Warm Up Add or subtract. 1. 6 + 104 2. 12(9) 3. 23 – 8 4

Holt Algebra 1

1-1 Variables and Expressions

Warm UpAdd or subtract.

1. 6 + 104 2. 12(9) 3. 23 – 8 4.

Multiply or divide.

5. 324 ÷ 18 6.

7. 13.5(10) 8. 18.2 ÷ 2

10811015

18135 9.1

6

Holt Algebra 1


Translate between words and algebra. Evaluate algebraic expressions.

Objectives

Holt Algebra 1


A variable is a letter or a symbol used to represent a value that can change.

A constant is a value that does not change.

A numerical expression contains only constants and operations.

An algebraic expression may contain variables, constants, and operations.

Holt Algebra 1


These expressions all mean “2 times y”:2y 2(y)2•y (2)(y)2 x y (2)y

Writing Math

Holt Algebra 1


Give two ways to write each algebra expression in words.

A. 9 + r B. q – 3 the sum of 9 and r9 increased by r

the product of m and 7m times 7

the difference of q and 33 less than q

the quotient of j and 6j divided by 6

Example 1: Translating from Algebra to Words

C. 7m D. j 6

Holt Algebra 1


1a. 4 - n 1b.

1c. 9 + q 1d. 3(h)

4 decreased by n

the sum of 9 and q

the quotient of t and 5

the product of 3 and h

Give two ways to write each algebra expression in words.

Check It Out! Example 1

n less than 4 t divided by 5

q added to 9 3 times h

Holt Algebra 1

1-1 Variables and ExpressionsTo translate words into algebraic expressions, look for words that indicate the action that is taking place.

Put together,

combine

Add Subtract

Multiply Divide

Find how much more or less

Put together equal groups

Separate into equal groups

Holt Algebra 1


John types 62 words per minute. Write an expression for the number of words he types in m minutes.

m represents the number of minutes that John types.62 · m or 62m Think: m groups of 62 words

Example 2A: Translating from Words to Algebra

Holt Algebra 1


Roberto is 4 years older than Emily, who is y years old. Write an expression for Roberto’s age

y represents Emily’s age.y + 4 Think: “older than” means “greater than.”

Example 2B: Translating from Words to Algebra

Holt Algebra 1


Joey earns $5 for each car he washes. Write an expression for the number of cars Joey must wash to earn d dollars.

d represents the total amount that Joey will earn.

Think: How many groups of $5 are in d?

Example 2C: Translating from Words to Algebra

Holt Algebra 1


To evaluate an expression is to find its value.

To evaluate an algebraic expression, substitute numbers for the variables in the expression and then simplify the expression.

Holt Algebra 1


Evaluate each expression for m = 3, n = 2, and p = 9.

a. mn

b. p – n

c. p ÷ m


mn = 3 · 2

p – n = 9 – 2 = 7

= 6

p ÷ m = 9 ÷ 3 = 3

Substitute 3 for m and 2 for n.Simplify.

Substitute 9 for p and 2 for n.Simplify.

Substitute 9 for p and 3 for m.Simplify.

Holt Algebra 1

1-1 Variables and ExpressionsExample 4a: Recycling Application

Approximately eighty-five 20-ounce plastic bottles must be recycled to produce the fiberfill for a sleeping bag.Write an expression for the number of bottles needed to make s sleeping bags.

The expression 85s models the number ofbottles to make s sleeping bags.

Holt Algebra 1


Find the number of bottles needed to make20, 50, and 325 sleeping bags.Evaluate 85s for s = 20, 50, and 325.

s 85s20

50

325

85(20) = 170085(50) = 4250

85(325) = 27,625

Example 4b: Recycling Application ContinuedApproximately eighty-five 20-ounce plastic bottles must be recycled to produce the fiberfill for a sleeping bag.

To make 20 sleeping bags 1700 bottles are needed.To make 50 sleeping bags 4250 bottles are needed.To make 325 sleeping bags 27,625 bottles are needed.

Holt Algebra 1


A replacement set is a set of numbers that canbe substituted for a variable. The replacementset in Example 4 is {20, 50, and 325}.

Helpful Hint

Holt Algebra 1

1-2 Adding and Subtracting Real Numbers

Warm UpSimplify.1.

3 2. –4Write an improper fraction to represent eachmixed number.

3. 4 23

143 4. 7 6

7557

Write a mixed number to represent each improper fraction.

5. 125 2 2

56. 24

9 223

|–3| –|4|

Holt Algebra 1


Add real numbers. Subtract real numbers.

Objectives

Holt Algebra 1


All the numbers on a number line are called realnumbers. You can use a number line to modeladdition and subtraction of real numbers.

AdditionTo model addition of a positive number, move right. To model addition of a negative number move left.SubtractionTo model subtraction of a positive number, move left. To model subtraction of a negative number move right.

Holt Algebra 1


Example 1A: Adding and Subtracting Numberson a Number line

Add or subtract using a number line.

Start at 0. Move left to –4.

11 10 9 8 7 6 5 4 3 2 1 0

+ (–7)

–4+ (–7) = –11

To add –7, move left 7 units. –4

–4 + (–7)

Holt Algebra 1


Add or subtract using a number line.–3 + 7

Check It Out! Example 1a

Start at 0. Move left to –3.To add 7, move right 7 units.

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

–3

+7

–3 + 7 = 4

Holt Algebra 1


The absolute value of a number is the distance from zero on a number line. The absolute value of 5 is written as |5|.

5 units 5 units

210123456 6543- - - - - -

|5| = 5|–5| = 5

Holt Algebra 1


Holt Algebra 1


Example 2A: Adding Real Numbers

Add.

Use the sign of the number with the greater absolute value.The sum is negative.

When the signs of numbers are different, find the difference of the

absolute values:

Holt Algebra 1


Example 2B: Adding Real Numbers

Add.y + (–2) for y = –6

y + (–2) = (–6) + (–2)

(–6) + (–2)

First substitute –6 for y.

When the signs are the same, find the sum of the absolute values: 6 + 2 = 8.

–8 Both numbers are negative, so the sum is negative.

Holt Algebra 1


Check It Out! Example 2b

Add.

–13.5 + (–22.3)

When the signs are the same, find the sum of the absolute values.

–13.5 + (–22.3)

–35.8 Both numbers are negative so,the sum is negative.

13.5 + 22.3

Holt Algebra 1


Two numbers are opposites if their sum is 0. A number and its opposite are on opposite sides of zero on a number line, but are the same distance from zero. They have the same absolute value.

Holt Algebra 1


A number and its opposite are additive inverses.To subtract signed numbers, you can use additiveinverses.

11 – 6 = 5 11 + (–6) = 5Additive inverses

Subtracting 6 is the sameas adding the inverse of 6.

Subtracting a number is the same as adding theopposite of the number.

Holt Algebra 1


Holt Algebra 1


Subtract.–6.7 – 4.1

–6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1.

When the signs of the numbersare the same, find the sum of theabsolute values: 6.7 + 4.1 = 10.8.

= –10.8 Both numbers are negative, so the sum is negative.

Example 3A: Subtracting Real Numbers

Holt Algebra 1


Subtract.

5 – (–4)

5 − (–4) = 5 + 4

9

To subtract –4 add 4.

Find the sum of the absolute values.

Example 3B: Subtracting Real Numbers

Holt Algebra 1


Subtract.

Example 3C: Subtracting Real Numbers

First substitute for z.

To subtract , add .

Rewrite with a denominator of 10.

Holt Algebra 1


Example 3C Continued

Write the answer in the simplest form. Both numbers are negative, so the sum is negative.

When the signs of the numbers arethe same, find the sum of the absolute values: .

Holt Algebra 1


Check It Out! Example 3bSubtract.

Both numbers are positive so, the sum is positive.

To subtract add .–3 12 3 1

2When the signs of the numbers are the same, find the sum of the absolute values: = 4.3 1

212+

4

Holt Algebra 1

1-2 Adding and Subtracting Real NumbersCheck It Out! Example 4

What if…? The tallest known iceberg in the North Atlantic rose 550 feet above the oceans surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of –12,468 feet?

elevation at top of iceberg

550

Minus elevation of the Titanic –12,468–

550 – (–12,468)550 – (–12,468) = 550 + 12,468

Distance from the iceberg to the Titanic is 13,018 feet.

To subtract –12,468, add 12,468.Find the sum of the absolute values.= 13,018

Holt Algebra 1

1-3 Multiplying and Dividing Real Numbers

Add or subtract. 1. –2 + 9 7 2. –5 – (–3) –2

Add or subtract.3. –23 + 42 19 4. 4.5 – (–3.7) 8.2

5.

Warm-Up

6. The temperature at 6:00 A.M. was –23°F.At 3:00 P.M. it was 18°F. Find the differencein the temperatures.41°F

Holt Algebra 1


Multiply real numbers. Divide real numbers.

Objectives

Holt Algebra 1


When you multiply two numbers, the signs of thenumbers you are multiplying determine whetherthe product is positive or negative.

Factors Product3(5) Both positive3(–5) One negative–3(–5) Both negative

15 Positive–15 Negative

15 Positive

This is true for division also.

Holt Algebra 1


WORDS

Multiplying and Dividing Numbers with the Same Sign If two numbers have the same sign, their product or quotient is positive.

NUMBERS 4 5 = 20

–15 ÷ (–3) = 5

Multiplying and Dividing Signed Numbers

Holt Algebra 1


WORDS

Multiplying and Dividing Numbers with Different Signs If two numbers have different signs, their product or quotient is negative.

NUMBERS6(–3) = –18

Multiplying and Dividing Signed Numbers

–18 ÷ 2 = –9

Holt Algebra 1


Find the value of each expression.

–5 The product of two numberswith different signs is negative.

Example 1: Multiplying and Dividing Signed Numbers

A.

12 The quotient of two numberswith the same sign is positive.

B.

Holt Algebra 1


Find the value of the expression.

The quotient of two numberswith different signs is negative.

Example 1C: Multiplying and Dividing Signed Numbers

First substitute for x.

Holt Algebra 1


Find the value of each expression.

–7 The quotient of two numberswith different signs is negative.

Check It Out! Example 1a and 1b

1a. 35 (–5)

44 The product of two numberswith the same sign is positive.

1b. –11(–4)

Holt Algebra 1


Two numbers are reciprocals if their product is 1.A number and its reciprocal are called multiplicative inverses. To divide by a number, you can multiply by its multiplicative inverse.

Dividing by a nonzero number is the same as Multiplying by the reciprocal of the number.

Holt Algebra 1


10 ÷ 5 = 2 10 ∙ = = 215

105

Multiplicative inverses

Dividing by 5 is the same as multiplying by thereciprocal of 5, .

Holt Algebra 1


You can write the reciprocal of a number by switching the numerator and denominator. A whole number has a denominator of 1.

Helpful Hint

Holt Algebra 1


Example 2 Dividing by FractionsExample 2A: Dividing by Fractions

Divide.

To divide by , multiply by .

Multiply the numerators and multiply the denominators.

and have the same sign, so the quotient is positive.

Holt Algebra 1


Example 2B: Dividing by FractionsDivide.

Write as an improper fraction.

To divide by , multiply by .

and have different signs, so the quotient is negative.

Holt Algebra 1


Check It Out! Example 2c

Divide.

Write as an improper fraction.

To divide by multiply by .

The signs are different so the quotient is negative.

Holt Algebra 1


No number can be multiplied by 0 to give a product of 1, so 0 has no reciprocal. Because 0 has no reciprocal, division by 0 is not possible. We say that division by zero is undefined.

Holt Algebra 1


Properties of Zero

WORDS

NUMBERS

ALGEBRA

Multiplication by ZeroThe product of any numberand 0 is 0.

13 · 0 = 0 0(–17) = 0

a · 0 = 0 0 · a = 0

Holt Algebra 1


Properties of Zero

WORDS

NUMBERS

ALGEBRA

Zero Divided by a NumberThe quotient of 0 and any nonzero number is 0.

06 = 0

0 ÷ a = 0

0 ÷ 23 = 0

0a = 0

Holt Algebra 1


Properties of Zero

WORDS

NUMBERS

ALGEBRA

Division by ZeroDivision by 0 is undefined.

a ÷ 0 a0

12 ÷ 0Undefined

–5 0

Undefined

Holt Algebra 1


Example 3: Multiplying and Dividing with Zero

Multiply or divide.

A. 150

B. –22 0undefined

C. –8.45(0)0

Zero is divided by a nonzero number.The quotient of zero and any nonzeronumber is 0.

A number is divided by zero.Division by zero is undefined.

A number is multiplied by zero.The product of an number and 0 is 0.

0

Holt Algebra 1


rate

334

times

time11

3

Example 4: Recreation Application

Find the distance traveled at a rate of 3 mi/h for 1 hour.To find distance, multiply rate by time.

34

13

The speed of a hot-air balloon is 3 mi/h. It travels in a straight line for 1 hour before landing. How many miles away from the liftoff site does the balloon land?

13

34

Holt Algebra 1


Example 4: Recreation Application

3 34 • 1 1

3 = 15 4 • 4

3 Write and as improper fractions.343 1 1

3

15(4) 4(3) = 60

12

= 5

Multiply the numerators andmultiply the denominators.

3 34 and have the same sign, so

the quotient is positive.1 1

3

The hot-air balloon lands 5 miles from the liftoff site.

Holt Algebra 1

1-4 Powers and Exponents

Find the value of each expression.1. 35

–7 –5 2. 2x for x = –6 – 12

Warm-Up

Multiply or divide if possible.

3. –3 ÷ 1 34 (0)4. –21

3 5. – 034– 12

7 0 undefined

15 miles

6. A cyclist traveled on a straight road for 1 hours at a speed of 12 mi/h. How many miles did the cyclist travel?

14

Holt Algebra 1

1-4 Powers and Exponents5 Minute Warm-Up

Directions: Solve the followingproblems.1. 3 (15) 2. (4) 5

5 12

3. 4 – (15) 4. 5 + |-12|

5. (-121) ÷ 11 6. 44 ÷ 4 11

Holt Algebra 1


Evaluate expressions containing exponents.Objective

Holt Algebra 1


A power is an expression written with an exponent and a base or the value of such an expression. 3² is an example of a power.

The base is thenumber that isused as a factor.

32 The exponent, 2 tellshow many times thebase, 3, is used as afactor.

Holt Algebra 1

1-4 Powers and ExponentsWhen a number is raised to the second power, we usually say it is “squared.” The area of a square is s s = s2, is the side length.

SS

When a number is raised to the third power, we usually say its “cubed.” The of volume of a cube is s s s = s3 is the side length.

SSS

Holt Algebra 1


Write the power represented by each geometric model.

a.

22

b.


x

x

x

The figure is 2 units long and 2 units wide. 2 2

The factor 2 is used 2 times.

The figure is x units long, x units wide, and x units tall. x x x

The factor x is used 3 times.x3

Holt Algebra 1

1-4 Powers and ExponentsThere are no easy geometric models for numbers raised to exponents greater than 3, but you can still write them using repeated multiplication or a base and exponent.

3 to the second power, or 3 squared

3 3 3 3 3

Multiplication Power ValueWords

3 3 3 3

3 3 3

3 3

33 to the first power

3 to the third power, or 3 cubed3 to the fourth power

3 to the fifth power

3

9

27

81

243

31

Reading Exponents

32

33

34

35

Holt Algebra 1


Caution!In the expression –52, 5 is the base because

the negative sign is not in parentheses. In the expression (–2), –2 is the base because of the parentheses.

Holt Algebra 1


Evaluate each expression.A. (–6)3

(–6)(–6)(–6)–216

B. –102

–1 • 10 • 10

–100

Use –6 as a factor 3 times.

Find the product of –1 andtwo 10’s.

Example 2: Evaluating Powers

Think of a negative sign in front of a power as multiplying by a –1.

Holt Algebra 1


Use as a factor 2 times.2 9

Evaluate the expression.

C.

29 2

9

Example 2: Evaluating Powers

= 481

29 2

9

Holt Algebra 1


Write each number as a power of the given base.

A. 64; base 88 882

B. 81; base –3(–3)(–3)(–3)(–3)

(–3)4

The product of two 8’s is 64.

The product of four –3’s is 81.

Example 3: Writing Powers

Holt Algebra 1


In case of a school closing, the PTApresident calls 3 families. Each of these families calls 3 other familiesand so on. How many families will have been called in the 4th round of calls?

The answer will be the number of familiescontacted in the 4th round of calls.

Example 4: Problem-Solving Application

Understand the problem1

List the important information:• The PTA president calls 3 families.• Each family then calls 3 more families.

Holt Algebra 1


Draw a diagram to show the number of Families called in each round of calls.

2 Make a Plan

Example 4 Continued

2nd round of calls

1st round of calls

PTA President

Holt Algebra 1


Notice that after each round of calls the number of families contacted is a power of 3.1st round of calls: 1 3 = 3 or 31 families contacted

So, in the 4th round of calls, 34 families will havebeen contacted.34 = 3 3 3 3 = 81

Multiply four 3’s.

In the fourth round of calls, 81 families will have been contacted.

2nd round of calls: 3 3 = 9 or 32 families contacted3rd round of calls: 9 3 = 27 or 33 families contacted

Solve3

Example 4 Continued

Holt Algebra 1

1-5 Square Roots and Real Numbers

1. Write the power represented by the geometric model.

nn n2

Simplify each expression.

2.

4. 6

3. –63

5. (–2)6

−216

216 64

Warm-Up

Write each number as a power of the given base.

6. 343; base 7 7. 10,000; base 1073 104

Holt Algebra 1


Evaluate expressions containing square roots.Classify numbers within the real number system.

Objectives

Holt Algebra 1


A number that is multiplied by itself to form aproduct is called a square root of that product.The operations of squaring and finding a squareroot are inverse operations.

The radical symbol , is used to represent square roots. Positive real numbers have twosquare roots.4 4 = 42 = 16 = 4 Positive square

root of 16

(–4)(–4) = (–4)2 = 16 = –4 Negative squareroot of 16–

Holt Algebra 1


A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table.

002

112

100422

932

1642

2552

3662

4972

6482

8192 102

The nonnegative square root is represented by . The negative square root is represented by – .

Holt Algebra 1


The expression does not representa real number because there is no real number that can be multiplied by itself to form a product of –36.

Reading Math

Holt Algebra 1

1-5 Square Roots and Real NumbersExample 1: Finding Square Roots of

Perfect Squares

Find each square root.

42 = 16

32 = 9

Think: What number squared equals 16?

Positive square root positive 4.

Think: What is the opposite of the square root of 9?

Negative square root negative 3.

A.

= 4B.

= –3

Holt Algebra 1


Find the square root.

Think: What number squared equals ?25

81

Positive square root positive .59

Example 1C: Finding Square Roots of Perfect Squares

Holt Algebra 1


Find the square root.Check It Out! Example 1

22 = 4 Think: What number squaredequals 4?

Positive square root positive 2. = 2

52 = 25 Think: What is the opposite of the square root of 25?

1a.

1b.

Negative square root negative 5.

Holt Algebra 1


The square roots of many numbers like , are not whole numbers. A calculator can approximate the value of as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.

Holt Algebra 1

1-5 Square Roots and Real NumbersAll numbers that can be represented on a number line are called real numbers and can be classified according to their characteristics.

Natural numbers are the counting numbers: 1, 2, 3, …

Whole numbers are the natural numbers and zero: 0, 1, 2, 3, …

Integers are whole numbers and their opposites: –3, –2, –1, 0, 1, 2, 3, …

Holt Algebra 1

1-5 Square Roots and Real NumbersRational numbers can be expressed in the form ,where a and b are both integers and b ≠ 0: , , .

ab

12

71

910

Holt Algebra 1


Terminating decimals are rational numbers in decimal form that have a finite number of digits: 1.5, 2.75, 4.0

Repeating decimals are rational numbers in decimal form that have a block of one or more digits that repeat continuously: 1.3, 0.6, 2.14

Irrational numbers cannot be expressed in the form . They include square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat: , ,

ba

Holt Algebra 1

1-5 Square Roots and Real NumbersExample 3: Classifying Real Numbers

Write all classifications that apply to each Real number.

A. –32

–32 = – = –32.0

32 1

32 can be written as a fraction and a decimal.

rational number, integer, terminating decimalB. 5

5 = = 5.051


rational number, integer, whole number, naturalnumber, terminating decimal

Holt Algebra 1


Write all classifications that apply to each real number.

3a. 7 49

rational number, repeating decimal3b. –12

rational number, terminating decimal, integer

irrational number


3c.

67 9 = 7.444… = 7.4

7 can be written as a repeating decimal.

49

–12 = – = –12.0 12 1


= 3.16227766… The digits continue with no pattern.

Holt Algebra 1

1-6 Order of Operations

Find each square root.1. 2. 3. 4.12 -8 3

7 – 12

5. The area of a square piece of cloth is 68 in2. How long is each side of the piece of cloth? Round your answer to the nearest tenth of aninch.

8.2 in.

Warm-Up

Write all classifications that apply to each real number.6. 1

7. –3.898.

rational, integer, whole number, natural number, terminating decimal

rational, repeating decimalirrational

Holt Algebra 1


Use the order of operations to simplify expressions.

Objective

Holt Algebra 1


When a numerical or algebraic expression containsmore than one operation symbol, the order of operations tells which operation to perform first.

First:

Second:

Third:

Fourth:

Perform operations inside grouping symbols.

Evaluate powers.

Perform multiplication and division from left to right.

Perform addition and subtraction from left to right.

Order of Operations

Holt Algebra 1


Grouping symbols include parentheses ( ), brackets [ ], and braces { }. If an expression contains more than one set of grouping symbols, evaluate the expression from the innermost set first.

Holt Algebra 1


Helpful Hint The first letter of these words can help you remember the order of operations.

PleaseExcuseMy DearAuntSally

ParenthesesExponentsMultiplyDivideAddSubtract

Holt Algebra 1


5.4 – 32 + 6.2

5.4 – 32 + 6.2 There are no groupingsymbols.

5.4 – 9 + 6.2 Simplify powers.

–3.6 + 6.2

2.6

Subtract

Add.

Check It Out! Example 1bSimplify the expression.

Holt Algebra 1


–20 ÷ [–2(4 + 1)]

–20 ÷ [–2(4 + 1)] There are two sets of groupingsymbols.

–20 ÷ [–2(5)] Perform the operations in theinnermost set.

–20 ÷ –10

2

Perform the operation insidethe brackets.

Divide.

Check It Out! Example 1cSimplify the expression.

Holt Algebra 1


(x · 22) ÷ (2 + 6) for x = 6

Check It Out! Example 2bEvaluate the expression for the given value of x.

(x · 22) ÷ (2 + 6)

(6 · 22) ÷ (2 + 6)

(6 · 4) ÷ (2 + 6)

(24) ÷ (8) 3

First substitute 6 for x.

Square two.

Perform the operations inside the parentheses.Divide.

Holt Algebra 1


Fraction bars, radical symbols, and absolute-value symbols can also be used as grouping symbols. Remember that a fraction bar indicates division.

Holt Algebra 1


Simplify.2(–4) + 22 42 – 92(–4) + 22 42 – 9 –8 + 22 42 – 9 –8 + 22 16 – 9

14 72

Example 3A: Simplifying Expressions with Other Grouping Symbols

The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing.

Multiply to simplify the numerator.

Evaluate the power in the denominator.

Add to simplify the numerator. Subtract to simplify the denominator.

Divide.

Holt Algebra 1

1-6 Order of Operations Example 3B: Simplifying Expressions with Other Grouping Symbols

Simplify.3|42 + 8 ÷ 2|

3|42 + 8 ÷ 2|

3|16 + 8 ÷ 2|3|16 + 4|

3|20|3 · 2060

The absolute-value symbols act as grouping symbols.

Evaluate the power.Divide within the absolute-value symbols.

Add within the absolute-symbols.Write the absolute value of 20.Multiply.

Holt Algebra 1

1-6 Order of OperationsExample 4: Translating from Words to Math

Translate each word phrase into a numerical or algebraic expression.A. the sum of the quotient of 12 and –3 and

the square root of 25

Show the quotient being added to .

B. the difference of y and the product of 4 and

Use parentheses so that the product is evaluated first.

Holt Algebra 1

1-6 Order of OperationsCheck It Out! Example 4

Translate the word phrase into a numerical or algebraic expression: the product of 6.2 and the sum of 9.4 and 8.

6.2(9.4 + 8)Use parentheses to show that the sum of 9.4 and 8 is evaluated first.

Holt Algebra 1

1-7 Simplifying ExpressionsWarm-Up

Simply each expression.

1. 2[5 ÷ (–6 – 4)] 2. 52 – (5 + 4)

|4 – 8|3. 5 8 – 4 + 16 ÷ 22

–1 4

40Translate each word phrase into a numerical or algebraic expression.4. 3 three times the sum of –5 and n 3(–5 + n) 5. the quotient of the difference of 34 and

9 and the square root of 25

6. the volume of a storage box can be found using the expression l · w(w + 2). Find the volume of the box if l = 3 feet and w = 2 feet.

24 cubic feet

Holt Algebra 1

1-7 Simplifying Expressions

Use the Commutative, Associative, and Distributive Properties to simplify expressions.Combine like terms.

Objectives

Holt Algebra 1


Holt Algebra 1


Helpful Hint

Compatible numbers help you do math mentally. Try to make multiples of 5 or 10. They are simpler to use when multiplying.

Holt Algebra 1

1-7 Simplifying ExpressionsCheck It Out! Example 1a

Simplify.

21

Use the Commutative Property.

Use the Associative Property to make groups of compatible numbers.

Holt Algebra 1

1-7 Simplifying ExpressionsCheck It Out! Example 1b

Simplify. 410 + 58 + 90 + 2

410 + 90 + 58 + 2

(410 + 90) + (58 + 2)

(500) + (60)

560

Use the Commutative Property.

Use the Associative Property to make groups of compatible numbers.

Holt Algebra 1


The Distributive Property is used with Addition to Simplify Expressions.

The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

Holt Algebra 1

1-7 Simplifying ExpressionsExample 2A: Using the Distributive Property with

Mental Math Write the product using the Distributive Property. Then simplify.

5(59)

5(50 + 9)5(50) + 5(9)

250 + 45

295

Rewrite 59 as 50 + 9.

Use the Distributive Property.

Multiply.

Add.

Holt Algebra 1


8(33)

8(30 + 3)

8(30) + 8(3)240 + 24

264

Rewrite 33 as 30 + 3.

Use the Distributive Property.

Multiply.

Add.

Example 2B: Using the Distributive Property with Mental Math

Write the product using the Distributive Property. Then simplify.

Holt Algebra 1


The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms.

4x – 3x + 2

Like terms Constant

Holt Algebra 1


A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1.

1x2 + 3x

Coefficients

Holt Algebra 1


Using the Distributive Property can help you combine like terms. You can factor out the common factors to simplify the expression.

7x2 – 4x2 = (7 – 4)x2

= (3)x2

= 3x2

Factor out x2 from both terms.

Perform operations in parenthesis.

Notice that you can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same.

Holt Algebra 1

1-7 Simplifying ExpressionsExample 3A: Combining Like Terms

Simplify the expression by combining like terms.

72p – 25p

72p – 25p

47p

72p and 25p are like terms.

Subtract the coefficients.

Holt Algebra 1

1-7 Simplifying ExpressionsExample 3B: Combining Like Terms

Simplify the expression by combining like terms.

A variable without a coefficient has a coefficient of 1.

Write 1 as .

Add the coefficients.

and are like terms.

Holt Algebra 1

1-7 Simplifying ExpressionsExample 4: Simplifying Algebraic Expressions

Simplify 14x + 4(2 + x). Justify each step.

14x + 4(2) + 4(x) Distributive Property

Multiply.Commutative Property

Associative PropertyCombine like terms.

14x + 8 + 4x

(14x + 4x) + 8 14x + 4x + 8

18x + 8

14x + 4(2 + x)1. 2. 3. 4. 5. 6.

Procedure Justification

Holt Algebra 1


–12x – 5x + x + 3a Commutative Property

Combine like terms.–16x + 3a

–12x – 5x + 3a + x1. 2. 3.

Procedure Justification

Check It Out! Example 4bSimplify −12x – 5x + 3a + x. Justify each step.

Holt Algebra 1

1-8 Introduction to Functions

Warm-UpSimplify each expression.1. 165 +27 + 3 + 5

2. Write each product using the Distributive Property. Then simplify. 3. 5($1.99)4. 6(13)

200

8

5($2) – 5($0.01) = $9.956(10) + 6(3) = 78

Holt Algebra 1


Graph ordered pairs in the coordinate plane.Graph functions from ordered pairs.

Objectives

Holt Algebra 1


The coordinate plane is formed by the intersection of two perpendicular number lines called axes. The point of intersection, called the origin, is at 0 on each number line. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis.

Holt Algebra 1


Points on the coordinate plane are described using ordered pairs. An ordered pair consists of an x-coordinate and a y-coordinate and is written (x, y). Points are often named by a capital letter.

The x-coordinate tells how many units to move left or right from the origin. The y-coordinate tells how many units to move up or down.

Reading Math

Holt Algebra 1

1-8 Introduction to FunctionsExample 1: Graphing Points in the Coordinate Plane

Graph each point.A. T(–4, 4)

Start at the origin.Move 4 units left and 4 units up.

B. U(0, –5)Start at the origin.Move 5 units down.

•T(–4, 4)

•U(0, –5)

C. V (–2, –3)Start at the origin.Move 2 units left and 3 units down.

•V(–2, −3)

Holt Algebra 1

1-8 Introduction to FunctionsExample 2: Locating Points in the Coordinate Plane

Name the quadrant in which each point lies.

A. EQuadrant ll

B. Fno quadrant (y-axis)

C. GQuadrant l

D. HQuadrant lll

•E•F

•H

•Gx

y

Holt Algebra 1

1-8 Introduction to FunctionsAn equation that contains two variables can be used as a rule to generate ordered pairs. When you substitute a value for x, you generate a value for y. The value substituted for x is called the input, and the value generated for y is called the output.

y = 10x + 5Output Input

In a function, the value of y (the output) is determined by the value of x (the input). All of the equations in this lesson represent functions.

Holt Algebra 1

1-8 Introduction to FunctionsExample 3: Art Application

An engraver charges a setup fee of $10 plus $2 for every word engraved. Write a rule for the engraver’s fee. Write ordered pairs for the engraver’s fee when there are 5, 10, 15, and 20 words engraved.

Let y represent the engraver’s fee and x represent the number of words engraved.

Engraver’s fee is $10 plus $2 for each word

y = 10 + 2 · x

y = 10 + 2x

Holt Algebra 1


The engraver’s fee is determined by the number of words in the engraving. So the number of words is the input and the engraver’s fee is the output.

Writing Math

Holt Algebra 1

1-8 Introduction to FunctionsExample 3 Continued

Number ofWordsEngraved

Rule Charges Ordered Pair

x (input) y = 10 + 2x y (output) (x, y)y = 10 + 2(5)5 20 (5, 20)

y = 10 + 2(10)10 30 (10, 30)

y = 10 + 2(15)15 40 (15, 40)

y = 10 + 2(20)20 50 (20, 50)

Holt Algebra 1


When you graph ordered pairs generated by a function, they may create a pattern.

Holt Algebra 1

1-8 Introduction to FunctionsExample 4A: Generating and Graphing Ordered Pairs Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern.y = 2x + 1; x = –2, –1, 0, 1, 2

–2–1012

2(–2) + 1 = –3 (–2, –3)(–1, –1)(0, 1)(1, 3)(2, 5)

2(–1) + 1 = –12(0) + 1 = 12(1) + 1 = 32(2) + 1 = 5

•

•

•

•

•

Input Output Ordered Pair

x y (x, y)

The points form a line.

Holt Algebra 1

1-8 Introduction to FunctionsExample 4B: Generating and Graphing Ordered Pairs y = x2 – 3; x = –2, –1, 0, 1, 2

–2–1012

(–2)2 – 3 = 1 (–2, 1)(–1, –2)(0, –3)(1, –2)(2, 1)

(–1)2 – 3 = –2(0)2 – 3 = –3(1)2 – 3 = –2(2)2 – 3 = 1


x y (x, y)

The points form a U shape.

Holt Algebra 1

1-8 Introduction to FunctionsExample 4C: Generating and Graphing Ordered Pairs y = |x – 2|; x = 0, 1, 2, 3, 4

01234

|0 – 2| = 2 (0, 2)(1, 1)(2, 0)(3, 1)(4, 2)

|1 – 2| = 1|2 – 2| = 0|3 – 2| = 1|4 – 2| = 2


x y (x, y)

The points form a V shape.

Holt Algebra 1


A(2,2) G(3,2) M(4,2) S(5,2) Y(6,2)B(2,3) H(3,3) N(4,3) T(5,3) Z(6,3)C(2,4) I(3,4) O(4,4) U(5,4)D(2,5) J(3,5) P(4,5) V(5,5)E(2,6) K(3,6) Q(4,6) W(5,6)F(2,7) L(3,7) R(4,7) X(5,7)

Documents

Warm Up Add or subtract. 1. 6 + 104 2. 12(9) 3. 23 – 8 4