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Prealgebra

2.2 Evaluate, Simplify, and Translate Expressions

Prealgebra2.2 Evaluate, Simplify, and Translate Expressions
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Evaluate algebraic expressions
  • Identify terms, coefficients, and like terms
  • Simplify expressions by combining like terms
  • Translate word phrases to algebraic expressions
Be Prepared 2.2

Before you get started, take this readiness quiz.

  1. Is n÷5n÷5 an expression or an equation?
    If you missed this problem, review Example 2.4.
  2. Simplify 45.45.
    If you missed this problem, review Example 2.7.
  3. Simplify 1+89.1+89.
    If you missed this problem, review Example 2.8.

Evaluate Algebraic Expressions

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

Example 2.13

Evaluate x+7x+7 when

  1. x=3x=3
  2. x=12x=12
Try It 2.25

Evaluate:

y+4wheny+4when

  1. y=6y=6
  2. y=15y=15
Try It 2.26

Evaluate:

a5whena5when

  1. a=9a=9
  2. a=17a=17

Example 2.14

Evaluate 9x2,when9x2,when

  1. x=5x=5
  2. x=1x=1
Try It 2.27

Evaluate:

8x3,when8x3,when

  1. x=2x=2
  2. x=1x=1
Try It 2.28

Evaluate:

4y4,when4y4,when

  1. y=3y=3
  2. y=5y=5

Example 2.15

Evaluate x2x2 when x=10.x=10.

Try It 2.29

Evaluate:

x2whenx=8.x2whenx=8.

Try It 2.30

Evaluate:

x3whenx=6.x3whenx=6.

Example 2.16

Evaluate2xwhenx=5.Evaluate2xwhenx=5.

Try It 2.31

Evaluate:

2xwhenx=6.2xwhenx=6.

Try It 2.32

Evaluate:

3xwhenx=4.3xwhenx=4.

Example 2.17

Evaluate3x+4y6whenx=10andy=2.Evaluate3x+4y6whenx=10andy=2.

Try It 2.33

Evaluate:

2x+5y4whenx=11andy=32x+5y4whenx=11andy=3

Try It 2.34

Evaluate:

5x2y9whenx=7andy=85x2y9whenx=7andy=8

Example 2.18

Evaluate2x2+3x+8whenx=4.Evaluate2x2+3x+8whenx=4.

Try It 2.35

Evaluate:

3x2+4x+1whenx=3.3x2+4x+1whenx=3.

Try It 2.36

Evaluate:

6x24x7whenx=2.6x24x7whenx=2.

Identify Terms, Coefficients, and Like Terms

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are 7,y,5x2,9a,and13xy.7,y,5x2,9a,and13xy.

The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term 3x3x is 3.3. When we write x,x, the coefficient is 1,1, since x=1x.x=1x. Table 2.5 gives the coefficients for each of the terms in the left column.

Term Coefficient
77 77
9a9a 99
yy 11
5x25x2 55
Table 2.5

An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. Table 2.6 gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

Expression Terms
77 77
yy yy
x+7x+7 x,7x,7
2x+7y+42x+7y+4 2x,7y,42x,7y,4
3x2+4x2+5y+33x2+4x2+5y+3 3x2,4x2,5y,33x2,4x2,5y,3
Table 2.6

Example 2.19

Identify each term in the expression 9b+15x2+a+6.9b+15x2+a+6. Then identify the coefficient of each term.

Try It 2.37

Identify all terms in the given expression, and their coefficients:

4x+3b+24x+3b+2

Try It 2.38

Identify all terms in the given expression, and their coefficients:

9a+13a2+a39a+13a2+a3

Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

5x,7,n2,4,3x,9n25x,7,n2,4,3x,9n2

Which of these terms are like terms?

  • The terms 77 and 44 are both constant terms.
  • The terms 5x5x and 3x3x are both terms with x.x.
  • The terms n2n2 and 9n29n2 both have n2.n2.

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms 5x,7,n2,4,3x,9n2,5x,7,n2,4,3x,9n2,

7and4are like terms.7and4are like terms.
5xand3xare like terms.5xand3xare like terms.
n2and9n2are like terms.n2and9n2are like terms.

Like Terms

Terms that are either constants or have the same variables with the same exponents are like terms.

Example 2.20

Identify the like terms:

  1. y3,7x2,14,23,4y3,9x,5x2y3,7x2,14,23,4y3,9x,5x2
  2. 4x2+2x+5x2+6x+40x+8xy4x2+2x+5x2+6x+40x+8xy
Try It 2.39

Identify the like terms in the list or the expression:

9,2x3,y2,8x3,15,9y,11y29,2x3,y2,8x3,15,9y,11y2

Try It 2.40

Identify the like terms in the list or the expression:

4x3+8x2+19+3x2+24+6x34x3+8x2+19+3x2+24+6x3

Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think 3x+6x3x+6x would simplify to? If you thought 9x,9x, you would be right!

We can see why this works by writing both terms as addition problems.

The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.

Add the coefficients and keep the same variable. It doesn’t matter what xx is. If you have 33 of something and add 66 more of the same thing, the result is 99 of them. For example, 33 oranges plus 66 oranges is 99 oranges. We will discuss the mathematical properties behind this later.

The expression 3x+6x3x+6x has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.

The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.

Now it is easier to see the like terms to be combined.

How To

Combine like terms.

  1. Step 1. Identify like terms.
  2. Step 2. Rearrange the expression so like terms are together.
  3. Step 3. Add the coefficients of the like terms.

Example 2.21

Simplify the expression: 3x+7+4x+5.3x+7+4x+5.

Try It 2.41

Simplify:

7x+9+9x+87x+9+9x+8

Try It 2.42

Simplify:

5y+2+8y+4y+55y+2+8y+4y+5

Example 2.22

Simplify the expression: 7x2+8x+x2+4x.7x2+8x+x2+4x.

Try It 2.43

Simplify:

3x2+9x+x2+5x3x2+9x+x2+5x

Try It 2.44

Simplify:

11y2+8y+y2+7y11y2+8y+y2+7y

Translate Words to Algebraic Expressions

In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in Table 2.7.

Operation Phrase Expression
Addition aa plus bb
the sum of aa and bb
aa increased by bb
bb more than aa
the total of aa and bb
bb added to aa
a+ba+b
Subtraction aa minus bb
the difference of aa and bb
bb subtracted from aa
aa decreased by bb
bb less than aa
abab
Multiplication aa times bb
the product of aa and bb
abab, abab, a(b)a(b), (a)(b)(a)(b)
Division aa divided by bb
the quotient of aa and bb
the ratio of aa and bb
bb divided into aa
a÷ba÷b, a/ba/b, abab, baba
Table 2.7

Look closely at these phrases using the four operations:

  • the sum of aa and bb
  • the difference of aa and bb
  • the product of aa and bb
  • the quotient of aa and bb

Each phrase tells you to operate on two numbers. Look for the words of and and to find the numbers.

Example 2.23

Translate each word phrase into an algebraic expression:

  1. the difference of 2020 and 44
  2. the quotient of 10x10x and 33
Try It 2.45

Translate the given word phrase into an algebraic expression:

  1. the difference of 4747 and 4141
  2. the quotient of 5x5x and 22
Try It 2.46

Translate the given word phrase into an algebraic expression:

  1. the sum of 1717 and 1919
  2. the product of 77 and xx

How old will you be in eight years? What age is eight more years than your age now? Did you add 88 to your present age? Eight more than means eight added to your present age.

How old were you seven years ago? This is seven years less than your age now. You subtract 77 from your present age. Seven less than means seven subtracted from your present age.

Example 2.24

Translate each word phrase into an algebraic expression:

  1. Eight more than yy
  2. Seven less than 9z9z
Try It 2.47

Translate each word phrase into an algebraic expression:

  1. Eleven more than xx
  2. Fourteen less than 11a11a
Try It 2.48

Translate each word phrase into an algebraic expression:

  1. 1919 more than jj
  2. 2121 less than 2x2x

Example 2.25

Translate each word phrase into an algebraic expression:

  1. five times the sum of mm and nn
  2. the sum of five times mm and nn
Try It 2.49

Translate the word phrase into an algebraic expression:

  1. four times the sum of pp and qq
  2. the sum of four times pp and qq
Try It 2.50

Translate the word phrase into an algebraic expression:

  1. the difference of two times xand 8xand 8
  2. two times the difference of xand8xand8

Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an algebraic expression. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples.

Example 2.26

The height of a rectangular window is 66 inches less than the width. Let ww represent the width of the window. Write an expression for the height of the window.

Try It 2.51

The length of a rectangle is 55 inches less than the width. Let ww represent the width of the rectangle. Write an expression for the length of the rectangle.

Try It 2.52

The width of a rectangle is 22 meters greater than the length. Let ll represent the length of the rectangle. Write an expression for the width of the rectangle.

Example 2.27

Blanca has dimes and quarters in her purse. The number of dimes is 22 less than 55 times the number of quarters. Let qq represent the number of quarters. Write an expression for the number of dimes.

Try It 2.53

Geoffrey has dimes and quarters in his pocket. The number of dimes is seven less than six times the number of quarters. Let qq represent the number of quarters. Write an expression for the number of dimes.

Try It 2.54

Lauren has dimes and nickels in her purse. The number of dimes is eight more than four times the number of nickels. Let nn represent the number of nickels. Write an expression for the number of dimes.

Media Access Additional Online Resources

Section 2.2 Exercises

Practice Makes Perfect

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.

69.

7x+8whenx=27x+8whenx=2

70.

9x+7whenx=39x+7whenx=3

71.

5x4whenx=65x4whenx=6

72.

8x6whenx=78x6whenx=7

73.

x2whenx=12x2whenx=12

74.

x3whenx=5x3whenx=5

75.

x5whenx=2x5whenx=2

76.

x4whenx=3x4whenx=3

77.

3xwhenx=33xwhenx=3

78.

4xwhenx=24xwhenx=2

79.

x2+3x7whenx=4x2+3x7whenx=4

80.

x2+5x8whenx=6x2+5x8whenx=6

81.

2x+4y5whenx=7,y=82x+4y5whenx=7,y=8

82.

6x+3y9whenx=6,y=96x+3y9whenx=6,y=9

83.

(xy)2whenx=10,y=7(xy)2whenx=10,y=7

84.

(x+y)2whenx=6,y=9(x+y)2whenx=6,y=9

85.

a2+b2whena=3,b=8a2+b2whena=3,b=8

86.

r2s2whenr=12,s=5r2s2whenr=12,s=5

87.

2l+2wwhenl=15,w=122l+2wwhenl=15,w=12

88.

2l+2wwhenl=18,w=142l+2wwhenl=18,w=14

Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.

89.

15x2+6x+215x2+6x+2

90.

11x2+8x+511x2+8x+5


91.

10y3+y+210y3+y+2

92.

9y3+y+59y3+y+5

In the following exercises, identify the coefficient of the given term.

93.

8a8a

94.

13m13m

95.

5r25r2

96.

6x36x3

In the following exercises, identify all sets of like terms.

97.

x3,8x,14,8y,5,8x3x3,8x,14,8y,5,8x3

98.

6z,3w2,1,6z2,4z,w26z,3w2,1,6z2,4z,w2

99.

9a,a2,16ab,16b2,4ab,9b29a,a2,16ab,16b2,4ab,9b2

100.

3,25r2,10s,10r,4r2,3s3,25r2,10s,10r,4r2,3s

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.

101.

10x+3x10x+3x

102.

15x+4x15x+4x

103.

17a+9a17a+9a

104.

18z+9z18z+9z

105.

4c+2c+c4c+2c+c

106.

6y+4y+y6y+4y+y

107.

9x+3x+89x+3x+8

108.

8a+5a+98a+5a+9

109.

7u+2+3u+17u+2+3u+1

110.

8d+6+2d+58d+6+2d+5

111.

7p+6+5p+47p+6+5p+4

112.

8x+7+4x58x+7+4x5

113.

10a+7+5a2+7a410a+7+5a2+7a4


114.

7c+4+6c3+9c17c+4+6c3+9c1

115.

3x2+12x+11+14x2+8x+53x2+12x+11+14x2+8x+5

116.

5b2+9b+10+2b2+3b45b2+9b+10+2b2+3b4

Translate English Phrases into Algebraic Expressions

In the following exercises, translate the given word phrase into an algebraic expression.

117.

The sum of 8 and 12

118.

The sum of 9 and 1

119.

The difference of 14 and 9

120.

8 less than 19

121.

The product of 9 and 7

122.

The product of 8 and 7

123.

The quotient of 36 and 9

124.

The quotient of 42 and 7

125.

The difference of xx and 44

126.

33 less than xx

127.

The product of 66 and yy

128.

The product of 99 and yy

129.

The sum of 8x8x and 3x3x

130.

The sum of 13x13x and 3x3x

131.

The quotient of yy and 33

132.

The quotient of yy and 88

133.

Eight times the difference of yy and nine

134.

Seven times the difference of yy and one

135.

Five times the sum of xx and yy

136.

Nine times five less than twice xx

In the following exercises, write an algebraic expression.

137.

Adele bought a skirt and a blouse. The skirt cost $15$15 more than the blouse. Let bb represent the cost of the blouse. Write an expression for the cost of the skirt.

138.

Eric has rock and classical CDs in his car. The number of rock CDs is 33 more than the number of classical CDs. Let cc represent the number of classical CDs. Write an expression for the number of rock CDs.

139.

The number of girls in a second-grade class is 44 less than the number of boys. Let bb represent the number of boys. Write an expression for the number of girls.

140.

Marcella has 66 fewer male cousins than female cousins. Let ff represent the number of female cousins. Write an expression for the number of boy cousins.

141.

Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let nn represent the number of nickels. Write an expression for the number of pennies.

142.

Jeannette has $5$5 and $10$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let tt represent the number of tens. Write an expression for the number of fives.

Everyday Math

In the following exercises, use algebraic expressions to solve the problem.

143.

Car insurance Justin’s car insurance has a $750$750 deductible per incident. This means that he pays $750$750 and his insurance company will pay all costs beyond $750.$750. If Justin files a claim for $2,100,$2,100, how much will he pay, and how much will his insurance company pay?

144.

Home insurance Pam and Armando’s home insurance has a $2,500$2,500 deductible per incident. This means that they pay $2,500$2,500 and their insurance company will pay all costs beyond $2,500.$2,500. If Pam and Armando file a claim for $19,400,$19,400, how much will they pay, and how much will their insurance company pay?

Writing Exercises

145.

Explain why “the sum of x and y” is the same as “the sum of y and x,” but “the difference of x and y” is not the same as “the difference of y and x.” Try substituting two random numbers for xx and yy to help you explain.

146.

Explain the difference between “4“4 times the sum of xx and yy and “the sum of 44 times xx and y.”y.”

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

After reviewing this checklist, what will you do to become confident for all objectives?

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