4 6 8
−2
2
4
6
8
10
x
y
−4 −2 2 4 6 8
−2
2
4
6
8
10
x
y
2 4 6 8 10
−8
−6
−4
−2
2
x
y
−4 −2 2 4 6
−8
−6
−4
−2
2
x
y
Odd-Numbered Answers to Exercise Set 2.3:
Slope and Intercepts of Lines
MATH 1300 Fundamentals of Mathematics 373
43. (a) x-intercept: 5−
y-intercept: 10 13 3
3=
(b) Coordinates of x-intercept: ( )5, 0−
Coordinates of y-intercept: ( )13
0, 3
(c), (d): See graph below.
45. (a) x-intercept: 21 15 5
4− = −
y-intercept: 7
(b) Coordinates of x-intercept: ( )15
4 , 0−
Coordinates of y-intercept: ( )0, 7
(c), (d): See graph below.
47. (a) x-intercept: 7 12 2
3=
y-intercept: 7 12 2
3=
(b) Coordinates of x-intercept: ( )12
3 , 0
Coordinates of y-intercept: ( )12
0, 3
(c), (d): See graph below.
49. (a) x-intercept: None
y-intercept: 3
(b) Coordinates of x-intercept: N/A
Coordinates of y-intercept: ( )0, 3
(c), (d): See graph below.
−6 −4 −2 2
−2
2
4
x
y
−8 −6 −4 −2 2 4
−2
2
4
6
8
x
y
−2 2 4 6
−2
2
4
x
y
−4 −2 2 4
−2
2
4
x
y
Odd-Numbered Answers to Exercise Set 2.3:
Slope and Intercepts of Lines
University of Houston Department of Mathematics 374
51. (a) x-intercept: 4−
y-intercept: None
(b) Coordinates of x-intercept: ( )4, 0−
Coordinates of y-intercept: N/A
(c), (d): See graph below.
53. (a) 2 8y x= −
(b)
(c) x-intercept: 4
y-intercept: 8−
(d) Slope: 2
55. Summary of y-intercepts from numbers 53 and 54:
2 8y x= − y-intercept: 8−
3y x= − + y-intercept: 3
The y-intercept is the constant term. The equation of
a line is often written in the form y mx b= + , and b
represents the y-intercept of the line.
x y
0 8−−−−
4 0
2 4−−−−
7 6
0.5− 9−−−−
−6 −4 −2 2
−4
−2
2
4
x
y
−6 −4 −2 2 4 6 8 10 12
−10
−8
−6
−4
−2
2
4
6
x
y
Odd-Numbered Answers to Exercise Set 2.4:
Equations of Lines
MATH 1300 Fundamentals of Mathematics 375
1. 43
2y x= −
3. 3−−= xy
5. (a) 52 +−= xy
(b) Slope: 2− ; y-intercept: 5
(c)
7. (a) 5 1y x= −
(b) Slope: 5; y-intercept: 1−
(c)
9. (a)4xy = −
(b) Slope: 14
− ; y-intercept: 0
(c)
11. (a) 54
3y x= −
(b) Slope: 54
; y-intercept: 3−
(c)
13. (a) 25
6y x= − +
(b) Slope: 25
− ; y-intercept: 6
(c)
15. (a) 52
2y x= −
(b) Slope: 52
; y-intercept: 2−
(c)
−2 2 4 6
−2
2
4
6
x
y
−6 −4 −2 2 4 6
−6
−4
−2
2
4
x
y
−4 −2 2 4 6
−6
−4
−2
2
4
x
y
−4 −2 2 4 6
−8
−6
−4
−2
2
x
y
−2 2 4 6 8 10 12 14 16
−6
−4
−2
2
4
6
8
x
y
−6 −4 −2 2 4 6
−8
−6
−4
−2
2
4
x
y
Odd-Numbered Answers to Exercise Set 2.4:
Equations of Lines
University of Houston Department of Mathematics 376
17. (a)
(b) ( ) ( )23
4 6y x− = +
(c) 23
8y x= +
19. (a)
(b) ( ) ( )34
2 8y x− = − + or ( ) ( )34
7 4y x+ = − −
(c) 34
4y x= − −
21. 47
3y x= − +
23. 45
7y x= −
25. 2 49 3
y x= − +
27. 35
4y x= +
29. 7 35 5
y x= − −
31. 57
5y x= −
33. 32
6y x= − +
35. 6200800 += xC
−12 −10 −8 −6 −4 −2 2
−4
−2
2
4
6
8
10
x
y
−10 −8 −6 −4 −2 2 4 6
−10
−8
−6
−4
−2
2
4
x
y
Odd-Numbered Answers to Exercise Set 2.5:
Parallel and Perpendicular Lines
MATH 1300 Fundamentals of Mathematics 377
1. Parallel
3. Perpendicular
5. Neither
7. Perpendicular
9. Parallel
11. Neither
13. Perpendicular
15. Parallel
17. Perpendicular
19. Parallel
21. (a) ( )7 2 4y x− = −
(b) 2 1y x= −
23. (a) ( )16
5 12y x− = +
(b) 16
7y x= +
25. (a) ( )54
7 3y x+ = − −
(b) 5 134 4
y x= − −
27. (a) ( )32
6 1y x− = − +
(b) 3 92 2
y x= − +
29. 4=y
31. 2x =
33. (a) 132
3 3y x= +
(b) 32
y x= −
35. 52
2y x= −
37. 192
5 5y x= − +
39. 186 +−= xy
41. 5 43 3
y x= −
Odd-Numbered Answers to Exercise Set 2.6:
An Introduction to Functions
University of Houston Department of Mathematics 378
1. This mapping does not make sense, since Erik could
not record two different temperatures at 9AM. The
mapping does not represent a function.
3. Yes, the mapping represents a function.
5. No, the mapping does not represent a function.
7. (a) ( ) 47
xf x = +
(b) 4
( )7
xf x
+=
9. (a) 2( ) 6 36f x x x= − = −
(b) ( )2
( ) 6f x x= −
11.
13. 0x ≠
Interval notation: ( ) ( ), 0 0,−∞ ∞∪
15. 3x ≠
Interval notation: ( ) ( ), 3 3,−∞ ∞∪
17. 4x ≠ −
Interval notation: ( ) ( ), 4 4,−∞ − − ∞∪
19. 52
t ≠ −
Interval notation: ( ) ( )5 52 2
, ,−∞ − − ∞∪
21. 94
x ≠
Interval notation: ( ) ( )9 94 4
, ,−∞ ∞∪
23. 3x ≠ − and 3x ≠
Interval notation: ( ) ( ) ( ), 3 3, 3 3,−∞ − − ∞∪ ∪
25. All real numbers
Interval Notation: ( ),−∞ ∞
27. All real numbers
Interval Notation: ( ),−∞ ∞
29. 0≥t
Interval Notation: [ )∞,0
31. 5≥x
Interval Notation: [ )∞,5
33. All real numbers
Interval Notation: ( ),−∞ ∞
35. 92
x ≥ −
Interval Notation: )92
,− ∞
37. 15
x ≤
Interval Notation: ( 15
, −∞
39. 52
x ≤ −
Interval Notation: ( 52
, −∞ −
41. 2x ≥ and 6x ≠
Interval notation: [ ) ( )2, 6 6, ∞∪
43. All real numbers
Interval Notation: ( ),−∞ ∞
45. 5t ≠ −
Interval notation: ( ) ( ), 5 5,−∞ − − ∞∪
47. All real numbers
Interval Notation: ( ),−∞ ∞
49. 53
x ≥
Interval Notation: )53
, ∞
51. All real numbers
Interval Notation: ( ),−∞ ∞
53. All real numbers
Interval Notation: ( ),−∞ ∞
x 3( ) 5f x x= −= −= −= −
2− 13−−−−
1− 6−−−−
0 5−−−−
1 4−−−−
2 3
Odd-Numbered Answers to Exercise Set 2.6:
An Introduction to Functions
MATH 1300 Fundamentals of Mathematics 379
55. 7x ≠
Interval notation: ( ) ( ), 7 7,−∞ ∞∪
57. 4x ≠ −
Interval notation: ( ) ( ), 4 4,−∞ − − ∞∪
59. (a) (3) 11f =
(b) 75
x =
(c) ( ) 1312 2
f − = −
(d) 7
10x =
(e) ( )0 4f = −
(f) 45
x =
61. (a) (1) 2h =
(b) 4, 2x x= =
(c) ( )2 5h − =
(d) No such value of x exists (since 3x − cannot
be negative).
(e) ( )7 4h =
(f) 10, 4x x= = −
63. (a) (7) 3h =
(b) (25) 27 9 3 3 3h = = =
(c) ( ) 39124 4
h = =
65. (a) (16) 1f =
(b) (12) 12 3 4 3 3 2 3 3f = − = − = −
(c) ( )9 0f =
67. (a) (3) 0g =
(b) ( )4 42g − =
(c) ( ) 3512 4
g − =
(d) ( )0 6g =
69. (a) 12
( 7)f − =
(b) 23
(0)f = −
(c) ( ) 72
5f =
(d) ( )3f is undefined.
(e) ( )2 0f − =
Odd-Numbered Answers to Exercise Set 2.7:
Functions and Graphs
University of Houston Department of Mathematics 380
1. No, the graph does not represent a function.
3. Yes, the graph represents a function.
5. Yes, the graph represents a function.
7. No, the graph does not represent a function.
9. Yes, the graph represents a function.
11. (a)
(b) No, the set of points does not represent a
function. The graph does not pass the vertical
line test at 2x = .
13. (a)
(b) Yes, the set of points represents a function. The
graph passes the vertical line test.
15. If each x value is paired with only one y value, then
the set of points represents a function. If an x value is
paired with more than one y value (i.e. two or more
coordinates have the same x value but different y
values), then the set of points does not represent a
function.
17. (a) Domain: [ ]4, 6−
(b) Range: [ ]3, 9−
(c) ( 2) 3f − =
(0) 3f = −
(4) 9f =
(6) 3f =
(d) 4x = − , 4x =
19. (a) Domain: ( ), 6−∞
(b) Range: ( ], 5−∞
(c) ( 2) 1g − =
(0) 5g =
(2) 1g = −
(4) 1g =
(6)g is undefined
(d) ( 2)g − is greater than (3)g , since 1 0> .
21. (a) Domain: ( ),−∞ ∞
(b)
x 3
2( ) 6f x x= − += − += − += − +
2− 9
1− 15
27.5====
0 6
1 9
24.5====
2 3
−4 −2 2 4
−2
2
4
6
x
y
−2 2 4 6
−4
−2
2
4
x
y
−6 −4 −2 2 4 6
−2
2
4
6
8
10
x
y
Odd-Numbered Answers to Exercise Set 2.7:
Functions and Graphs
MATH 1300 Fundamentals of Mathematics 381
23. (a) Domain: [ )1, 3−
(b)
25. (a) Domain: ( ),−∞ ∞
(b)
27. (a) Domain: [ )3, ∞
(b)
29. (a) Domain: ( ),−∞ ∞
(b)
x ( ) 3 5h x x= −= −= −= −
1− 8−−−−
0 5−−−−
1 2−−−−
2 1
3 4 (open circle)
x ( ) 3g x x= −= −= −= −
1 2
2 1
3 0
4 1
5 2
x ( ) 3f x x= −= −= −= −
3 0
4 1
5 2 1.4≈≈≈≈
7 2
12 3
x 2( ) 4F x x x= −= −= −= −
1− 5
0 0
1 3−−−−
2 4−−−−
3 3−−−−
−6 −4 −2 2 4 6 8
−10
−8
−6
−4
−2
2
4
x
y
−2 2 4 6 8 10 12
−6
−4
−2
2
4
6
8
x
y
−4 −2 2 4 6 8
−6
−4
−2
2
4
6
x
y
−2 2 4 6 8
−2
2
4
x
y
Odd-Numbered Answers to Exercise Set 2.7:
Functions and Graphs
University of Houston Department of Mathematics 382
31. (a) 5 8 5 8
3 3 3xy x+= = +
(b) Yes, the equation defines y as a function of x.
33. (a) 2 23 7 3 72 2 2
xy x− += = − +
(b) Yes, the equation defines y as a function of x.
35. (a) 3y x= ± +
(b) No, the equation does not define y as a function
of x.
37. (a) ( )2y x= ± +
(b) No, the equation does not define y as a function
of x.
39. (a) 5 7 5 7
2 2 2
xy x
+= = +
(b) Yes, the equation defines y as a function of x.
Odd-Numbered Answers to Exercise Set 3.1:
An Introduction to Polynomial Functions
MATH 1300 Fundamentals of Mathematics 383
1. (a) Yes
(b) Degree: 3
(c) Binomial
3. (a) Yes
(b) Degree: 1
(c) Binomial
5. (a) No
(b) N/A
(c) N/A
7. (a) No
(b) N/A
(c) N/A
9. (a) No
(b) N/A
(c) N/A
11. (a) No
(b) N/A
(c) N/A
13. (a) Yes
(b) Degree: 6
(c) Monomial
15. (a) No
(b) N/A
(c) N/A
17. (a) Yes
(b) Degree: 0
(c) Monomial
19. (a) Yes
(b) Degree: 5
(c) None of these
21. (a) Yes
(b) Degree: 7
(c) Binomial
23. (a) No
(b) N/A
(c) N/A
25. (a) Yes
(b) Degree: 9
(c) Trinomial
27. (a) False
(b) True
(c) True
(d) False
29. (a) True
(b) True
(c) False
(d) False
31. (a) True
(b) False
(c) False
(d) True
33. x-intercepts: 1, 2, 4−
y-intercept: 8
35. x-intercepts: 1, 0, 3−
y-intercept: 0
37. (a) Quadratic
(b) x-intercepts: 8, 8−
y-intercept: 64−
(c) ( 4) 48f − = −
( 1) 63f − = −
(6) 28f = −
39. (a) Cubic
(b) x-intercept: 2
y-intercept: 32
(c) ( 4) 288f − =
( 1) 36f − =
(6) 832f = −
41. (a) Linear
(b) x-intercept: 125
y-intercept: 12
(c) ( 4) 32f − =
( 1) 17f − =
(6) 18f = −
43. (a) Quadratic
(b) y-intercept: 28−
(c) ( 4) 0f − =
( 1) 24f − = −
(6) 10f = −
Odd-Numbered Answers to Exercise Set 3.2:
Adding, Subtracting, and Multiplying Polynomials
University of Houston Department of Mathematics 384
1. 2 3 40x x+ −
3. 2 4 3x x− +
5. 2 15 36x x+ +
7. 2 16 80x x− −
9. 2 16x −
11. 2 14 49x x− +
13. 22 5 3x x+ −
15. 220 39 7x x+ +
17. 23 14 8x x− +
19. 4 22 35x x+ −
21. 7 6 57 37 10x x x− + −
23. (a) 3x−
(b) 7x−
(c) 210x−
25. (a) 4 35 4x x− −
(b) 4 35 28x x− −
(c) 1060x−
27. (a) 5 327 6x x−
(b) 3 29 3 2x x x− −
(c) 315x−
29. (a) 4 370 35x x− −
(b) 23 5x x+
(c) 5350x−
31. (a) 2 10x +
(b) 4−
(c) 2 10 21x x+ +
33. (a) 2 7x x+ −
(b) 2 7x x+ +
(c) 3 22 21x x x− −
35. (a) 2 10 4x x− + −
(b) 2 20x− −
(c) 3 25 25 20x x x− +
37. (a) 2 2 9x x− −
(b) 2 6 15x x− + +
(c) 3 22 5 36 36x x x− − −
39. (a) 3 24 13x x x− −
(b) 4 3 216 6 40x x x− −
(c) 5 4 3 22 16 40 10x x x x x+ − − −
41. (a) 25 3 2x x− +
(b) 2 7 4x x+ −
(c) 4 3 26 11 3 11 3x x x x− − + −
43. (a) 5 4 32 7 4x x x x− + −
(b) 5 4 32 3 2x x x x− − + −
(c) 9 8 7 6 5 4 22 5 4 7 2 11 3x x x x x x x− + − + + − +
45. (a) 3 22 4 2 9x x x− − + +
(b) 3 24 4 2 1x x x− + + −
(c) 6 5 4 3 23 12 2 19 16 10 20x x x x x x− + + − − + +
47. (a) 2 15 48x x+ +
(b) 2 13 50x x− − −
(c) 3 213 35 49x x x+ + −
Odd-Numbered Answers to Exercise Set 3.3
Dividing Polynomials
MATH 1300 Fundamentals of Mathematics 385
1. Quotient: 4x − ;
Remainder: 3
3. Quotient: 6x + ;
Remainder: 8−
5. Quotient: 452
−− xx ;
Remainder: 0
7. Quotient: 5432
++ xx ;
Remainder: 7−
9. Quotient: 72 +x ;
Remainder: 145 +x
11. Quotient: 1823
21 −+ xx ;
Remainder: 8−
13. Quotient: 1532
−− xx ;
Remainder: 605 +x
15. Quotient: 2+x ;
Remainder: 24
17. Quotient: 4232
+− xx ;
Remainder: 8
19. Quotient: 4423
−+− xxx ;
Remainder: 0
21. Quotient: 1774323
−−+ xxx ;
Remainder: 75−
23. Quotient: 422
+− xx ;
Remainder: 0
25. Quotient: 6242
−+ xx ;
Remainder: 2
27. (a) Using substitution, 4)2( =P
(b) The remainder is 4 , so 4)2( =P .
29. (a) 7)1( −=−P
(b) The remainder is 7− , so 7)1( −=−P .
31. 97)5( −=P
33. ( ) 1043 −=−P
35. ( )( )2 11 24 8 3x x x x− + = − −
37. ( )( )2 7 18 2 9x x x x− − = + −
39. ( )( )24 25 21 7 4 3x x x x− − = − +
41. ( )( )22 7 5 1 2 5x x x x+ + = + +
Odd-Numbered Answers to Exercise Set 3.4
Quadratic Functions
University of Houston Department of Mathematics 386
1. (a) Vertex: ( )3, 2− −
(b) The parabola opens upward.
(c) y-intercept: 7
(d) Axis of symmetry: 3x = −
(e)
3. (a) Vertex: ( )1, 1−
(b) The parabola opens upward.
(c) y-intercept: 0
(d) Axis of symmetry: 1x =
(e)
5. (a) Vertex: ( )2, 3
(b) The parabola opens upward.
(c) y-intercept: 11
(d) Axis of symmetry: 2x =
(e)
7. (a) Vertex: ( )7, 0
(b) The parabola opens upward.
(c) y-intercept: 49
(d) Axis of symmetry: 7x =
(e)
−10 −8 −6 −4 −2 2 4
−4
−2
2
4
6
8
10
x
y
−2 2 4
−2
2
4
x
y
−6 −4 −2 2 4 6 8 10
−2
2
4
6
8
10
12
x
y
−7 7 14 21
−14
−7
7
14
21
28
35
42
49
56
63
x
y
Odd-Numbered Answers to Exercise Set 3.4
Quadratic Functions
MATH 1300 Fundamentals of Mathematics 387
9. (a) Vertex: ( )4, 7−
(b) The parabola opens downward.
(c) y-intercept: 9−
(d) Axis of symmetry: 4x = −
(e)
11. (a) Vertex: ( )0, 5−
(b) The parabola opens downward.
(c) y-intercept: 5−
(d) Axis of symmetry: 0x =
(e)
13. (a) Vertex: ( )5,15
(b) The parabola opens upward.
(c) y-intercept: 115
(d) Axis of symmetry: 5x =
(e)
15. (a) Vertex: ( )2, 6− −
(b) The parabola opens downward.
(c) y-intercept: 14−
(d) Axis of symmetry: 2x = −
(e)
−14 −12 −10 −8 −6 −4 −2 2 4 6
−10
−8
−6
−4
−2
2
4
6
8
x
y
−8 −6 −4 −2 2 4 6 8
−20
−15
−10
−5
5
x
y
−4 −2 2 4 6 8 10 12
−20
20
40
60
80
100
120
140
x
y
−8 −6 −4 −2 2 4 6
−24
−20
−16
−12
−8
−4
4
x
y
Odd-Numbered Answers to Exercise Set 3.4
Quadratic Functions
University of Houston Department of Mathematics 388
17. (a) Vertex: ( )5 132 4
, −
(b) The parabola opens upward.
(c) y-intercept: 3
(d) Axis of symmetry: 52
x =
(e)
19. (a) Vertex: ( )3 418 16
,−
(b) The parabola opens downward.
(c) y-intercept: 2
(d) Axis of symmetry: 38
x = −
(e)
21. (a) 2( ) 2 15f x x x= + −
(b) ( )1, 16− −
(c) Graph III
23. (a) 2( ) 2 12 10f x x x= − + −
(b) ( )3, 8
(c) Graph II
−6 −4 −2 2 4 6 8 10
−8
−6
−4
−2
2
4
6
x
y
−2 2
−2
2
4
x
y
Odd-Numbered Answers to Exercise Set 4.1:
Greatest Common Factor and Factoring by Grouping
MATH 1300 Fundamentals of Mathematics 389
1. 26xy
3. 4 4a b
5. 3 54a c
7. 3 5 4x y z
9. ( )5 2a +
11. ( )3 5b− −
13. ( )3 3 8x y−
15. ( )2 3 4x y−
17. ( )22 3 1ab a b +
19. ( )5 3 4rt r t−
21. ( )22 2 4x x x+ −
23. ( )3 2 3 55 3 7x y xy x y− − +
25. ( )2 4 5 8 5 57 5 4 3a b c a c b ab c− +
27. ( )5 3 2 4 2 3 310 21 49c a b a c b c− − +
29. (a) ( )5y x −
(b) ( )( )4 5x x− −
31. (a) ( )3b a+
(b) ( ) ( )5 3c a+ +
33. ( )( )5 3 4a a b+ +
35. ( )( )8 2 1x x+ +
37. ( ) ( )5 2 1x x+ −
39. ( ) ( )2 3 11a a− −
41. ( )( )2b c a+ +
43. ( )( )5y z x+ −
45. ( )( )3x x y− +
47. ( )( )c d a b− −
49. ( ) ( )4 1y x− +
51. ( )( )1y x y+ −
53. ( )( )3 2 4b a+ +
55. ( ) ( )3 2 1t x t− +
57. ( )( )4 3 2c d a b− −
59. ( )( )d e f a b− − +
61. ( )( )4 5 3 2x z x y+ − −
63. ( )( )3 2x x− +
65. ( )( )4 3x x− −
67. ( )( )3 5 2 3x x+ +
69. ( )( )3 7 3 2x x+ −
71. ( ) ( )2 7 2 7x x+ + , or ( )2
2 7x +
Odd-Numbered Answers to Exercise Set 4.2:
Factoring Special Binomials and Trinomials
University of Houston Department of Mathematics 390
1. (a) 2 16x −
(b) 2 8 16x x+ +
(c) 2 8 16x x− +
3. False
5. True
7. True
9. False
11. (a) ( )( )3 3x x+ −
(b) 2 9x +
(c) ( )2
3x +
(d) ( )2
3x −
13. ( )( )7 7x x+ −
15. 2 144x +
17. ( ) ( )1 1p p+ −
19. ( )( )10 10x x+ −
21. ( )( )5 5c c+ −
23. ( ) ( )2 3 2 3b b+ −
25. ( )( )4 1 4 1x x+ −
27. ( )( )7 10 7 10x y x y+ −
29. 2 225 16c d+
31. 2 23 3
x x + −
33. x a x a
y b y b
+ −
35. 4 4
5 3 5 3
xy xy + −
37. ( )2
10x −
39. ( )2
1x +
41. ( )2
9x +
43. ( )2
2 3x −
45. ( )2
5 4x +
47. ( )2
x b−
49. ( )2
2 5bc d−
51. ( )( )3 28 2 2 4x x x x− = − + +
53. ( ) ( )24 4 16x x x+ − +
55. ( )( )23 3 9p p p− + +
57. ( )( )2 2x y x xy y− + +
59. ( )( )2 25 2 25 10 4a b a ab b− + +
Odd-Numbered Answers to Exercise Set 4.3:
Factoring Polynomials
MATH 1300 Fundamentals of Mathematics 391
1. (a) 13
(b) No
3. (a) 100
(b) Yes
5. (a) 36
(b) Yes
7. (a) 81
(b) Yes
9. (a) 36−
(b) No
11. ( )( )5 1x x+ −
13. ( )( )3 2x x− −
15. 2 7 12x x− −
17. ( )( )10 2x x+ +
19. ( )( )3 8x x+ −
21. ( )2
8x +
23. ( )( )7 8x x− −
25. ( )( )4 15x x+ −
27. ( ) ( )3 14x x+ +
29. ( )( )7 7x x+ −
31. 2 3x −
33. 29 25x +
35. ( )( )2 1 3x x+ −
37. ( )( )2 1 4 3x x+ −
39. ( )( )3 4 3 1x x+ −
41. ( )( )4 5 2x x+ −
43. ( )( )4 3 3 2x x− −
45. ( )9x x +
47. ( )5 4x x− −
49. ( )( )2 3 3x x+ −
51. ( )( )25 2 2x x x− + −
53. ( )( )2 4 1x x+ +
55. ( )( )10 7 6x x− − +
57. ( )( )11 2x x x+ −
59. ( )2
2x x− +
61. ( )2 2 6 6x x x+ +
63. ( )( )3 3 10 3 10x x x+ −
65. ( )( )5 2 1 5 3x x+ +
67. ( )( )( )2 5 5x x x+ + −
69. ( )( )25 4x x− +
71. ( )( )( )9 2 1 2 1x x x+ + −
Odd-Numbered Answers to Exercise Set 4.4:
Using Factoring to Solve Equations
University of Houston Department of Mathematics 392
1. 3,7 == xx
3. 6,2 −=−= xx
5. 7,5 =−= xx
7. 4,18 =−= xx
9. 32
, 5x x= − =
11. 3 42 3
,x x= − = −
13. 23
1,x x= =
15. 5,5 =−= xx
17. 3 32 2
,x x= − =
19. 8,0 == xx
21. 4, 9x x= = −
23. 7,0 == xx
25. 2,2 −== xx
27. 3,6 =−= xx
29. 18
, 3x x= = −
31. 0, 3, 2x x x= = − = −
33. (a) x-intercepts: 4, 2
(b) ( )4, 0 , ( )2, 0
(c) y-intercept: 8
Coordinates of y-intercept: ( )0, 8
(d) Vertex: ( )3, 1−
(e) The parabola opens upward.
(f) Axis of symmetry: 3x =
(g)
35. (a) x-intercept: 4
(b) ( )4, 0
(c) y-intercept: 16
Coordinates of y-intercept: ( )0, 16
(d) Vertex: ( )4, 0
(e) The parabola opens upward.
(f) Axis of symmetry: 4x =
(g)
37. (a) x-intercepts: 7, 3−
(b) ( )7, 0− , ( )3, 0
(c) y-intercept: 21
Coordinates of y-intercept: ( )0, 21
(d) Vertex: ( )2, 25−
(e) The parabola opens downward.
(f) Axis of symmetry: 2x = −
(g)
−4 −2 2 4 6 8 10
−2
2
4
6
8
10
x
y
−6 −4 −2 2 4 6 8 10 12 14
−2
2
4
6
8
10
12
14
16
x
y
−12 −10 −8 −6 −4 −2 2 4 6 8 10
−10
−5
5
10
15
20
25
30
x
y
Odd-Numbered Answers to Exercise Set 4.4:
Using Factoring to Solve Equations
MATH 1300 Fundamentals of Mathematics 393
39. (a) x-intercepts: 6, 2−
(b) ( )6, 0− , ( )2, 0
(c) y-intercept: 36−
Coordinates of y-intercept: ( )0, 36−
(d) Vertex: ( )2, 48− −
(e) The parabola opens upward.
(f) Axis of symmetry: 2x = −
(g)
41. (a) x-intercepts: 4, 4−
(b) ( )4, 0− , ( )4, 0
(c) y-intercept: 16−
Coordinates of y-intercept: ( )0, 16−
(d) Vertex: ( )0, 16−
(e) The parabola opens upward.
(f) Axis of symmetry: 0x =
(g)
43. (a) x-intercepts: 3 32 2
, −
(b) ( )32
, 0− , ( )32
, 0
(c) y-intercept: 9
Coordinates of y-intercept: ( )0, 9
(d) Vertex: ( )0, 9
(e) The parabola opens downward.
(f) Axis of symmetry: 0x =
(g)
45. 0, 2, 5− −
47. 0, 5, 5−
49. 2, 3, 3−
Note: In the following problems, there are other correct
ways of modeling the situation in part (a). The final
answer in part (b), however, is unique, regardless of the
method used for solving the problem.
51. (a) Let the width of the rectanglew =
5 the length of the rectanglew + =
Equation: 36)5( =+ww
(b) 4=w (Note that 9w = − does not make sense as
the width of the rectangle.)
Answer: The length of the rectangle is 9 cm,
and the width of the rectangle is 4
cm.
−12 −10 −8 −6 −4 −2 2 4 6 8 10
−60
−54
−48
−42
−36
−30
−24
−18
−12
−6
6
12
18
x
y
−12 −10 −8 −6 −4 −2 2 4 6 8 10 12
−18
−16
−14
−12
−10
−8
−6
−4
−2
2
x
y
−8 −6 −4 −2 2 4 6 8
−4
−2
2
4
6
8
10
x
y
Odd-Numbered Answers to Exercise Set 4.4:
Using Factoring to Solve Equations
University of Houston Department of Mathematics 394
53. (a) Let the base of the trianglex =
3 the height of the trianglex − =
))(( trianglea of Area21 heightbase=
Equation: 90)3)((21 =−xx
(b) 15=x (Note that 12x = − does not make sense
as the base of the triangle.)
Answer: The base of the triangle is 15 cm, and
the height of the triangle is 12 cm.
Odd-Numbered Answers to Exercise Set 5.1:
Simplifying Rational Expressions
MATH 1300 Fundamentals of Mathematics 395
1. 3
5
3. 3
4−
5.
2
3
5
4
y
x
7. ( )
4
22
x y
x
+−
9. 1−
11. ( )
, or 3 3
c d d c− −−
13. 4
15. 1
5x −
17.
2 2a b
a b
+
+
19. 7 7 7
, or , or 2 2 2
c c c
c c c
+ − − +−
− − −
21. 5
7
x
x
−
+
23.
2
2
5 6
12
x x
x x
+ +
+ −
25. 2
7
x
x
−
−
27. 6
6
x
x
−
+
29. 9
x
31. 2( 3)
2
x
x
−
+
33. ( 6)
8
x
x
+
35. 3( 1)
2( 5)
x
x
+
−
37. 4 5
5 2
x
x
+
−
39. 3 4
5 2
x
x
−
+
41. 2 7
4 1
x
x
+
−
43. 1m
m n
+
+
45. 2
5
x
z
+
−
47. 2 2 4x x+ +
49. 3x +
Odd-Numbered Answers to Exercise Set 5.2:
Multiplying and Dividing Rational Expressions
University of Houston Department of Mathematics 396
1. 2
3
3. 4
5.
2
5 5 5
c
a b d
7.
3 3p t
n−
9. 3
1
x−
11. 5
10
x
x
+
−
13. 5x −
15. 5x
17. ( )3 2 , or 3 2x x− + − −
19. 3
5
21. ( )2 3 4
3
x −
23. 4
25. 2
5x−
27. 2
4
x
x
−
+
29. ( )
( )
22
3 4
x
x
−
−
31. ( )( )
2
4 3
16
x x
x
+ +−
+
33. 4
1
x
x
+
−
35. 3
x y
x
+
−
37. 4
3
39. 4
5
41. 1
9−
43.
3
3 10
y
x z
45.
6
2 7
b
c d
47. 3
5
x
x
+
−
49. 1
1x −
51. ( )3 1x +
53. ( )
1
2 4 x+
55. ( )( )
( )( )
29 1
1 3
x x
x x
+ −
+ +
57. 5
7
x
x
−
−
59. 1
61. a b
a b
+
−
Odd-Numbered Answers to Exercise Set 5.3:
Adding and Subtracting Rational Expressions
MATH 1300 Fundamentals of Mathematics 397
1. 14 15
35
x y+
3. 27 8
36
b a
ab
−
5.
3
2 5
7 2y x
x y
+
7. 2 15
5
x
x
+
+
9. 1
5
11. ( )( )
5 13
1 5
x
x x
−
− −
13. ( )
11 3
1
x
x x
+
+
15. ( )( )
2
1 2
x
x x
− +
+ +
17. ( )
( ) ( )
4 12
3 7
x
x x
+
− +
19. 1 1
, or 3 3
x x
x x
− −
− −
21. 3 13
5
x
x
+
+
23. 2 3
2
x
x
+
−
25. ( )( )
( )( )
5 2
1 3
x x
x x
+ −
+ −
27. ( )( )
( )
4 1
2
x x
x x
− +
+
29. ( )( )
22
1 2 1
x
x x+ +
31. ( )
( )( )
22 2 7
2 4
x x
x x
− +
+ −
33. ( )( )
7
4 3x x
−
+ −
35. ( )( )
41 9
12 2 3 1
x
x x
+
+ −
37. ( )( )
219 6
1 2
x x
x x x
− −
− +
39. ( )( )
210 37
4 2
x x
x x
− −
− +
41. ( ) ( )
( )( )
6 3
4 2
x x
x x x
+ −
+ −
43. ( )( )
2
4 6
x
x x
+
+ +
Odd-Numbered Answers to Exercise Set 5.4:
Complex Fractions
University of Houston Department of Mathematics 398
1. 14
15
3.
2
3
20y
x
5.
3 3
2 4
5
4
b d
a c
7.
6
2
xy z
9. 5
54−
11. 5
7
13. 3
2−
15. ( )2 2
3
x
x
−
+
17. ( )2
ab a b
b a
+
−
19. ( )
( )
33 3
4 5
x x
x
+
−
21. 5−
23. ( )2 1x −
25. 2 3
5 4
b a
b a
+
−
27. ( )
( )( )
2
1 3
x x
x x
+
− +
29. 25
2
x−
31. ( )( )
( )( )
4 5 1
4 3
x x
x x
+ +
− −
33. 5
4
x
x
+
−
35. (a)
1
11
x
x+
(b) 1
1 x+
37. (a)
1 1
1 1
x y
x y
+
−
(b) y x
y x
+
−
39. (a)
2 2
1 1
1 1
x y
x y
−
+
(b) y x
xy
−
41. (a)
3 3
1 1
1 1c d
c d
−
−
(b)
2 2
2 2
c d
d cd c+ +
43. (a) 3 3
2 2
1 1
1 1a b
a b
+
+
(b)
( )
( )( )
( )
2 23 3
2 2 2 2
b a b ab bb a
ab b a ab b a
+ + ++=
+ +
Odd-Numbered Answers to Exercise Set 5.4:
Complex Fractions
MATH 1300 Fundamentals of Mathematics 399
45. (a) 1
11
1x
+
+
(b) 2 1
1
x
x
+
+
47. (a) 5
41
5x
−
+
(b) 15 4
5 1
x
x
+
+
Odd-Numbered Answers to Exercise Set 5.5:
Solving Rational Equations
University of Houston Department of Mathematics 400
1. 30x =
3. 20c = −
5. 3017
x = −
7. 103
x = −
9. No solution.
( 1x = − is an extraneous solution.)
11. 4360
x = −
13. 54
x =
15. No solution.
17. 9x = −
19. 4, 4x x= − =
21. 1, 7x x= =
23. 4t =
25. 8x =
27. 352
x = −
29. 2w =
31. 1x = −
33. 25x =
35. 5a =
37. 114
x =
39. No solution.
( 1x = − is an extraneous solution.)
41. 22x = −
43. 2, 3x x= − =
45. 12
4,x x= − =
47. 1, 4x x= − =
49. 2, 2x x= − =
51. 12, 2x x= − = −
53. 192
0,x x= = −
55. 43
5,x x= − =
57. 29
x = −
( 1x = is an extraneous solution.)
Odd-Numbered Answers to Exercise Set 5.6:
Rational Functions
MATH 1300 Fundamentals of Mathematics 401
1. (a) (0) 0f =
(b) 14
( 1)f − =
(c) ( ) 1183
f = −
3. (a) 27
(0)f =
(b) 1110
( 3)f − =
(c) ( ) 24315
f = −
5. (a) ( 2)f − is undefined.
(b) 13
(0)f = −
(c) 17
(5)f =
7. (a) 3
112( 3)f − = −
(b) (0) 0f =
(c) 1223
(12)f = −
9. (a) (3) 0f =
(b) ( 4)f − is undefined.
(c) 3
28(0)f = −
11. (a) Domain: 3x ≠
(b) Vertical asymptote at 3x =
(c) x-intercept: 4
y-intercept: 43
(d) 32
(1)f = ; 54
( 1)f − =
(e) Graph IV
13. (a) Domain: 3x ≠
(b) Vertical asymptote at 3x =
(c) x-intercept: 6−
y-intercept: 2−
(d) 72
(1)f = − ; 54
( 1)f − = −
(e) Graph II
15. (a) Domain: 2x ≠
(b) Vertical asymptote at 2x =
(c) x-intercept: None
y-intercept: 2
(d) (1) 4f = ; 43
( 1)f − =
(e) Graph III
17. (a) Domain: 0x ≠
(b) Vertical asymptote at 0x =
(c) x-intercept: None
y-intercept: None
(d) (1) 4f = ; ( 1) 4f − = −
(e) Graph II
19. (a) Domain: 5x ≠ −
( ) ( ), 5 5,−∞ − − ∞∪
(b) Vertical asymptote: 5x = −
(c) x-intercept: None
y-intercept: 2
21. (a) Domain: 2x ≠ −
( ) ( ), 2 2,−∞ − − ∞∪
(b) Vertical asymptote: 2x = −
(c) x-intercept: 6
y-intercept: 3−
23. (a) Domain: 0x ≠
( ) ( ), 0 0,−∞ ∞∪
(b) Vertical asymptote: 0x =
(c) x-intercept: 3−
y-intercept: None
25. (a) Domain: 3, 3x x≠ − ≠
( ) ( ) ( ), 3 3, 3 3,−∞ − − ∞∪ ∪
(b) Vertical asymptotes: 3, 3x x= − =
(c) x-intercept: 9
y-intercept: 1−
Odd-Numbered Answers to Exercise Set 5.6:
Rational Functions
University of Houston Department of Mathematics 402
27. (a) Domain: 2, 6x x≠ ≠
( ) ( ) ( ), 2 2, 6 6,−∞ ∞∪ ∪
(b) Vertical asymptotes: 2, 6x x= =
(c) x-intercept: None
y-intercept: 2−
29. (a) Domain: 1x ≠
( ) ( ), 1 1,−∞ ∞∪
(b) Vertical asymptote: 1x =
(c) x-intercept: 5−
y-intercept: 5
31. (a) Domain: 0, 8x x≠ ≠ −
( ) ( ) ( ), 8 8, 0 0,−∞ − − ∞∪ ∪
(b) Vertical asymptotes: 0, 8x x= = −
(c) x-intercept: None
y-intercept: None
33. (a) Domain: 5x ≠
( ) ( ), 5 5,−∞ ∞∪
(b) Vertical asymptote: 5x =
(c) x-intercept: 5−
y-intercept: 5
35. (a) Domain: 5, 5x x≠ − ≠
( ) ( ) ( ), 5 5, 5 5,−∞ − − ∞∪ ∪
(b) Vertical asymptotes: 5, 5x x= − =
(c) x-intercept: 0
y-intercept: 0
37. (a) Domain: 75
x ≠ −
( ) ( )7 75 5
, ,−∞ − − ∞∪
(b) Vertical asymptote: 75
x = −
(c) x-intercepts: 7, 2−
y-intercept: 2−
39. (a) Domain: 1, 4x x≠ ≠
( ) ( ) ( ), 1 1, 4 4,−∞ ∞∪ ∪
(b) Vertical asymptotes: 1, 4x x= =
(c) x-intercepts: 6 65 5
,−
y-intercept: 9−