Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Foundations of AlgebraCHAPTER
1.1 Number Sets and the Structure of Algebra1.2 Fractions1.3 Adding and Subtracting Real Numbers;
Properties of Real Numbers1.4 Multiplying and Dividing Real Numbers;
Properties of Real Numbers1.5 Exponents, Roots, and Order of Operations1.6 Translating Word Phrases to Expressions1.7 Evaluating and Rewriting Expressions
11
Copyright © 2011 Pearson Education, Inc.
Number Sets and the Structure of Algebra1.11.1
1. Understand the structure of algebra.2. Classify number sets.3. Graph rational numbers on a number line.4. Determine the absolute value of a number.5. Compare numbers.
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Objective 1
Understand the structure of algebra.
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Definitions
Variable: A symbol that can vary in value.Constant: A symbol that does not vary in value.
Variables are usually letters of the alphabet, like x or y.Usually constants are symbols for numbers, like 1, 2, ¾, 6.74.
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Expression: A constant, variable, or any combination of constants, variables, and arithmetic operations that describe a calculation.
Examples of expressions:
2 + 6 or 4x 5 or 21
3 r h
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Equation: A mathematical relationship that contains an equal sign.
Examples of equations:
2 + 6 = 8 or 4x 5 = 12 or 21
3V r h
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Inequality: A mathematical relationship that contains an inequality symbol (, <, >, , or ).
Symbolic form Translation
8 3 Eight is not equal to three.
5 < 7 Five is less than seven.
7 > 5 Seven is greater than five.
x 3 x is less than or equal to three.
y 2 y is greater than or equal to two.
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Objective 2
Classify number sets.
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Set: A collection of objects.
Braces are used to indicate a set. For example, the set containing the numbers 1, 2, 3, and 4 would be written {1, 2, 3, 4}.
The numbers 1, 2, 3, and 4 are called the members or elements of this set.
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Writing SetsTo write a set, write the members or elements of the set separated by commas within braces, { }.
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Example 1
Write the set containing the first four days of the week.
Answer{Sunday, Monday, Tuesday, Wednesday}
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Numbers are classified using number sets.
Natural numbers contain the counting numbers 1, 2, 3, 4, …and is written {1, 2, 3, …}. The three dots are called ellipsis and indicate that the numbers continue forever in the same pattern.
Whole numbers: natural numbers and 0 {0, 1, 2, 3,…}Integers: whole numbers and the opposite (or negative) of every natural number {…, 3, 2, 1, 0, 1, 2, 3…}
Rational: every real number that can be expressed as a ratio of integers.
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Rational number: Any real number that can be expressed in the form , where a and b are integers and b 0.
a
b
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Example 2
Determine whether the given number is a rational number.a. b. 0.8 c.
Answera.
5
60.3
5
6
Yes, because 5 and 6 are integers.
b. 0.8
Yes, 0.8 can be expressed as a fraction 8 over 10, and 8 and 10 are integers.
c. 0.3
The bar indicates that the digit repeats. This is the decimal equivalent of 1 over 3. Yes this is a rational number.
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Irrational number: Any real number that is not rational.
Examples:
Real numbers: The union of the rational and irrational numbers.
2, 3,
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Objective 3
Graph rational numbers on a number line.
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Example 3
Graph on a number line.
Answer The number is located 4/5 of the way between 2 and 3.
42
5
31-1 20 42
5
Between 2 and 3, we divide the number line into 5 equally spaced divisions. Place a dot on the 4th mark.
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Objective 4
Determine the absolute value of a number.
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Absolute value: A given number’s distance from 0 on a number line.
The absolute value of a number n is written |n|.The absolute value The absolute valueof 5 is 5. of 5 is 5.|5| = 5 |5| = 5
5 units from 0 5 units from 0
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Absolute ValueThe absolute value of every real number is either positive or 0.
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Example 4
Simplify.a. |9.4| b.
Answer
a. |9.4| = 9.4
b.
2
9
2
9
2
9
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Objective 5
Compare numbers.
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Comparing NumbersFor any two real numbers a and b, a is greater than b if a is to the right of b on a number line. Equivalently, b is less than a if b is to the left of a on a number line.
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Example 5
Use =, <, or > to write a true statement.a. 3 ___ 3 b. 1.8 ___ 1.6
Answera. 3 ___ 3
3 > 3because 3 is farther right on a number line.
b. 1.8 ___ 1.6 1.8 < 1.6because –1.8 is further to the left on a number line.
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To which set of numbers does 6 belong?
a) Irrational
b) Natural and whole numbers
c) Natural numbers, whole numbers,
and integersd) Integers and rational numbers
1.1
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To which set of numbers does 6 belong?
a) Irrational
b) Natural and whole numbers
c) Natural numbers, whole numbers,
and integersd) Integers and rational numbers
1.1
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Simplify |7|.
a) 7
b) 7
c) 0
d) 1/7
1.1
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Simplify |7|.
a) 7
b) 7
c) 0
d) 1/7
1.1
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Which statement is false?
a) 7 > 4
b) 2.4 > 1.4
c) 10 < 22
d) 3.6 > 6.4
1.1
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Which statement is false?
a) 7 > 4
b) 2.4 > 1.4
c) 10 < 22
d) 3.6 > 6.4
1.1
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Fractions1.21.2
1. Write equivalent fractions.2. Write equivalent fractions with the LCD.3. Write the prime factorization of a number.4. Simplify a fraction to lowest terms.
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Fraction: A quotient of two numbers or expressions a and b having the form where b 0.
The top number in a fraction is called the numerator.The bottom number is called the denominator.Fractions indicated part of a whole.
,a
b
3
4
Numerator
Denominator
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Objective 1
Write equivalent fractions.
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Writing Equivalent FractionsFor any fraction, we can write an equivalent fraction by multiplying or dividing both its numerator and denominator by the same nonzero number.
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Example 1
Find the missing number that makes the fractions equivalent.a. b.
Solutiona. b.
9
15 5
?
4
?18
36 2
9 ?
15 45
9 3
315
27
45
18 ?
36 2
18
36
18
18
1
2
Multiply the numerator and denominator by 3.
Divide the numerator and denominator by 6.
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Objective 2
Write equivalent fractions with the LCD.
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Multiple: A multiple of a given integer n is the product of n and an integer.
We can generate multiples of a given number by multiplying the given number by the integers.
2 1 2
2 2 4
2 3 6
2 4 8
2 5 10
2 6 12
3 1 3
3 2 6
3 3 9
3 4 12
3 5 15
3 6 18
Multiples of 2 Multiples of 3
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Least common multiple (LCM): The smallest number that is a multiple of each number in a given set of numbers.
Least common denominator (LCD): The least common multiple of the denominators of a given set of fractions.
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Example 2
Write as equivalent fractions with the LCD.
SolutionThe LCD of 8 and 6 is 24.
7 5 and
8 6
3
7 7=
8
3
8
21
24
4
5 5=
6
4
6
20
24
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Objective 3
Write the prime factorization of a number.
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Factors: If a b = c, then a and b are factors of c.
Example: 6 7 = 42, 6 and 7 are factors of 42
Prime number: A natural number that has exactly two different factors, 1 and the number itself.
Example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,…
Prime factorization: A factorization that contains only prime factors.
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Example 3
Find the prime factorization of 420.Solution 420
42
10
6
7
2
5
2
3
Factor 420 to 10 and 42. (Any two factors will work.)
Factor 10 to 2 and 5, which are primes. Then factor 42 to 6 and 7.
7 is prime and then factor 6 into 2 and 3, which are primes.
Answer 2 2 3 5 7
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Objective 4
Simplify a fraction to lowest terms.
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Lowest terms: Given a fraction and b 0, if the only factor common to both a and b is 1, then the fraction is in lowest terms.
a
b
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Simplifying a Fraction with the Same Nonzero Numerator and Denominator
Eliminating a Common Factor in a Fraction
11, when 0.
1
nn
n
1, when 0 and 0.
1
an a ab n
bn b b
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These rules allow us to write fractions in lowest terms using prime factorizations. The idea is to replace the numerator and denominator with their prime factorizations and then eliminate the prime factors that are common to both the numerator and denominator.
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Simplifying a Fraction to Lowest Terms
To simplify a fraction to lowest terms:1. Replace the numerator and denominator with their
prime factorizations.2. Eliminate (divide out) all prime factors common to the numerator and denominator.3. Multiply the remaining factors.
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Example 4a
Simplify to lowest terms.
Solution
30
42
30
42
2 3 5
2 3 7
Replace the numerator and denominator with their prime factorizations; then eliminate the common prime factors.
5
7
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Example 4b
Simplify to lowest terms.
Solution
220
2380
220
2380
2 2 5 11
2 2 5 7 17
Replace the numerator and denominator with their prime factorizations; then eliminate the common prime factors.
11
119
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Example 5
At a company, 225 of the 1050 employees have optional eye insurance coverage as part of their benefits package. What fraction of the employees have optional eye insurance coverage?Solution
Answer 3 out of 14 employees have optional eye insurance.
225
1050
3 3 5 5
2 3 5 5 7
3
14
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What is the prime factorization of 360?
a) 6 6 5
b) 23 32 5
c) 22 32 5
d) 32 5 7
1.2
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What is the prime factorization of 360?
a) 6 6 5
b) 23 32 5
c) 22 32 5
d) 32 5 7
1.2
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Simplify to lowest terms:
a)
b)
c)
d)
112
280
14
35
2
5
1
4
21
23
1.2
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Simplify to lowest terms:
a)
b)
c)
d)
112
280
14
35
2
5
1
4
21
23
1.2
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Adding and Subtracting Real Numbers; Properties of Real Numbers1.31.3
1. Add integers.2. Add rational numbers.3. Find the additive inverse of a number.4. Subtract rational numbers.
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Objective 1
Add integers.
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Parts of an addition statement: The numbers added are called addends and the answer is called a sum.
2 + 3 = 5
Addends Sum
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Properties of Addition
Symbolic Form Word Form
Additive Identity
a + 0 = a The sum of a number and 0 is that number.
Commutative Property of Addition
a + b = b + a Changing the order of addends does not affect the sum.
Associative Property of Addition
a + (b + c) = (a + b) + c Changing the grouping of three or more addends does not affect the sum.
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Example 1Indicate whether each equation illustrates the additive identity, commutative property of addition, or the associative property of addition.a. (5 + 6) + 3 = 5 + (6 + 3)Answer Associative property of addition
b. 0 + (9) = 9Answer Additive identity
c. (9 + 6) + 4 = 4 + (9 + 6) Answer Commutative property of addition
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Adding Numbers with the Same Sign
To add two numbers that have the same sign, add their absolute values and keep the same sign.
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Example 2
Add.a. 27 + 12 b. –16 + (– 22)
Solutiona. 27 + 12 = 39
b. –16 + (–22) = –38
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Adding Numbers with Different Signs
To add two numbers that have different signs, subtract the smaller absolute value from the greater absolute value and keep the sign of the number with the greater absolute value.
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Example 3
Add.a. 35 + (–17) b. –29 + 7
Solutiona. 35 + (–17) = 18
b. –29 + 7 = –22
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Example 3 continued
Add.c. 15 + (–27) d. –32 + 6
Solutionc. 15 + (–27) = –12
d. –32 + 6 = –26
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Objective 2
Add rational numbers.
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Adding Fractions with the Same Denominator
To add fractions with the same denominator, add the numerators and keep the same denominator; then simplify.
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Example 4
Add.a. b.
Solutiona.
2 4
9 9
2 4
9 9
2 3 2
3 3 3
4 5
12 12
4 5b.
12 12
3 3 3
3 2 2 4
Replace 6 and 9 with their prime
factorizations, divide out the common factor, 3, then multiply the remaining factors.
Simplify to lowest terms by dividing out the common factor, 3.
2 4 6
9 9
4 5 9
12 12
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Example 4 continued
Add.c.
Solutiona.
7 3
10 10
7 3
10 10
2 2 2
2 5 5
Simplify to lowest terms by dividing out the common factor, 2.
7 ( 3) 4
10 10
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Adding Fractions
To add fractions with different denominators:1. Write each fraction as an equivalent fraction with
the LCD.2. Add the numerators and keep the LCD.3. Simplify.
Solution
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1 4 1(3)
3 4 4(3)
Example 5a1 1
Add: 3 4
1 1
3 4 Write equivalent fractions
with 12 in the denominator.
4 3
12 12 Add numerators and keep
the common denominator.
7
12
Because the addends have the same sign, we add and keep the same sign.
Solution
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5 2 3(3)
6 2 4(3)
Example 5b5 3
Add: 6 4
5 3
6 4 Write equivalent fractions
with 12 in the denominator.
10 9
12 12 Add numerators and keep
the common denominator.
10 9
12
Because the addends have different signs, we subtract and keep the sign of the number with the greater absolute value.
1
12
Solution
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7 15 9(4)
8 15 30(4)
Example 5c7 9
Add: 8 30
7 9
8 30 Write equivalent fractions
with 120 in the denominator.
105 36
120 120 Add numerators and keep
the common denominator.
105 36
120
Reduce to lowest terms.
69
120
3 23
2 2 2 3 5
23
40
Anna has a balance of $378.45 and incurs a debt of $85.42. What is Anna’s new balance?
Solution
A debt of $85.42 is $85.42. Her balance is 378.45 + (– 85.42) = $293.03
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Example 6
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Objective 3
Find the additive inverse of a number.
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Additive inverses: Two numbers whose sum is 0.
What happens if we add two numbers that have the same absolute value but different signs, such as 5 + (–5)? In money terms, this is like making a $5 payment towards a debt of $5. Notice the payment pays off the debt so that the balance is 0.
5 + (–5) = 0
Because their sum is zero, we say 5 and –5 are additive inverses, or opposites.
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Example 7
Find the additive inverse of the given number.a. 8 b. –2 c. 0
Answersa. –8 because 8 + (–8) = 0
b. 2 because – 2 + 2 = 0
c. 0 because 0 + 0 = 0
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Example 8
Simplify.a. – (–5) b. –|2| c. –| –9|
Answersa. – (–5) = 5
b. –|2| = –2
c. –| –9| = –9
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Objective 4
Subtract rational numbers.
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Parts of a subtraction statement:
8 – 5 = 3
Minuend
Subtrahend
Difference
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Rewriting Subtraction
To write a subtraction statement as an equivalent addition statement, change the operation symbol from a minus sign to a plus sign, and change the subtrahend to its additive inverse.
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Example 9a
Subtracta. –17 – (–5)
SolutionWrite the subtraction as an equivalent addition.
–17 – (–5)
= –17 + 5 = –12
Change the operation from minus to plus.
Change the subtrahend to its additive inverse.
Solution
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Example 9b3 1
Subtract: 8 4
3 1
8 4
1
4
3
8
Write equivalent fractions with the common denominator, 8.
3 1(2)
8 4(2)
3 2
8 8
1
4
3
8
5
8
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Example 9c
c. 4.07 – 9.03
Solution Write the equivalent addition statement.
4.07 – 9.03 = 4.07 + (– 9.03) = –4.96
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Example 10
In an experiment, a mixture begins at a temperature of 52.6C. The mixture is then cooled to a temperature of 29.4C. Find the difference between the initial and final temperatures.
Solution 52.6 – (–29.4) = 52.6 + 29.4 = 82
Answer The difference between the initial and final temperatures is 82C.
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Add –6 + (–9).
a) –15
b) 3
c) 3
d) 15
1.3
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Add –6 + (–9).
a) –15
b) 3
c) 3
d) 15
1.3
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Subtract 5 – (–8).
a) –13
b) 3
c) 3
d) 13
1.3
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Subtract 5 – (–8).
a) –13
b) 3
c) 3
d) 13
1.3
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Subtract
3 1.
7 3
a)
b)
c)
d)
16
21
1
2
2
21
2
21
1.3
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Subtract
3 1.
7 3
a) 16
21
b) 1
2
c) 2
21
d) 2
21
1.3
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Multiplying and Dividing Real Numbers; Properties of Real Numbers1.41.4
1. Multiply integers.2. Multiply more than two numbers.3. Multiply rational numbers.4. Find the multiplicative inverse of a number.5. Divide rational numbers.
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Objective 1
Multiply integers.
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In a multiplication statement, factors are multiplied to equal a product.
ProductFactors
2 3 = 6
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Properties of Multiplication
Symbolic Form Word Form
Multiplicative Property of 0
The product of a number multiplied by 0 is 0.
Multiplicative Identity
The product of a number multiplied by 1 is the number.
Commutative Property of
Multiplication
ab=ba Changing the order of factors does not affect the product.
Associative Property of
Multiplication
a(bc) = (ab)c Changing the grouping of three or more factors does not affect the product.
Distributive Property of
Multiplication over Addition
a(b + c) =ab + ac A sum multiplied by a factor is equal to the sum of that factor multiplied by each addend.
0 0 a
1 a a
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Example 1Give the name of the property of multiplication that is illustrated by each equation.a. 6(3) = 3 6Answer Commutative property of multiplication
b. 3(9 5) = [3(9)] 5Answer Associative property of multiplication
c. 4(4 – 2) = 4 4 – 4 2 Answer Distributive property of multiplication over addition
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Multiplying Two Numbers with Different Signs
When multiplying two numbers that have different signs, the product is negative.
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Example 2
Multiply.
a. 7(–4) b. (–15)3
Solutiona. 7(–4) =
b. (–15)3 =
Warning: Make sure you see the difference between 7(–4), which indicates multiplication, and 7 – 4, which indicates subtraction.
–28
–45
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Multiplying Two Numbers with the Same Sign
When multiplying two numbers that have the same sign, the product is positive.
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Example 3
Multiply.
a. –5(–9) b. (–6)(–8)
Solutiona. –5(–9) =
b. (–6)(–8) =
45
48
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Objective 2
Multiply more than two numbers.
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Multiplying with Negative Factors
The product of an even number of negative factors is positive, whereas the product of an odd number of negative factors is negative.
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Example 4
Multiply.a. (–1)(–3)(–6)(7) Solution Because there are three negative factors (an
odd number of negative factors), the result is negative. (–1)(–3)(–6)(7) = –126
b. (–2)(–4)(2)(–5)(–3)Solution Because there are four negative factors(an
even number of negative factors), the result is positive. (–2)(–4)(2)(–5)(–3) = 240
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Objective 3
Multiply rational numbers.
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Multiplying Fractions
, where 0 and 0.a c ac
b db d bd
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Example 5a
Multiply
Solution
3 4 .
5 9
3 4 3 2 2
5 9 5 3 3
4
15
Divide out the common factor, 3.
Because we are multiplying two numbers that have different signs, the product is negative.
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Example 5b
Multiply
Solution
6 6 12
15 16 15
3
25
Divide out the common factors.
Because there are an even number of negative factors, the product is positive.
6 6 12
15 16 15
2 3 2 3 2 2 3
3 5 2 2 2 2 3 5
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Multiplying Decimal Numbers
To multiply decimal numbers:1. Multiply as if they were whole numbers.2. Place the decimal in the product so that it has the
same number of decimal places as the total number of decimal places in the factors.
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Example 6a
Multiply (–7.6)(0.04).Solution First, calculate the value and disregard signs for now.
0.04 2 places 7.6 + 1 place 0 2 4+ 0 2 8 0 0.3 0 4
Answer –0.304
When we multiply two numbers with different signs, the product is negative.
3 places
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Example 6b
Multiply (3)(5.2)(1.4)(6.1).Solution First, calculate the value and disregard signs for now.
Multiply from left to right.
(3)(5.2)(1.4)(6.1) = (15.6)(1.4)(6.1) = 21.84(6.1) = 133.224
Answer 133.224
15.6 = (3)(5.2)
21.84 = 15.6(1.4)
The product of an even number of negative factors is positive. The factors have a total of 3 decimal places, so the product has three decimal places.
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Objective 4
Find the multiplicative inverse of a number.
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Multiplicative inverses: Two numbers whose product is 1.
2
3 and are multiplicative inverses because their product is 1.
3
2
2 3 6 1
3 2 6
Notice that to write a number’s multiplicative inverse, we simply invert the numerator and denominator. Multiplicative inverses are also known as reciprocals.
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Example 7
Find the multiplicative inverse.a. b. c. 9
Answera. The multiplicative inverse is
b. The multiplicative inverse is 8.
c. The multiplicative inverse is
2
7
7.
2
1.
9
1
8
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Objective 5
Divide rational numbers.
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Dividend
8 2 = 4
Divisor
Quotient
Parts of a division statement:
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Dividing Signed Numbers
When dividing two numbers that have the same sign, the quotient is positive.When dividing two numbers that have different signs, the quotient is negative.
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Example 8
Divide.a. b.
Solutiona. b.
56 ( 8)
56 ( 8) 7
72 6
72 6 12
Slide 1- 118Copyright © 2011 Pearson Education, Inc.
Division Involving 0
0 0 when 0.n n
0 is undefined when 0.n n
0 0 is indeterminate.
Slide 1- 119Copyright © 2011 Pearson Education, Inc.
Dividing Fractions
, where 0, 0, and 0.a c a d
b c db d b c
Slide 1- 120Copyright © 2011 Pearson Education, Inc.
Example 9
Divide
Solution
3 4.
10 5
3 4 3 5
10 5 10 4 Write an equivalent multiplication.
3 5
5 2 2 2
Divide out the common factor, 5.
3
8
Because we are dividing two numbers that have different signs, the result is negative.
Slide 1- 121Copyright © 2011 Pearson Education, Inc.
Dividing Decimal NumbersTo divide decimal numbers, set up a long division
and consider the divisor.Case 1: If the divisor is an integer, divide as if the
dividend were a whole number and place the decimal point in the quotient directly above its position in the dividend.
Case 2: If the divisor is a decimal number, 1. Move the decimal point in the divisor to the
right enough places to make the divisor an integer.
2. Move the decimal point in the dividend the same number of places.
Slide 1- 122Copyright © 2011 Pearson Education, Inc.
Dividing Decimal Numbers continued
3. Divide the divisor into the dividend as if both numbers were whole numbers. Make sure you align the digits in the quotient properly.
4. Write the decimal point in the quotient directly above its new position in the dividend.
In either case, continue the division process until you get a remainder of 0 or a repeating digit (or block of digits) in the quotient.
Slide 1- 123Copyright © 2011 Pearson Education, Inc.
Example 10
Divide 44.64 ÷ (3.6)
Solution Because the divisor is a decimal number, we move the decimal point enough places to the right to create an integer—in this case, one place. Then we move the decimal point one place to the right in the dividend. Because we are dividing two numbers with the same sign, the result is positive.
Slide 1- 124Copyright © 2011 Pearson Education, Inc.
Example 10 continued
Divide 44.64 ÷ (3.6)
Solution 12.436 446.4
36
86
72
144
144
0
Slide 1- 125Copyright © 2011 Pearson Education, Inc.
Example 11
Martha was decorating cookies. She used 2/3 of a container of frosting that was 3/4 full. What fractional part of the container did she use?
Solution To find 2/3 of 3/4, multiply.
2 3 2 3 1=
3 4 2 2 3 2
Slide 1- 126Copyright © 2011 Pearson Education, Inc.
Multiply (–6)(–3)(7).
a) 126
b) 126
c) –63
d) 63
1.4
Slide 1- 127Copyright © 2011 Pearson Education, Inc.
Multiply (–6)(–3)(7).
a) 126
b) 126
c) –63
d) 63
1.4
Slide 1- 128Copyright © 2011 Pearson Education, Inc.
Divide
a)
b)
c)
d)
14.6 0.03 .
486.6
48.6
48.6
486.6
1.4
Slide 1- 129Copyright © 2011 Pearson Education, Inc.
Divide
a)
b)
c)
d)
14.6 0.03 .
486.6
48.6
48.6
486.6
1.4
Copyright © 2011 Pearson Education, Inc.
Exponents, Roots, and Order of Operations1.51.5
1. Evaluate numbers in exponential form.2. Evaluate square roots.3. Use the order-of-operations agreement to simplify numerical expressions.4. Find the mean of a set of data.
Slide 1- 131Copyright © 2011 Pearson Education, Inc.
Objective 1
Evaluate numbers in exponential form.
Slide 1- 132Copyright © 2011 Pearson Education, Inc.
Sometimes problems involve repeatedly multiplying the same number. In such problems, we can use an exponent to indicate that a base number is repeatedly multiplied.
Exponent: A symbol written to the upper right of a base number that indicates how many times to use the base as a factor.
Base: The number that is repeatedly multiplied.
Slide 1- 133Copyright © 2011 Pearson Education, Inc.
When we write a number with an exponent, we say the expression is in exponential form. The expression is in exponential form, where the base is 2 and the exponent is 4. To evaluate , write 2 as a factor 4 times, then multiply.
4242
Exponent
42 2 2 2 2 = 16
Base
Four 2s
Slide 1- 134Copyright © 2011 Pearson Education, Inc.
Evaluating an Exponential Form
To evaluate an exponential form raised to a natural number exponent, write the base as a factor the number of times indicated by the exponent; then multiply.
Slide 1- 135Copyright © 2011 Pearson Education, Inc.
Example 1a
Evaluate. (–9)2
SolutionThe exponent 2 indicates we have two factors of –9. Because we multiply two negative numbers, the result is positive.
(–9)2 = (–9)(–9) = 81
Slide 1- 136Copyright © 2011 Pearson Education, Inc.
Example 1b
Evaluate.
SolutionThe exponent 3 means we must multiply the base by itself three times.
33
5
33
5
3 3 3
5 5 5
27
125
Slide 1- 137Copyright © 2011 Pearson Education, Inc.
Evaluating Exponential Forms with Negative Bases
If the base of an exponential form is a negative number and the exponent is even, then the product is positive.
If the base is a negative number and the exponent is odd, then the product is negative.
Slide 1- 138Copyright © 2011 Pearson Education, Inc.
Example 2
Evaluate.a. b. c. d.Solutiona.
b.
c.
d.
4( 3)
4( 3) ( 3)( 3)( 3)( 3) 81
43 3( 2) 32
43 3 3 3 3 81
3( 2) ( 2)( 2)( 2) 8
32 2 2 2 8
Slide 1- 139Copyright © 2011 Pearson Education, Inc.
Objective 2
Evaluate square roots.
Slide 1- 140Copyright © 2011 Pearson Education, Inc.
Roots are inverses of exponents. More specifically, a square root is the inverse of a square, so a square root of a given number is a number that, when squared, equals the given number.
Square RootsEvery positive number has two square roots, a
positive root and a negative root. Negative numbers have no real-number square
roots.
Slide 1- 141Copyright © 2011 Pearson Education, Inc.
Example 3
Find all square roots of the given number.
Solutiona. 49Answer 7
b. 81Answer No real-number square roots exist.
Slide 1- 142Copyright © 2011 Pearson Education, Inc.
The symbol, called the radical, is used to indicate finding only the positive (or principal) square root of a given number. The given number or expression inside the radical is called the radicand.
,
25 5
Radicand
RadicalPrincipal Square Root
Slide 1- 143Copyright © 2011 Pearson Education, Inc.
Square Roots Involving the Radical Sign
The radical symbol denotes only the positive (principal) square root.
, where 0 and 0. a a
a bb b
Slide 1- 144Copyright © 2011 Pearson Education, Inc.
Example 4
Evaluate the square root.a. b. c. d.
Solutiona.
c.
169 250.64
169 13
0.64 0.8
64
81
64 8b.
81 9
d. 25 not a real number
Slide 1- 145Copyright © 2011 Pearson Education, Inc.
Objective 3
Use the order-of-operations agreement to simplify numerical expressions.
Slide 1- 146Copyright © 2011 Pearson Education, Inc.
Order-of- Operations Agreement
Perform operations in the following order:1. Within grouping symbols: parentheses ( ),
brackets [ ], braces { }, above/below fraction bars, absolute value | |, and radicals .
2. Exponents/Roots from left to right, in order as they occur.
3. Multiplication/Division from left to right, in order as they occur.
4. Addition/Subtraction from left to right, in order as they occur.
Slide 1- 147Copyright © 2011 Pearson Education, Inc.
Example 5a
Simplify.
Solution
26 15 ( 5) 2
26 15 ( 5) 2
26 ( 3) 2
26 ( 6)
32
Divide 15 ÷ (5) = –3
Multiply (–3) 2 = –6
Add –26 + (–6) = –32
Slide 1- 148Copyright © 2011 Pearson Education, Inc.
Example 5b
Simplify.
Solution
43 2 12 20
43 2 12 20 43 2 8
43 2 8
81 2 8
Subtract inside the absolute value.
Simplify the absolute value.
Evaluate the exponent.
81 16
65
Multiply.
Add.
Slide 1- 149Copyright © 2011 Pearson Education, Inc.
Example 5c
Simplify.
Solution
23 5 6 2 1 49
23 5 6 2 1 49
23 5 6 3 49
9 5 3 7
9 15 7
24 7
17
Calculate within the innermost parenthesis.
Evaluate the exponential form, brackets, and square root.
Multiply 5(3).
Add 9 + 15.
Subtract 24 – 7.
Slide 1- 150Copyright © 2011 Pearson Education, Inc.
Square Root of a Product or QuotientIf a square root contains multiplication or division, we can multiply or divide first, then find the square root of the result, or we can find the square roots of the individual numbers, then multiply or divide the square roots.
Square Root of a Sum or DifferenceWhen a radical contains addition or subtraction, we must add or subtract first, then find the root of the sum or difference.
Slide 1- 151Copyright © 2011 Pearson Education, Inc.
Example 6a
Simplify.
Solution
213.5 5 4 3 142 21
10.2
Subtract within the radical.
Evaluate the exponential form and root.
Divide.
Multiply.
Subtract.
213.5 5 4 3 142 21
213.5 5 4 3 121
13.5 5 16 3(11)
2.7 16 3 11
43.2 33
Slide 1- 152Copyright © 2011 Pearson Education, Inc.
Sometimes fraction lines are used as grouping symbols. When they are, we simplify the numerator and denominator separately, then divide the results.
Slide 1- 153Copyright © 2011 Pearson Education, Inc.
Example 7a
Simplify.
Solution
38( 5) 2
4(8) 8
Evaluate the exponential form in the numerator and multiply in the denominator.
Multiply in the numerator and subtract in the denominator.
Subtract in the numerator.
Divide.
38( 5) 2
4(8) 8
8( 5) 8
4(8) 8
40 8
32 8
48
24
2
Slide 1- 154Copyright © 2011 Pearson Education, Inc.
Example 7b
Simplify.
Solution
3
9(4) 12
4 (8)( 8)
Because the denominator or divisor is 0, the answer is undefined.
3
9(4) 12
4 (8)( 8)
36 12
64 (8)( 8)
48
64 ( 64)
48
0
Slide 1- 155Copyright © 2011 Pearson Education, Inc.
Objective 4
Find the mean of a set of data.
Slide 1- 156Copyright © 2011 Pearson Education, Inc.
Finding the Arithmetic MeanTo find the arithmetic mean, or average, of n numbers, divide the sum of the numbers by n.
Arithmetic mean = 1 2 ... nx x x
n
Slide 1- 157Copyright © 2011 Pearson Education, Inc.
Example 8
Bruce has the following test scores in his biology class: 92, 96, 81, 89, 95, 93. Find the average of his test scores.
Solution
92 96 81 89 95 93
6
546
6
91
Divide the sum of the 6 scores by 6.
Slide 1- 158Copyright © 2011 Pearson Education, Inc.
Simplify using order of operations.
a) 18
b) 6
c) 30
d) 36
26 18 9 6
1.5
Slide 1- 159Copyright © 2011 Pearson Education, Inc.
Simplify using order of operations.
a) 18
b) 6
c) 30
d) 36
26 18 9 6
1.5
Slide 1- 160Copyright © 2011 Pearson Education, Inc.
Simplify using order of operations.
a)
b)
c)
d) undefined
3
2
2 4 2
6 30 2 4
8
300
250
361
2
11
1.5
Slide 1- 161Copyright © 2011 Pearson Education, Inc.
Simplify using order of operations.
a)
b)
c)
d) undefined
3
2
2 4 2
6 30 2 4
8
300
250
361
2
11
1.5
Copyright © 2011 Pearson Education, Inc.
Translating Word Phrases to Expressions1.61.6
1. Translate word phrases to expressions.
Slide 1- 163Copyright © 2011 Pearson Education, Inc.
Objective 1
Translating word phrases to Expressions
Slide 1- 164Copyright © 2011 Pearson Education, Inc.
Translating Basic PhrasesAddition Translation Subtraction Translation
The sum of x and three
x + 3 The difference of x and three
x – 3
h plus k h + k h minus k h – k
seven added to t 7 + t seven subtracted from t
t – 7
three more than a number
n + 3 three less than a number
n – 3
y increased by two
y + 2 y decreased by two y – 2
Note: Since addition is a commutative operation, it does not matter in what order we write the translation.
Note: Subtraction is not a commutative operation; therefore, the way we write the translation matters.
Slide 1- 165Copyright © 2011 Pearson Education, Inc.
Translating Basic PhrasesMultiplication Translation Division Translation
The product of x and three
3x The quotient of x and three
x 3 or
h times k hk h divided by k h k or
Twice a number 2n h divided into k k h or
Triple the number
3n The ratio of a to b a b or
Two-thirds of a number
Note: Like addition, multiplication is a commutative operation: it does not matter in what order we write the translation.
Note: Division is like subtraction in that it is not a commutative operation; therefore, the way we write the translation matters.
2
3n
3
x
h
kk
ha
b
Slide 1- 166Copyright © 2011 Pearson Education, Inc.
Translating Basic PhrasesExponents Translation Roots Translation
c squared c2 The square root of x
The square of b b2
k cubed k3
The cube of b b3
n to the fourth power
n4
y raised to the fifth power
y5
x
Slide 1- 167Copyright © 2011 Pearson Education, Inc.
The key words sum, difference, product, and quotient indicate the answer for their respective operations.
sum of x and 3
x + 3
difference of x and 3
product of x and 3 quotient of x and 3
x – 3
x 3 x 3
Slide 1- 168Copyright © 2011 Pearson Education, Inc.
Example 1
Translate to an algebraic expression.a. five more than two times a numberTranslation: 5 + 2n or 2n + 5
b. seven less than the cube of a numberTranslation: n3 – 7
c. the sum of h raised to the fourth power and twelveTranslation: h4 + 12
Slide 1- 169Copyright © 2011 Pearson Education, Inc.
Translating Phrases Involving Parentheses
Sometimes the word phrases imply an order of operations that would require us to use parentheses in the translation.
These situations arise when the phrase indicates that a sum or difference is to be calculated before performing a higher-order operation such as multiplication, division, exponent, or root.
Slide 1- 170Copyright © 2011 Pearson Education, Inc.
Example 2
Translate to an algebraic expression.a. seven times the sum of a and bTranslation: 7(a + b)
b. the product of a and b divided by the sum of w2 and 4
Translation: ab (w2 + 4) or 2 4
ab
w
Slide 1- 171Copyright © 2011 Pearson Education, Inc.
Translate the phrase to an algebraic expression. Twelve less than three times a number
a) 3n + 12
b) 12 – 3n
c) 3n – 12
d) 3n 12
1.6
Slide 1- 172Copyright © 2011 Pearson Education, Inc.
Translate the phrase to an algebraic expression.
Twelve less than three times a number
a) 3n + 12
b) 12 – 3n
c) 3n – 12
d) 3n 12
1.6
Slide 1- 173Copyright © 2011 Pearson Education, Inc.
Translate the phrase to an algebraic expression.
The difference of a and b decreased by the sum of w and z
a) (a – b) – (w + z)
b) a – b – (w + z)
c) ab – (w + z)
d) (b – a) – (w + z)
1.6
Slide 1- 174Copyright © 2011 Pearson Education, Inc.
Translate the phrase to an algebraic expression.
The difference of a and b decreased by the sum of w and z
a) (a – b) – (w + z)
b) a – b – (w + z)
c) ab – (w + z)
d) (b – a) – (w + z)
1.6
Copyright © 2011 Pearson Education, Inc.
Evaluating and Rewriting Expressions1.71.7
1. Evaluate an expression.2. Determine all values that cause an expression to be undefined.3. Rewrite an expression using the distributive property.4. Rewrite an expression by combining like terms.
Slide 1- 176Copyright © 2011 Pearson Education, Inc.
Objective 1
Evaluate an expression.
Slide 1- 177Copyright © 2011 Pearson Education, Inc.
Evaluating an Algebraic Expression
To evaluate an algebraic expression:1. Replace the variables with their corresponding
given values.2. Calculate the numerical expression using the order
of operations.
Slide 1- 178Copyright © 2011 Pearson Education, Inc.
Example 1a
Evaluate 3w – 4(a – 6) when w = 5 and a = 7.
Solution3w – 4(a 6)
3(5) – 4(7 – 6)= 3(5) – 4(1)= 15 – 4= 11
Replace w with 5 and a with 7.
Simplify inside the parentheses first.
Multiply.
Subtract.
Slide 1- 179Copyright © 2011 Pearson Education, Inc.
Example 1b
Evaluate x2 – 0.4xy + 9, when x = 7 and y = –2.
Solutionx2 – 0.4xy + 9
(7)2 – 0.4(7)(–2) + 9= 49 – 0.4(7)(–2) + 9= 49 – (–5.6) + 9= 49 + 5.6 + 9= 63.6
Replace x with 7 and y with –2.
Begin calculating by simplifying the exponential form.
Multiply.
Write the subtraction as an equivalent addition.
Add from left to right.
Slide 1- 180Copyright © 2011 Pearson Education, Inc.
Objective 2
Determine all values that cause an expression to be undefined.
Slide 1- 181Copyright © 2011 Pearson Education, Inc.
When evaluating a division expression in which the divisor or denominator contains a variable or variables, we must be careful about what values replace the variable(s).
We often need to know what values could replace the variable(s) and cause the expression to be undefined or indeterminate.
Slide 1- 182Copyright © 2011 Pearson Education, Inc.
Example 2Determine all values that cause the expression to be undefined.a. b.
Answera. If x = 4, we have which is undefined because the denominator is 0.b. If x = 2 or 9 the fraction will be undefined since the denominator will = 0.
8
4x 2
( 2)( 9)x x
8 8,
4 4 0
2 2
( 2 2)( 2 9) 0
2 2
(9 2)(9 9) 0
Slide 1- 183Copyright © 2011 Pearson Education, Inc.
Objective 3
Rewrite an expression using the distributive property.
Slide 1- 184Copyright © 2011 Pearson Education, Inc.
The Distributive Property of Multiplication over Addition
a(b + c) = ab + ac
This property gives us an alternative to the order of operations.
2(5 + 6) = 2(11) 2(5 + 6) = 25 + 26
= 22 = 10 + 12
= 22
Slide 1- 185Copyright © 2011 Pearson Education, Inc.
Example 3
Use the distributive property to write an equivalent expression and simplify.a. 3(x + 3) b. –5(w – 4)
Solutiona. 3(x + 3) = 3 x + 3 3
= 3x + 9
b. –5(w – 4) = –5 w – (–5) 4= –5w + 20
Slide 1- 186Copyright © 2011 Pearson Education, Inc.
Objective 4
Rewrite an expression by combining like terms.
Slide 1- 187Copyright © 2011 Pearson Education, Inc.
Terms: Expressions that are the addends in an expression that is a sum.
Coefficient: The numerical factor in a term.The coefficient of 5x3 is 5.The coefficient of –8y is –8.
Like terms: Variable terms that have the same variable(s) raised to the same exponents, or constant terms.Like terms Unlike terms4x and 7x 2x and 8y different variables
5y2 and 10y2 7t3 and 3t2 different exponents
8xy and 12xy x2y and xy2 different exponents
7 and 15 13 and 15x different variables
Slide 1- 188Copyright © 2011 Pearson Education, Inc.
Combining Like TermsTo combine like terms, add or subtract the coefficients and keep the variables and their exponents the same.
Slide 1- 189Copyright © 2011 Pearson Education, Inc.
Example 4
Combine like terms.a. 10y + 8y
Solution 10y + 8y = 18y
b. 8x – 3x Solution 8x – 3x = 5x
c. 13y2 – y2
Solution 13y2 – y2 = 12y2
Slide 1- 190Copyright © 2011 Pearson Education, Inc.
Example 5
Combine like terms in 5y2 + 6 + 4y2 – 7.
Solution 5y2 + 6 + 4y2 – 7= 5y2 + 4y2 + 6 – 7 Combine like terms.
= 9y2 – 1
Slide 1- 191Copyright © 2011 Pearson Education, Inc.
Example 6
Combine like terms in 18y + 7x – y – 7x.
Solution 18y + 7x – y – 7x
= 17y + 0
= 17y
Slide 1- 192Copyright © 2011 Pearson Education, Inc.
Example 7
Combine like terms inSolution
1 14 3 .
12 6a b a b
1 14 3
12 6a b a b
1 14 3
12 6a a b b Collect like terms.
Write the fraction coefficients as equivalent fractions with their LCD, 12.
1 1(2)4 3
12 6(2)a a b b
1 24 3
12 12a a b b
33 3
12a b
13 3
4a b Combine like terms.
Slide 1- 193Copyright © 2011 Pearson Education, Inc.
Evaluate the expression 4(a + b) when a = 3 and b = –2.
a) 4
b) 4
c) 12
d) 20
1.7
Slide 1- 194Copyright © 2011 Pearson Education, Inc.
Evaluate the expression 4(a + b) when a = 3 and b = –2.
a) 4
b) 4
c) 12
d) 20
1.7
Slide 1- 195Copyright © 2011 Pearson Education, Inc.
For which values is the expression undefined?
a) 8
b) 2
c) 2 and 5
d) 2 and 5
8
( 2)( 5)
m
m m
1.7
Slide 1- 196Copyright © 2011 Pearson Education, Inc.
For which values is the expression undefined?
a) 8
b) 2
c) 2 and 5
d) 2 and 5
8
( 2)( 5)
m
m m
1.7
Slide 1- 197Copyright © 2011 Pearson Education, Inc.
Simplify: 7x + 8 – 2x – 4
a) 9x – 4
b) 9x + 4
c) 5x – 4
d) 5x + 4
1.7
Slide 1- 198Copyright © 2011 Pearson Education, Inc.
Simplify: 7x + 8 – 2x – 4
a) 9x – 4
b) 9x + 4
c) 5x – 4
d) 5x + 4
1.7
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