# chapter 10 more equations and inequalities solving two-step equations 10-2 solving multistep ......

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10-1 Solving Two-StepEquations

10-2 Solving MultistepEquations

LAB Model Equations withVariables on Both Sides

10-3 Solving Equations withVariables on Both Sides

Mid-Chapter Quiz

Focus on Problem Solving

10-4 Solving MultistepInequalities

10-5 Solving for a Variable10-6 Systems of Equations Problem Solving on Location

Math-Ables

Technology Lab

Study Guide and Review

Chapter 10 Test

Performance Assessment

Standardized Test Prep

Chapter

10

Chapter Opener Onlinego.hrw.comKEYWORD: MP4 Ch10

10

496 Chapter 10

HydrologistHydrologists measure water

flow between rivers, streams, lakes,and oceans. They map their resultsto record locations and movementof water above and below theearths surface.

Hydrologists are involved inprojects such as water-resourcestudies, field irrigation, floodmanagement, soil-erosionprevention, and the study of water discharge from creeks,streams, and rivers. The tableshows the rate of water discharge for four U.S. rivers.

Discharge

River Location (

m3/s)

Glen Canyon

314.6

Hells Canyon

Snake Dam, ID

726.04

Missouri St. Joseph, MO

1751.4

Columbia The Dalles, OR

6331.65

Chapter

More Equations and InequalitiesMore Equations and Inequalities

• More Equations and Inequalities 497

Choose the best term from the list to complete each sentence.

1. A letter that represents a value that can change is called a(n) __?__.

2. A(n) __?__ has one or more variables.

3. The algebraic expression 5x 2 3y 4x 2 7 has four __?__. Since they have the same variable raised to the same power, 5x 2 and 4x 2 are __?__.

4. When you individually multiply the numbers inside parentheses by the factor outside the parentheses, you are applying the __?__.

Complete these exercises to review skills you will need for this chapter.

Distribute MultiplicationReplace each with a number so that each equation illustrates theDistributive Property.

5. 6 (11 8) 6 11 6 6. 7 (14 12) 14 12

7. 9 (6 ) 9 6 9 2 8. 14 ( 7) 14 20 14 7

Simplify Algebraic ExpressionsSimplify each expression by applying the Distributive Property andcombining like terms.

9. 3(x 2) 7x 10. 4(y 3) 8y 11. 2(z 1) 3z

12. 4(t 6) t 13. (r 3) 8r 14. 5(4 2m) 7

Connect Words and EquationsWrite an equation to represent each situation.

15. The perimeter P of a rectangle is the sum of twice the length andtwice the width w.

16. The volume V of a rectangular prism is the product of its threedimensions: length , width w, and height h.

17. The surface area S of a sphere is the product of 4 and the square ofthe radius r.

18. The cost c of a telegram of 18 words is the cost f of the first 10 wordsplus the cost a of each additional word.

algebraic expression

Distributive Property

like terms

terms

variable

• 498 Chapter 10 More Equations and Inequalities

10-1 Solving Two-Step Equations

Learn to solve two-step equations.

Sometimes more than one inverse operationis needed to solve an equation. Beforesolving, ask yourself, What is being done to the variable, and in what order? Thenwork backward to undo the operations.

Landscapers charge an hourly rate for labor,plus the cost of the plants. The number ofhours a landscaper worked can be found bysolving a two-step equation.

PROBLEM SOLVING APPLICATION

Chriss landscaping bill is \$380. The plants cost \$212, and the laborcost \$48 per hour. How many hours did the landscaper work?

4. Understand the Problem

The answer is the number of hours the landscaper worked on the yard.List the important information: The plants cost \$212, the labor cost\$48 per hour, and the total bill is \$380.

Let h represent the hours the landscaper worked.

Total bill Plants Labor

380 212 48h

4. Make a Plan

Think: First the variable is multiplied by 48, and then 212 is added tothe result. Work backward to solve the equation. Undo the operationsin reverse order: First subtract 212 from both sides of the equation,and then divide both sides of the new equation by 48.

4. Solve

14688

4488h

Divide to undo multiplication.

3.5 h

The landscaper worked 3.5 hours.

4. Look Back

If the landscaper worked 3.5 hours, the labor would be \$48(3.5) \$168.The sum of the plants and the labor would be \$212 \$168 \$380.

212 48h 212

48h

380 212

168

1

1

2

3

4

E X A M P L E

• 10-1 Solving Two-Step Equations 499

Solving Two-Step Equations

Solve.

A p4 5 13

Think: First the variable is divided by 4, and then 5 is added.To isolate the variable, subtract 5, and then multiply by 4.

4 p4 4 8 Multiply to undo division.

p 32

Check p4 5

?13

342 5

?13 Substitute 32 into the original equation.

8 5 ?

13

B 1.8 2.5m 1.7

Think: First the variable is multiplied by 2.5, and then 1.7 is subtracted. To isolate the variable, add 1.7, and thendivide by 2.5.

2

2.5

.5m

Divide to undo multiplication.

1.4 m

C k 94

6

Think: First 4 is added to the variable, and then the result isdivided by 9. To isolate the variable, multiply by 9, and thensubtract 4.

k

94

6

9 k 94

9 6 Multiply to undo division.k 4 54

4 4 Subtract to undo addition. k 50

3.52.5

2.5m 1.7 1.7

2.5m

1.8 1.7

3.5

E X A M P L E 2

p4 5

5p4

13 5

8

Think and Discuss

1. Describe how you would solve 4(x 2) 16.

• 500 Chapter 10 More Equations and Inequalities

10-1 ExercisesExercises Homework Help Onlinego.hrw.com Keyword: MP4 10-1

FOR EXTRA PRACTICE

see page 750

1. Joe is paid a weekly salary of \$520. He is paid an additional \$21 forevery hour of overtime he works. This week his total pay, includingregular salary and overtime pay, was \$604. How many hours ofovertime did Joe work this week?

Solve.

2. 9t 12 75 3. 2.4 1.2x 1.8 4. 7r

11 25

5. b 224

13 6. 14q 17 39 7. a 28

3 3

8. The cost of a family membership at a health club is \$58 per monthplus a one-time \$129 start-up fee. If a family spent \$651, how manymonths is their membership?

Solve.

9. m

3 2 8 10. c

21

12 11. 15g 4 46

12. h 1919

2 13. 6y 3 27 14. 9.2 4.4z 4

Solve.

15. 5w 3.8 16.3 16. 15 3x 6 17. m5 6 9

18. 2.3a 8.6 5.2 19. q

74

1 20. 9 5g 23

21. 6z 2 0 22. 52d 32

12 23. 47k 83 318

24. 8 6 p4 25. 46 3n 23 26.

7 5

s 4

27. 9y 7.2 4.5 28. 23 6h 161 29. 1 35b

15

Write an equation for each sentence, and then solve it.

30. The quotient of a number and 2, minus 9, is 14.

31. A number increased by 5 and then divided by 7 is 12.

32. The sum of 10 and 5 times a number is 25.

See Example 1

See Example 2

GUIDED PRACTICEGUIDED PRACTICE

PRACTICE AND PROBLEM SOLVINGPRACTICE AND PROBLEM SOLVING

INDEPENDENT PRACTICEINDEPENDENT PRACTICESee Example 1

See Example 2

• 10-1 Solving Two-Step Equations 501

Simplify. (Lesson 1-6)

37. x 4x 3 7x 38. 2m 4 2m 39. w 17 2 40. 5s 3r s 5r

41. TEST PREP Find the area of the parallelogram. (Lesson 6-1)

A 38 cm2 B 76 cm2 C 288 cm2 D 336 cm2

Life Science

About 20% of the more than 2500 species of snakes are venomous. The United States has 20 domestic venomous snake species, including coral snakes, rattlesnakes, copperheads, and cottonmouths.

33. The inland taipan of central Australia is the worlds most toxic venomous snake. Just 1 mg of its venom is enough to kill 1000 mice. One bite contains up to 110 mg of venom. About how many mice could be killed with the venom contained in just one inland taipan bite?

34. A rattlesnake grows a new rattle segment each time it sheds its skin. Rattlesnakes shed their skin an average of three times per year. However, segments often break off. If a rattlesnake had 44 rattle segments break off in its lifetime and it had 10 rattles when it died, approximately howmany years did the rattlesnake live?

35. All snakes shed their skin as they grow. The shed skin of a snake is an average of 10% longer than the actual snake. If the shed skin of a coral snake is 27.5 inches long, estimate the length of the coral snake.

36. CHALLENGE Black mambas feedmainly on small rodents and birds. Suppose a black mamba is 100 feet away from an animal that is running at 8 mi/h. About how long will it take for the mamba to catch the animal? (Hint: 1 mile 5280 feet)

14 cm

24 cm

12 cm

Records of Worlds Most Venomous Snakes

Category Record Type of Snake

Fastest 12 mi/h Black mamba

Longest 18 ft 9 in. King cobra

Heaviest 34 lb Eastern diamondback rattlesnake

Longest fangs 2 in. Gaboon viper

Venom is collected from snakes and injected into horses,which develop antibodies. The horses blood is sterilized to make antivenom.

MSPA_10L01_498_501 8/14/06 2:53 PM Page 501

KEYWORD: MP4 Snakes

• 502 Chapter 10 More Equations and Inequalities

10-2 Solving Multistep Equations

Learn to solve multistep equations.

To solve a complicated equation, you may have to simplify theequation