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Prealgebra

5.4 Solve Equations with Decimals

Prealgebra5.4 Solve Equations with Decimals
  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:
  • Determine whether a decimal is a solution of an equation
  • Solve equations with decimals
  • Translate to an equation and solve
Be Prepared 5.4

Before you get started, take this readiness quiz.

  1. Evaluate x+23whenx=14.x+23whenx=14.
    If you missed this problem, review Example 4.77.
  2. Evaluate 15y15y when y=−5.y=−5.
    If you missed this problem, review Example 3.41.
  3. Solve n7=42.n7=42.
    If you missed this problem, review Example 4.99.

Determine Whether a Decimal is a Solution of an Equation

Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals.

Now that we’ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We’ll list these steps here again for easy reference.

How To

Determine whether a number is a solution to an equation.

  1. Step 1. Substitute the number for the variable in the equation.
  2. Step 2. Simplify the expressions on both sides of the equation.
  3. Step 3. Determine whether the resulting equation is true.
    • If so, the number is a solution.
    • If not, the number is not a solution.

Example 5.40

Determine whether each of the following is a solution of x0.7=1.5:x0.7=1.5:

x=1x=1x=−0.8x=−0.8x=2.2x=2.2

Try It 5.79

Determine whether each value is a solution of the given equation.

x0.6=1.3:x0.6=1.3:x=0.7x=0.7x=1.9x=1.9x=−0.7x=−0.7

Try It 5.80

Determine whether each value is a solution of the given equation.

y0.4=1.7:y0.4=1.7:y=2.1y=2.1y=1.3y=1.3−1.3−1.3

Solve Equations with Decimals

In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.

Properties of Equality

Subtraction Property of Equality
For any numbers a,b,andc,a,b,andc,
If a=b,a=b, then ac=bc.ac=bc.
Addition Property of Equality
For any numbers a,b,andc,a,b,andc,
If a=b,a=b, then a+c=b+c.a+c=b+c.
The Division Property of Equality
For any numbers a,b,andc,andc0a,b,andc,andc0
If a=b,a=b, then ac=bcac=bc
The Multiplication Property of Equality
For any numbers a,b,andc,a,b,andc,
If a=b,a=b, then ac=bcac=bc

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

Example 5.41

Solve: y+2.3=−4.7.y+2.3=−4.7.

Try It 5.81

Solve: y+2.7=−5.3.y+2.7=−5.3.

Try It 5.82

Solve: y+3.6=−4.8.y+3.6=−4.8.

Example 5.42

Solve: a4.75=−1.39.a4.75=−1.39.

Try It 5.83

Solve: a3.93=−2.86.a3.93=−2.86.

Try It 5.84

Solve: n3.47=−2.64.n3.47=−2.64.

Example 5.43

Solve: −4.8=0.8n.−4.8=0.8n.

Try It 5.85

Solve: −8.4=0.7b.−8.4=0.7b.

Try It 5.86

Solve: −5.6=0.7c.−5.6=0.7c.

Example 5.44

Solve: p1.8=−6.5.p1.8=−6.5.

Try It 5.87

Solve: c−2.6=−4.5.c−2.6=−4.5.

Try It 5.88

Solve: b−1.2=−5.4.b−1.2=−5.4.

Translate to an Equation and Solve

Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

Example 5.45

Translate and solve: The difference of nn and 4.34.3 is 2.1.2.1.

Try It 5.89

Translate and solve: The difference of yy and 4.94.9 is 2.8.2.8.

Try It 5.90

Translate and solve: The difference of zz and 5.75.7 is 3.4.3.4.

Example 5.46

Translate and solve: The product of −3.1−3.1 and xx is 5.27.5.27.

Try It 5.91

Translate and solve: The product of −4.3−4.3 and xx is 12.04.12.04.

Try It 5.92

Translate and solve: The product of −3.1−3.1 and mm is 26.66.26.66.

Example 5.47

Translate and solve: The quotient of pp and −2.4−2.4 is 6.5.6.5.

Try It 5.93

Translate and solve: The quotient of qq and −3.4−3.4 is 4.5.4.5.

Try It 5.94

Translate and solve: The quotient of rr and −2.6−2.6 is 2.5.2.5.

Example 5.48

Translate and solve: The sum of nn and 2.92.9 is 1.7.1.7.

Try It 5.95

Translate and solve: The sum of jj and 3.83.8 is 2.6.2.6.

Try It 5.96

Translate and solve: The sum of kk and 4.74.7 is 0.3.0.3.

Section 5.4 Exercises

Practice Makes Perfect

Determine Whether a Decimal is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

285.

x0.8=2.3x0.8=2.3
x=2x=2x=−1.5x=−1.5x=3.1x=3.1

286.

y+0.6=−3.4y+0.6=−3.4
y=−4y=−4y=−2.8y=−2.8y=2.6y=2.6

287.

h1.5=−4.3h1.5=−4.3
h=6.45h=6.45h=−6.45h=−6.45h=−2.1h=−2.1

288.

0.75k=−3.60.75k=−3.6
k=−0.48k=−0.48k=−4.8k=−4.8k=−2.7k=−2.7

Solve Equations with Decimals

In the following exercises, solve the equation.

289.

y+2.9=5.7y+2.9=5.7

290.

m+4.6=6.5m+4.6=6.5

291.

f+3.45=2.6f+3.45=2.6

292.

h+4.37=3.5h+4.37=3.5

293.

a+6.2=−1.7a+6.2=−1.7

294.

b+5.8=−2.3b+5.8=−2.3

295.

c+1.15=−3.5c+1.15=−3.5

296.

d+2.35=−4.8d+2.35=−4.8

297.

n2.6=1.8n2.6=1.8

298.

p3.6=1.7p3.6=1.7

299.

x0.4=−3.9x0.4=−3.9

300.

y0.6=−4.5y0.6=−4.5

301.

j1.82=−6.5j1.82=−6.5

302.

k3.19=−4.6k3.19=−4.6

303.

m0.25=−1.67m0.25=−1.67

304.

q0.47=−1.53q0.47=−1.53

305.

0.5x=3.50.5x=3.5

306.

0.4p=9.20.4p=9.2

307.

−1.7c=8.5−1.7c=8.5

308.

−2.9x=5.8−2.9x=5.8

309.

−1.4p=−4.2−1.4p=−4.2

310.

−2.8m=−8.4−2.8m=−8.4

311.

−120=1.5q−120=1.5q

312.

−75=1.5y−75=1.5y

313.

0.24x=4.80.24x=4.8

314.

0.18n=5.40.18n=5.4

315.

−3.4z=−9.18−3.4z=−9.18

316.

−2.7u=−9.72−2.7u=−9.72

317.

a0.4=−20a0.4=−20

318.

b0.3=−9b0.3=−9

319.

x0.7=−0.4x0.7=−0.4

320.

y0.8=−0.7y0.8=−0.7

321.

p5=−1.65p5=−1.65

322.

q4=−5.92q4=−5.92

323.

r1.2=−6r1.2=−6

324.

s1.5=−3s1.5=−3

Mixed Practice

In the following exercises, solve the equation. Then check your solution.

325.

x5=−11x5=−11

326.

25=x+3425=x+34

327.

p+8=−2p+8=−2

328.

p+23=112p+23=112

329.

−4.2m=−33.6−4.2m=−33.6

330.

q+9.5=−14q+9.5=−14

331.

q+56=112q+56=112

332.

8.615=d8.615=d

333.

78m=11078m=110

334.

j6.2=−3j6.2=−3

335.

23=y+3823=y+38

336.

s1.75=−3.2s1.75=−3.2

337.

1120=f1120=f

338.

−3.6b=2.52−3.6b=2.52

339.

−4.2a=3.36−4.2a=3.36

340.

−9.1n=−63.7−9.1n=−63.7

341.

r1.25=−2.7r1.25=−2.7

342.

14n=71014n=710

343.

h3=−8h3=−8

344.

y7.82=−16y7.82=−16

Translate to an Equation and Solve

In the following exercises, translate and solve.

345.

The difference of nn and 1.91.9 is 3.4.3.4.

346.

The difference nn and 1.51.5 is 0.8.0.8.

347.

The product of −6.2−6.2 and xx is −4.96.−4.96.

348.

The product of −4.6−4.6 and xx is −3.22.−3.22.

349.

The quotient of yy and −1.7−1.7 is −5.−5.

350.

The quotient of zz and −3.6−3.6 is 3.3.

351.

The sum of nn and −7.3−7.3 is 2.4.2.4.

352.

The sum of nn and −5.1−5.1 is 3.8.3.8.

Everyday Math

353.

Shawn bought a pair of shoes on sale for $78$78. Solve the equation 0.75p=780.75p=78 to find the original price of the shoes, p.p.

354.

Mary bought a new refrigerator. The total price including sales tax was $1,350.$1,350. Find the retail price, r,r, of the refrigerator before tax by solving the equation 1.08r=1,350.1.08r=1,350.

Writing Exercises

355.

Think about solving the equation 1.2y=60,1.2y=60, but do not actually solve it. Do you think the solution should be greater than 6060 or less than 60?60? Explain your reasoning. Then solve the equation to see if your thinking was correct.

356.

Think about solving the equation 0.8x=200,0.8x=200, but do not actually solve it. Do you think the solution should be greater than 200200 or less than 200?200? Explain your reasoning. Then solve the equation to see if your thinking was correct.

Self Check

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

.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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