mmulti-step ulti-step florida eequations and quations and

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Name

Class Date

Chapter at a Glance

Benchmark Lesson WorktextStudent

Textbook

Remember It? 227 –228

Review of MA.7.A.3.3

7-1 Solving Two-Step Equations 296 –299

Review of MA.7.A.3.3

7-2 Simplifying Algebraic Expressions

300–303

MA.8.A.4.1 7-3 Solving Literal Equations for a Variable

229 –236 304 –307

MA.8.A.4.2 7-4 Solving Inequalities by Adding or Subtracting

237–244 310 –313

MA.8.A.4.2 7-5 Solving Inequalities by Multiplying or Dividing

245 –252 314 –317

MA.8.A.4.2 7-6 Solving Two-Step Inequalities 253 –260 318 –321

Study It! 263–265

Write About It! 266C

HA

PT

ER

7

Chapter 7 Multi-Step Equations and Inequalities 225

Multi-Step Multi-Step Equations and Equations and InequalitiesInequalities

FLORIDA

7C H A P T E R

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Vocabulary ConnectionsVocabulary Connections

Key Vocabulary Vocabulario Vokabilèalgebraic inequality desigualdad algebraica inegalite varyab

inequality desigualdad inegalite

literal equation ecuación literal ekwasyon literal

solution set conjunto solución ansanm solisyon

To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like.

1. The word literal can mean “expressed in letters.” What would you expect to see in a literal equation?

2. The prefix in- means “not.” What do you think may be true about two quantities connected in a mathematical sentence by an inequality?

3. You know that the solution of an equation is a value that makes the equation true. What do you think makes up the solution set of an inequality?

LA.8.1.6.5 The student will relate new vocabulary to familiar words….

CH

AP

TE

R 7

226 Chapter 7 Multi-Step Equations and Inequalities


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Review skills and prepare for future lessons.Review skills and prepare for future lesson

Remember It?Remember It?Name Class Date

Rev MA.7.A.3.3Solving Two-Step Equations (Student Textbook pp. 296–299)

Solve.

7x + 12 = 33 Think: First the variable is multiplied by 7, and then 12 is added. To isolate the variable, subtract 12, and then divide by 7.

7x + 12 = 33

−−−−−

-12 −−−−

-12

7x = 21

7x __ 7 = 21 __ 7

x = 3

z __ 3 - 8 = 5 Think: First the variable is divided by 3, and then 8 is subtracted. To isolate the variable, add 8, and then multiply by 3.

z __ 3 - 8 = 5

−−−−

+8 −−−

8

z __ 3 = 13

3 · z __ 3 = 3 · 13

z = 39

Solve.

1. 3m + 5 = 35 2. 55 = 7 - 6y 3. 5r + 15 = 0

4. t __ 2 + 7 = 15 5. w __ 4 - 5 = 11 6. x + 2 _____ 3 = 18

7. Jake weighed 150.7 pounds with his army boots on, and 144.9 pounds without them. What is the weight of each boot?

8. A music service charges a $2.99 monthly membership fee plus $0.05 for each song purchased. If Naomi’s charge for the month was $10.89, how many songs did she purchase?

S l

Lesson 7-1

Lesson Tutorial Videos Chapter 7 Multi-Step Equations and Inequalities 227

7-1T H R O U G H

7-2

Subtract 12 fromboth sides.

Divide both sidesby 7.

Add 8 to both sides.

Multiply both sidesby 3.


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Lesson Tutorial Videos 228 Chapter 7 Multi-Step Equations and Inequalities

Rev. MA.7.A.3.3

Simplifying Algebraic Expressions (Student Textbook pp. 300–303)

Simplify.

3(z - 6) + 2z

3z - 3(6) + 2z

3z - 18 + 2z

5z - 18

Solve.

14p - 8p = 54

6p = 54

6p

___ 6 = 54 __ 6

p = 9

Simplify.

9. 5(3m - 2) + 4m 10. 12w + 2 (w + 3)

11. 4x + 3y - 2x 12. 2 r 2 - 4t + 3 t 3

Solve.

13. 7y + y = 48 14. 8z - 2z = 42

15. 6y + y = 35 16. 9z - 3z = 48

17. The width of a soccer field should be 60% of its length. Write and simplify an expression for the perimeter of a soccer field with a length of x feet.

Si lif

Lesson 7-2

Distributive Property

3z and 2z are like terms.

Combine coefficients.

Combine like terms.

Divide both sides by 6.

NotesNotes


Go to thinkcentral.com

Tools

x 42 =+ x =+ 4y2y

2x 4x1 =+ 2x 4x=+ y

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Explore It!Explore It! Learn It!Learn It! Practice It!Practice It! Apply It!Apply It!Summarize It!Summarize It!

Explore It !Explore It !Name Class Date

7-3 Solving Literal Equations for a Variable 229

Activity 1Use algebra tiles to model each equation.

Previously, you have used algebra tiles to model one- and two-step equations with one variable. An algebra tile is just a symbol—one thing that represents something else—so if you change what the algebra tiles represent, you can use them to model equations with two variables.

1 Model x + 2y = 4y.

This equation is similar to x + 2 = 4. To To model x + 2y = 4y, use the same model, see how, model the equation x + 2 = 4. but change what each represents.

2 Model 2x + y = 4x.

First model 2x + 1 = 4x. Next sketch a model of 2x + y = 4x.

Try ThisSketch a model of each equation.

1. x + 6y = 5y 2. 4x = 6y 3. x + 3y = 2x + 4y

Model Equations with Two VariablesModel Equations with Two VariablesYou can use what you know about combining like terms to solve equations with two or more variables for one of the variables.

Solving Literal Equations for a Variable

MA.8.A.4.1 Solve literal equations for a specified variable.

7-3

Let each represent y instead of the value 1.

Each represents y instead of 1.

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x 83 =+ x 8y3y =+

x 5= x 5y=

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Activity 2Use algebra tiles to model and solve x + 3y = 8y for x.

1 Think: This equation is similar to x + 3 = 8. To solve x + 3y = 8y for x, use a model to solve x + 3 = 8.

2 To solve x + 3 = 8: To solve x + 3y = 8y:

3

Remove 3 from each side.

Remove 3y from each side.

4

Try ThisUse a model to solve each equation for the given variable. 4. x + 4y = 6y for x 5. 2x = 8y for x 6. 3x + y = x + 2y for y

Draw Conclusions 7. The solution to the equation in Activity 2 is x = 5y. How could you solve that

equation for y?

230 7-3 Solving Literal Equations for a Variable

Each represents y instead of 1.

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Learn It !Learn It !Name Class Date


Example 1

Solve each equation for the given variable.SCheck It Out!

1a. A = ℓw for w. 1b. I = Prt for P.

Solve each equation for the given variable.

A. A = ℓw for ℓ

A __ w = ℓw ___ w To isolate ℓ, divide both sides ofthe equation by w.

A __ w = ℓ

B. I = Prt for t

I _____ = Prt _____ To isolate t, divide both sides of

the equation by .

I __ Pr = t

Lesson ObjectiveSolve equations with more than one variable for a given variable.

Vocabulary

literal equation


Solving Literal Equations for a Variable (Student Textbook pp. 304–307)

7-3

Lesson Tutorial Videos


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2. The formula V = Bh gives the volume V of a rectangular prism with base area B and height h. The volume of a rectangular prism is 96 cm3. What is the height of the prism if base area is 12 cm?

Check It Out!

The area of a rectangular table is 30 ft2. What is the width of the table if the length is 7.5 feet?

A = �w Use the formula for area of a rectangle.

A ______ = �w _____ Solve the formula for w by dividing both sides by .

A __ � = w

30 ___ 7.5 = w To find the width, substitute for A and for �.

= w Simplify.

The width of the table is feet.

Example 2


Example 3


A. c = d - e for d

c = d - e

−−

+e −−−−

−−

+e To solve for d, add e to both sides of the equation.

c + e = d



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Learn It!Learn It! Practice It!Practice It! Apply It!Apply It!Summarize It!Summarize It!Explore It!Explore It!

3a. 3x - 2y = y + z for z 3b. 4t + 2s = 2t + r - s for r

Solve each equation for the given variable.SCheck It Out!



B. 4x = y + 1 - 3x for y

4x = y + 1 - 3x

−−−

+ 3x −−−−−−

−−−

+ 3x Combine like terms to simplify the equation by

7x = y + 1 adding 3x to both sides of the equation.

To solve for y, add to both sides. 7x - 1 = y

Example 4

The equation t = m + 10e gives the test score t for a student who answers m multiple-choice questions and e essay questions correctly. Solve this equation for e.

t = m + 10e

t -m = 10e

t -m _____ = 10e _____

t-m ____ 10 = e


Check It Out!

4. Solve 2k - 4d = r for k


NonexamplesExamples

Definition

LiteralEquation

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Name Class Date

Summarize It !Summarize It !Think and Discuss1. Describe a situation where solving the perimeter formula P = 2ℓ + 2w for

ℓ or w would be helpful.

2. The formula for the surface area S of a cylinder is S = 2πrh + 2πr 2 , where r is the radius and h is the height. Explain how to solve this equation for h.

3. Get Organized Complete the graphic organizer. Fill in the sections by writing the definition of a literal equation. Then give examples and nonexamples of literal equations.



7-3

LA.8.2.2.3 The student will organize information to show understanding or relationships…

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Name Class Date

Practice It !Practice It !Solving Literal Equations for a Variable



1. k = xy for x 2. F = ma for a 3. D = m __ v for m

4. 2f + g = f + h for h 5. 8 + p + 2r = 3r for p 6. y = 2x + b for b

7. 2x + 4y = C for x 8. 4s - 3t = I for s 9. Q = 1 _ 2 c + 1 _ 2 d for d

10. j = 2.5k + l for l 11. 3a - 4b = 5b + c for c 12. b + 2c - 6 = 3c for b

13. m + 2n = n + 9 for m 14. a = 3.6b + c for c 15. P = 2ℓ + 2w for ℓ

16. 6g - 4h = k for g 17. x = y + 4

____ 2 for y 18. Q = 2a + 4ac for a

19. The formula s = d __ t is used to find the speed s of an object traveling a distance d in an amount of time t. How long does it take an object to travel 300 meters if it moves at a speed of 15 m/s?

20. The formula C = 2πr is used to find the circumference of a circle. What is the radius of a circle if the circumference is 18 meters?

21. The area of a sector of a circle is given by the formula A = π r 2 S

____ 360 , where r is the radius of the circle and S is the angle measure of the sector. Solve this formula for S and then find the angle measure of a sector of a circle circle with an area of 4π cm 2 and a radius of 4 cm.

22. The equation M = 0.25q + 0.10d + 0.05n + 0.01p gives the total amount of money M in Paolo’s coin jar. Solve this equation for p.


7-3

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Apply It !Apply It !Name Class Date


1. The painting Starry Night by Vincent van Gogh is a rectangle with an area of6,716 cm 2 . The width of the painting is73 cm. What is the length of the painting?

2. The rectangular screen of a computer monitor has a perimeter of 51 inches. The length of the screen is 14 inches.

a. Write an equation that relates the perimeter P, length ℓ, and width w of a rectangle.

b. Solve the equation for w.

c. Find the width of the screen.

3. The equation I = Prt gives the amount I of simple interest where P is the principal, r is the interest rate, and t is the time in years.

a. Solve the equation for r.

b. Stephanie invests $500 in an account with simple interest for 6 years. During that time, she earns $120 in interest. What is the account’s interest rate?

c. How long does it take the account to earn $200 in interest?

4. In baseball, a player’s batting average B is the number of hits H divided by the number of at-bats A.

a. Write a literal equation for calculating the batting average. Then solve the equation for A.

b. The table shows statistics for three players with the Tampa Bay Rays during the 2007 season. Which player had the greatest number of at-bats? How many at-bats did he have?

Player Hits Batting Average

Carlos Peña 138 0.282

Carl Crawford 184 0.315

Brendan Harris 149 0.286

5. Short Response Women’s shoe sizes are related to the length of the foot as follows. The size S is 22.5 less than 3 times the length of the foot F in inches. Explain how you can estimate the length of a woman’s foot if you know her shoe size.



7-3


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Activity 1Consider the balance-scale model below.

Model Words Symbols

Three triangles weigh less than one triangle and one circle.

3t < t + c

For each balance-scale model below, use the information above to decide whether each pictured inequality will always, sometimes, or never be true. Then explain how you know. (Assume that all similarly shaped objects weigh the same.)

ScaleAlways, sometimes,

or never?Explanation

1

always

One triangle has been added to each side, which will keep the right side heavier than the left side.

2

3

4

5

Model InequalitiesModel InequalitiesYou have used balance scales to model equations. You can also use balance scales to model inequalities.

MA.8.A.4.2 Solve and graph one- and two-step inequalities in one variable.

Solving Inequalities by Adding or Subtracting

7-4 Solving Inequalities by Adding or Subtracting 237

7-4


x + 3 7>

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238 7-4 Solving Inequalities by Adding or Subtracting

Try This 1. Use the relationship in Activity 1 to draw 2. Use the relationship in Activity 1 to draw another model that will always be true. another model that will never be true.

Activity 2Use a balance-scale model to solve the inequality x + 3 > 7.

1 Model the inequality x + 3 > 7 with a balance scale.

2 To solve, isolate x on the left side by removing 3 unit squares from the left. To keep the equation unbalanced in the same direction, remove 3 unit squares from the right side too.

3 The scale shows the variable x isolated on the left and shows the solution of the inequality, x > 4.

Try ThisRedraw each model with the variable x isolated on one side of the inequality. Then write the solution of the inequality.

3. 4. 5.

Draw Conclusions 6. What is the effect of adding the same amount to, or subtracting the same

amount from, both sides of an inequality?


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Example 1

Compare. Write < or >.

A. 23 - 14 6

6 9 is greater than 6.

B. 5(12) 70

70 60 is less than 70.

Lesson ObjectiveSolve and graph inequalities using addition or subtraction

Vocabulary

inequality

algebraic inequality

solution set

MA.8.A.4.2 Solve and graph one-and two-step inequalities in one variable.

7-4

Solving Inequalities by Adding or Subtracting (Student Textbook pp. 310–313)


1a. 15 + 4 18 1b. 3(12) 26

Compare. Write < or >.CCheck It Out!


6 7 8 9 10 11

4 5 6

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1c. 32 - 8 25 1d. 3(4) 15

Solve and graph each inequality.

A. x + 2.5≤ 8

-

-

Subtract from both sides.

x ≤

The graph starts at 5.5 and extends to the left on thenumber line because all values less than 5.5 are part ofthe solution set. There is a closed circle at 5.5 because 5.5is also part of the solution set.

Check

According to the graph, 4 should be a solution, since 4 5.5, and 7 should not be a

solution because 7 5.5.

x + 2.5 ≤ 8

4 + 2.5 � 8 Substitute 4 for x.

6.5 ≤ 8 4

So 4 is a solution.

B. w - 1 < 8

Add to both sides.

w <

The graph starts at and extends to the on the number line because all values less than 9 are partof the solution set. There is an open circle at 9 because 9 is not part of the solution set.

S l d

Example 2


x + 2.5 ≤ 8

7 + 2.5 � 8 Substitute for x.

9.5 � 8 7

And is not a solution.


-12-11-10 -9 -8 -7

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C. r + 7 ≥ -3


r ≥

The graph starts at and extends to the

on the number line because all values than

are part of the solution set. There is a(n)

circle at because is part of the solution set.


Check It Out!

2a. h + 6 ≤ -2


2b. p - 4 < 2

2c. h - 8 ≥ - 4

2d. t + 14 > 22


SolvingInequalities

Solve UsingAddition

Solve UsingSubtraction

Inequality Inequality

Solution Solution

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Name Class Date

Summarize It !Summarize It !Think and Discuss1. Give all the symbols that make 5 + 8 13 true. Explain.

2. Compare and contrast equations and inequalities.

3. Get Organized Complete the graphic organizer. In each box, write an inequality you can solve using the given operation. Then show the steps needed to solve each inequality in the space provided.



7-4



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Name Class Date

Practice It !Practice It ! MA.8.A.4.2 Solve and graph one- and two-step inequalities in one variable.



Compare. Write <, >, or =.

1. 8 + 13 20 2. 23 3(7) 3. 28 - 9 18

4. 67 9(8) 5. 52 - 37 15 6. 78 12(6)


7. x + 3 > 7 8. y - 5 ≤ 1 9. n + 7.4 ≥ 12.4

10. h - 1 ≥ 6 11. x - 13 > -28 12. t - 7 ≥ 5

13. d + 6 < 1 14. 17 + m < 23 15. 7.75 ≥ x -0.25

16. In a homerun derby, contestants must hit at least a total of 34 homeruns in the fi rst three rounds in order to qualify for the fi nal round. Phillippe hit 16 homeruns in the fi rst round and 11 in the second round. In order to qualify for the fi nal round, how many homeruns must Phillippe hit in the third round? Write and solve an inequality to answer the question.

17. Sara has $40 to spend on a shopping trip for a friend’s birthday. She spends $28.50 on a gift and $8.72 on wrapping paper and ribbon. What is the most she can spend on a birthday card? Write and solve an inequality to answer the question.

7-4


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1. In 2000, the population of Gainesville, Florida, was at least 111,000. For the next 7 years, the town’s population increased by an average of 375 residents per year. What inequality gives the town’s population in 2007?

2. Jared has a $100 gift card for a store. He buys 3 DVDs, 4 t-shirts, and a flashlight. The table shows the prices of the items. (All prices include sales tax.)

ShopMart Prices

Item Price

DVD $12.99

T-Shirt $7.49

Flashlight $8.50

Jared also wants to buy a houseplant for his sister. Write and solve an inequality that Jared can use to determine how much he can spend on the houseplant.

3. An athlete is eligible for a track and field tournament if the sum of the distances of four long jumps is greater than 25 m. Keiko has already made long jumps of 6.2 m, 5.9 m, and 6.8 m. What distance does Keiko need on her fourth jump to make the tournament?

4. At JQX Software, expenses e for January were $593,000 and the month’s profit p was at most $750,000.

a. A company’s profit p is its revenue r less its expenses e. What inequality gives the company’s revenue r for January?

b. During the same month, revenue at MacroMode Software was expressed by the inequality r ≤ 1,200,000. Grace claims that revenue at MacroMode must have been less than revenue at JQX. Do you agree? Justify your reasoning.

5. Gridded Response During a 3-day camping trip, Terrence hiked no more than 18 mi. On the first day he hiked 9 1 _ 3 mi. On the second day he hiked 7 1 _ 2 mi. What is the greatest number of miles he could have hiked on the third day?



7-4



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Activity 1Complete the table.

Trial 1 Trial 2

1 Start with the column labeled Trial 1. Choose a value for x that makes the given inequality true. Write your value of x in the box.

x > 0

> 0

x < 0

< 0

2 Choose a value for a such that a is positive. Write the product a · x in the box. Is the resulting inequality true? If so, circle True below the inequality. If not, circle False.

a · x

> 0

True False

a · x

< 0

True False

3 Choose a value for b such that b is negative. Write the product b · x in the box. Is the resulting inequality true? Circle True or False.

b · x

> 0

True False

b · x

< 0

True False

4 Write the quotient x __ a in the box. (Round to the nearest tenth if necessary.) Is the resulting inequality true? Circle True or False.

x ÷ a

> 0

True False

x ÷ a

< 0

True False

5 Write the quotient x _ b in the box. (Round to the

nearest tenth if necessary.) Is the resulting inequality true? Circle True or False.

x ÷ b

> 0

True False

x ÷ b < 0

True False

6 Move to the column labeled Trial 2. Repeat Steps 1–5 using new values for a, b, and x.

Explore Properties of InequalitiesExplore Properties of InequalitiesFrom the Multiplication and Division Properties of Equality, you know that if you multiply or divide both sides of an equation by the same number, the two sides of the equation will remain equal. In this activity, you will investigate whether there are similar properties of inequality.


Solving Inequalities by Multiplying or Dividing

7-5 Solving Inequalities by Multiplying or Dividing 245

7-5

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246 7-5 Solving Inequalities by Multiplying or Dividing

Try ThisAnswer true or false.

1. If 3n > 0, then n > 0. 2. If -2n < 0, then n < 0.

3. If -5n > 0, then n < 0. 4. If 4n < 0, then n < 0.

Activity 2Complete the table.

Trial 1 Trial 2

1 Start with the column labeled Trial 1. Write a number in each box that makes the given inequality true. Do not choose 0 for either value.

< <

2 Multiply both numbers by the same positive number. Write the products in the boxes. Is the resulting inequality true? Circle True or False.

<

True False

<

True False

3 Multiply both numbers by the same negative number. Write the products in the boxes. Is the resulting inequality true? Circle True or False.

<

True False

<

True False

4 Divide both numbers by the same positive number. Write the quotients in the boxes. Is the resulting inequality true? Circle True or False.

<

True False

<

True False

5 Divide both numbers by the same negative number. Write the quotients in the boxes. Is the resulting inequality true? Circle True or False.

<

True False

<

True False

6 Choose a new pair of numbers and repeat Steps 1–5. Record your results in the column labeled Trial 2 above.

Try ThisUse the inequality 4 < 8 for Exercises 5–8. Complete each inequality using < or >.

5. (3)4 (3)8 6. (-5)4 (-5)8 7. 4 ÷ (2) 8 ÷ (2) 8. 4 ÷ (-4) 8 ÷ (-4)

Draw Conclusions 9. What is the effect of multiplying or dividing both sides of an inequality by the

same positive number? By the same negative number?


47 48 49 50 5146

-7 -6 -5 -4 -3 -2

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Example 1

Solve and graph.

A. 12 < a __ 4

· 12 < · a __ 4 Multiply both sides by .

, or

Check According to the graph, 49 should be a solution because 49 > 48, and 47

should not be a solution because 47 < 48.

12 � a __ 4 12 � a __ 4

12 < 49 __ 4 Substitute 49 for a. 12 < 47 __ 4 Substitute 47 for a.

12 < ✔ 12 < ✗

So 49 a solution. So 47 a solution.

B. -9b ≤ 45

-9b ______

≥ 45 ______

Divide both sides by ; change ≤ to ≥.

Lesson ObjectiveSolve and graph inequalities using multiplication or division.

7-5


Solving Inequalities by Multiplying or Dividing (Student Textbook pp. 314–317)

7-5 Solving Inequalities by Multiplying or Dividing 247Lesson Tutorial Videos


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Lesson Tutorial Videos 248 7-5 Solving Inequalities by Multiplying or Dividing

Example 2

A geology club needs to raise at least $500. They are buying gemstones for $2.50 and selling them for $4.00. How many gemstones must the club sell to make its goal?

Understand the ProblemFind the least number of gemstones the geology club must sell to make at least $500. List the important information:

• The geology club needs to make at least $ .

• They are buying gemstones for $ . They are selling gemstones for

$ .

Show the relationship of the information:

(the price for - the price for · the number of ≥ $500selling a gemstone buying a gemstone) gemstones sold

1

Problem Solving Application

Check It Out!

1a. q

___ -8 > 1 1b. 3 < t __ 7

1c. 5 ≤ 2z 1d. -4x ≤ 17

Solve and graph.


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Lesson Tutorial Videos 7-5 Solving Inequalities by Multiplying or Dividing 249

Make a PlanWrite an inequality. Let x represent the number of gemstones sold.

($ - $ ) · x ≥ $500

Solve($4.00 - $2.50) · x ≥ $500

$1.50 · x ≥ $500

$1.50 x ________

$

≥ $500 ________

$

x ≥ At least gemstones must be sold to meet the goal.

Look BackSelling one gemstone would make $ for the geology club; selling

334 gemstones would make $1.50 · 334 = $ .

2

3

4

Check It Out!

2. An entomologist is collecting insects in the tropics. Once she captures a specimen, she puts it in a display case. On average, each insect occupies 2 square inches. She has 3 square feet of empty display left to use. How many insects can she fit in that space?



by a Negative Number

Multiply

Divide

by a Positive Number

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Name Class Date

Summarize It !Summarize It !Think and Discuss1. Give all the symbols that make 5 · -3 15 true. Explain.

2. Explain how you would solve the inequality -4x ≤ 24.

3. Get Organized Complete the graphic organizer. Fill in the table by writing an inequality that can be solved by using the given operation. Show the solution of each inequality.



7-5

LA.8.2.2.3 The student will organize information to show understanding or relationships …


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Name Class Date

Practice It !Practice It !Solving Inequalities by Multiplying or Dividing

7-5


7-5 Solving Inequalities by Multiplying or Dividing 251

Solve and graph.

1. -9c ≤ 90 2. m ___

-8 < -4 3. z _ 4 ≤ -7

4. - 15f ≥ 75 5. n ___

-6 > -3 6. 15r ≥ -45

7. h ___

-1 < 6 8. b

_ 9 ≥ 5 _ 3 9. -

3 _ 4 s > 6

10. It costs Elizabeth $220 to make the shirts that she sells in her store. How many shirts must she sell at $15 each to make a profit?

11. The distance from Martin’s house to the sub shop is no more than 5 times the distance from his house to the mall. If Martin’s house is 9 miles from the sub shop, what is the least distance his house could be from the mall?

12. The maximum weight an elevator can safely lift is 2200 pounds. If the average weight of the passengers on the elevator is 172 pounds, how many people can ride in the elevator at one time?

13. Melissa spends no less than 1 _ 4 the amount of time on her math homework as Jane does. If Jane spends 1 1 _ 2 hours a night on her math homework, how much time does Melissa spend working on her math homework a night?


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Polette participates in her school’s summer reading program. Use the table for 1–3.

Summer Reading Program

Grade Minimum Number of Pages

6 800

7 1000

8 1250

1. At the beginning of the summer, Polette said she wanted to read at least 12% more pages than the 8 th grade target. What inequality can she use to find the number of pages she must read to achieve her goal?

2. Summer vacation is ten weeks. Over the first nine weeks, Polette read an average of 150 pages per week. What is the greatest percent of decrease in pages read per week that Polette can have for the tenth week while still achieving her goal?

3. By the end of summer, Polette had read more than 150% of the 7 th grade goal but less than 200% of the 6 th grade goal. What are the least and greatest numbers of pages Polette could have read?

4. Marcus invested $480 in a savings account that paid 3% simple annual interest. He earned more than $72 in interest. For how many years did he have the account?

5. A rectangular flower bed is 9.2 m long. The area of the rectangular flower bed is at most 80.5% of the area of a square flower bed. The square flower bed has sides that are 8 m long. What inequality can you use to find the width w of the rectangular flower bed?

6. Extended Response Hannah and Diego both took road trips. The table shows the distance each person traveled and the greatest speed at which he or she drove. Is it possible to determine who drove for a greater amount of time? If so, tell who drove for a greater amount of time. If not, explain why not.

DriverDistance

Traveled (mi)Greatest

Speed (mi/h)

Hannah 348 58

Diego 325 65




7-5


0 1 2 3 4 5-1-2-3-4-5

0 1 2 3 4 5-1-2-3-4-5

0 1 2 3 4 5-1-2-3-4-5

0 1 2 3 4 5-1-2-3-4-5

0 1 2 3 4 5-1-2-3-4-5

0 1 2 3 4 5-1-2-3-4-5

0 1 2 3 4 5-1-2-3-4-5

0 1 2 3 4 5-1-2-3-4-5

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7-6 Solving Two-Step Inequalities 253

ActivityIn each row of the table below, match the two-step inequality and the graph of its solution with an equivalent one-step inequality from the box at right. Use each one-step inequality just once. (Hint: Solve each one-step inequality and match its solution to the appropriate graph.)

One-Step Inequalities

2x > - 4 4x < 16 - 1 _ 2 x < -2

-2x < -7 -5x > 10 4x < 12

-2x < 2 4x < -4

Two-StepInequality

Equivalent One-Step Inequality

Graph

1 4x - 21 < -5

2 -2x + 6 < 8

3 -5x - 1 > 9

4 4x - 10 < -14

5 2x + 16 > 12

6 4x + 7 < 19

7 - 1 __ 2 x + 2 < 0

8 3 -2x < -4

Transform Two-Step InequalitiesTransform Two-Step InequalitiesBy adding or subtracting the same number to both sides of a two-step inequality, you can transform it to an equivalent one-step inequality. Then you can use what you know about one-step inequalities to solve the original two-step inequality.

Solving Two-Step Inequalities


7-6


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254 7-6 Solving Two-Step Inequalities

For Exercises 1-8, explain how you can transform each two-step inequality in theActivity to its equivalent one-step inequality.

Two-Step Inequality Explanation

1. 4x - 21 < -5 Add 21 to both sides to get .

2. -2x + 6 < 8

3. -5x - 1 > 9

4. 4x -10 < -14

5. 2x + 16 > 12

6. 4x + 7 < 19

7. - 1 __ 2 x + 2 < 0

8. 3 -2x < -4

Try ThisUse addition or subtraction to change each two-step inequality to anequivalent one-step inequality.

9. 2x -9 > 11 10. -6x + 7 ≤ 31 11. - x _ 5 + 4 > -5 12. 7 + 9x ≥ -2

Draw Conclusions 13. Think about the procedures you follow when solving two-step equations.

How do you think the procedures for solving two-step inequalities may be similar?

14. Based on what you learned about solving one-step inequalities involving multiplication or division, what may be one important difference between solving two-step equations and solving two-step inequalities?

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2 3 4 5 6 71

0-6 -5 -4 -3 -2 -1

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Lesson Tutorial Videos @ thinkcentral.com

Solve and graph.

A. 4x + 1 > 13

4x + 1 > 13

- - Subtract from both sides.

4x >

4x > 12 Divide both sides by .

x >

B. -9x + 7 ≥ 25

-9x + 7 ≥ 25

-

-


-9x ≥

-9x 18

x

S l d

Example 1

Lesson ObjectiveSolve two-step inequalities and graph the solutions


Solving Two-Step Inequalities (Student Textbook pp. 318–321)

7-6


The graph begins at , has a(n)

circle, and extends to the .

Divide both sides by ; change to .

The graph begins at , has a(n)

circle, and extends to the .


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1a. -4x + 2 ≥ 18

1b. 14 > 3.1y -13.9

Check It Out! Solve and Graph.


Solve and graph.

A. 14 > 22 - 2 __ 3 m

14 > 22 - 2 __ 3 m

> Add to both sides.

> - 2 __ 3 m

(-8) ( - 2 __ 3 m ) Multiply both sides by ;

change to .

m

S l d

Example 2


0 138

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2a. 16 > 7 + 3b __ 5 2b. -3x ____ 5 +

1 __ 4 ≥ 5

__ 10 3b

Check It Out!


B. Solve 2x __ 5 + 3 __ 4 ≥

9 __ 10

( 2x __ 5 + 3

__ 4 ) ≥ ( 9 __ 10 ) Multiply by the LCD, .

( 2x __ 5 ) + ( 3 __ 4 ) ≥ ( 9 __ 10 ) Distributive Property.

+ ≥

- - Subtract from both sides.

≥

8x ≥ 3 Divide both sides by .

x ≥

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SolvingTwo-Step Equations

SolvingTwo-Step Inequalities

Similarities Differences

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Name Class Date

Summarize It !Summarize It !


Think and Discuss1. Compare solving a two-step equation with solving a two-step inequality.

2. Describe two situations in which you would have to reverse the inequality symbol when solving a two-step inequality.

3. Get Organized Complete the graphic organizer. In each box, write similarities and differences in solving two-step equations and two-step inequalities.

7-6




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Name Class Date

Practice It !Practice It ! MA.8.A.4.2 Solve and graph one- and two-step inequalities in one variable.



Solve and graph.

1. 5x + 3 ≥ 28 2. 10 - 1 __ 2 w > 12

3. 7y - 15 - 4y ≤ - 18 4. 7a < 12a + 10

5. 19 - 4m > 4m + 11 6. - 5 __ 6 d + 8 > 13

7. 2(3x + 1) ≤ 4 (2x - 3) 8. 14 - 2 __ 5 h > 13.9 - 3 __ 10 h

9. Six more than three-fourths a number is greater than one more than twice the number. Find the number.

10. Ms. Garcia wants to carpet her bedroom which requires 24 square yards. Her budget will allow her to spend no more than $600 for the project. If the installation of the carpet will cost $162, what is the most she can pay per square yard for the carpeting?

11. Tina wants to make flyers promoting a library book sale. The printer charges $40 plus $0.03 per flyer. How many flyers can Tina have made without spending more than the library’s $54 budget?

7-6

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Stage 1Stage 2

Stage 3

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1. The figure shows the first 3 stages of a pattern made from tiles.

a. Write an expression for the total number of tiles required to make Stage n of the pattern.

b. Miguel has a box that contains at most 3 dozen tiles. What is the greatest stage of the pattern he can make?

2. Karl is selling tickets to a school play. Adult tickets cost $6.75 and children’s tickets cost $4.75. Karl has already sold 7 children’s tickets. At least how many adult tickets must he sell in order to bring in at least $100 from his ticket sales?

3. Julia sells kitchen appliances. She receives a monthly salary of $2250 plus a 4% commission on her sales. What minimum sales total must she achieve for the month if she wants her total pay for the month to be at least $4000?

4. A dogwood tree grows as much as 15 inches per year. The Harrison family has a dogwood tree in their yard that is currently 6 feet tall. What inequality can be used to determine the minimum number of years it will take for the tree’s height to exceed 21 feet? How many years is this?

5. Celia’s long-distance phone company charges $5.95 per month plus $0.06 per minute. If Celia has budgeted $30 for her monthly long-distance phone bill, at most how minutes can she call long distance per month?

6. The ratio of boys to girls in Ms. Cheng’s class is 3:2. There are at least 30 students in Ms. Cheng’s class. Write and solve an inequality to find the number of boys in the class.

7. Gridded Response Troy is hiking a trail that is at least 13.4 miles long. He has already hiked 2.9 miles in one hour and he plans to continue hiking at a rate of 3 mi/h. What is the minimum time, in hours, that the entire hike will take?




7-6


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7-3T H R O U G H

7-6Got It?Got It?Quiz for Lessons 7-3 through 7-67-3 Solving Literal Equations for a Variable (Student Textbook pp. 304–307)Solve each equation for the given variable.

1. A = bh for h 2. I = Prt for r

3. g + 2h = 3j - h for g 4. ax + by = c for y

5. The length of a garden is 12 m. What is the widthif the area of the garden is 48 square meters?

7-4 Solving Inequalities by Adding or Subtracting (Student Textbook pp. 310–313)Solve and graph each inequality.

6. t - 12 < - 4 7. x + 3 ≥ 9

8. Barbara is saving money so that she can buy some new clothes. She knows that she needs at least $60, and she has saved $22 so far. At least how much more money does Barbara need to save?

7-5 Solving Inequalities by Multiplying or Dividing (Student Textbook pp. 314–317)Solve and graph each inequality.

9. - 5x > 15 10. 9 ≥ k

__ 3

11. Rachael is serving lemonade from a pitcher that holds 60 ounces. What are the possible numbers of 7-ounce juice glasses she can fill from one pitcher?

7-6 Solving Two-Step Inequalities (Student Textbook pp. 318–321)Solve and graph each inequality.

12. 2r + 4 > 10 13. 1 __ 3 - 3x __ 4 ≥ 5 __ 6

14. Jillian must average at least 90 on two quiz scores before she can move to the next skill level. Jillian got a 92 on her first quiz. What scores could Jillian get on her second quiz in order to move to the next skill level?


Ready to Go On?


Width = 2

Length = 5

0 1 2 3 4 5-1-2-3-4-5

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Connect It!Connect It!Name Class Date

MA8.A.1.5; MA8.A.4.1;MA8.A.4.2


Perimeter Patterns1. Use square tiles to make rectangles that have a

width of 2 and a length that is given in the table. For example, the rectangle shown here has length 5. The perimeter of this rectangle is 14. Complete the table by writing the perimeter of each rectangle.

Length 3 4 5 6 7

Perimeter 14

2. Look for patterns in the table. Then write an equation that gives the perimeter P

of a rectangle with length �.

3. Solve your equation for �.

4. Suppose a rectangle has a perimeter of 26. What is the length?

5. Suppose a rectangle has a length greater than 42. What can you say about the perimeter? Write and solve a two-step inequality to justify your answer.

Cover Up1. Choose two or more of the equations and inequalities shown below. Graph the solutions

or solution sets on the number line. Your goal is for the solutions to completely cover the number line, but each value on the number line can only be covered once.

3x + 4 = - 2 - 5x > 10 x - 8 ≥ - 6

- 7x + 20 = 6 - 4x + 1 ≥ - 7 2x + 5 > 1

2. Circle the equations or inequalities that you used to solve the puzzle.

Think About The Puzzler3. What process did you use to solve the puzzle?

Connect the Concepts of Lessons 7-3 through 7-6

7-3T H R O U G H

7-6


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Study It!Study It!


7C H A P T E R

FLORIDA

Vocabulary (Student Textbook page references)

algebraic inequality . . . . . (310)

inequality . . . . . . . . . . . . . . (310)

literal equation . . . . . . . . . (304)

solution set . . . . . . . . . . . . . (310)

Complete the sentences below with vocabulary words from the list above.

1. A(n) is a mathematical sentence that contains variables and states that two quantities are not necessarily equal.

2. The is the collection of values that make an inequality true.

3. A is an equation with two or more variables.

Solving a Literal Equation for a Variable (Student Textbook pp. 304–307)


Solve each equation for the given variable. 4. p = mv for m 5. E = Pt for P

6. 2w + x = 6y for x 7. de + f = 2 for f

8. F = 9 __ 5 C + 32 for C 9. 6g - 4h = k for g

SLesson 7-3


MA.8.A.4.1

P = a + b + c for a P = a + b + c

−−−−− -b - c

−−−−− - b - c

P - b - c = a

3k + 2e = C for e 3k + 2e = C −−−

-3k −−−

-3k 2e = C - 3k

e = C - 3k ______ 2

Subtract b and c from both sides of the equation.


Multi-Language Glossary

Subtract 3k from both sides. Then divide both sides by 2 to isolate e.


0 1 2 3 4 5

19 20 21 22 23 24

-1.9-2.1-2.3

110 130 150

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MA.8.A.4.2Solving Inequalities by Adding or Subtracting (Student Textbook pp. 310–313)

Solve and graph.

Solve and graph. 10. y - 2 > 5 11. w + 2 ≥ 4

12. 3 + q ≤ 0 13. m - 2 ≤ 46

14. 4 < x + 1 15. 8 ≥ 4 + x

Solving Inequalities by Multiplying or Dividing (Student Textbook pp. 314–317)

S(

Lesson 7-4

S(

Lesson 7-5


x + 5 ≤ 8 x + 5 ≤ 8

−−

-5 −−

-5 x ≤ 3

w - 3 ≥ 18 w - 3 ≥ 18

−−

−− 3

w ≥ 21

Subtract 5 from both sides.


MA.8.A.4.2

Solve and graph. z ____

-13 ≤ -10

(-13) z ____ -13 ≥ (-13) -10

z ≥ 130

10k > -21

10k ___ 10 > -21 ____ 10

k > -2.1

Divide both sides by 10.

Multiply both sides by -13.change ≤ to ≥.


-2-4-6

66 67 68 69 70 71

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MA.8.A.4.2

Solve and graph.

16. m __ 6 ≥ 3 17. 4n ≤ -12

18. -8 < t __ 2 19. -5p > 15

20. 9 ≥ - b __ 3 21. -6a < -48

Solving Two-Step Inequalities (Student Textbook pp. 318–321)

Solve and graph.

22. 5z - 12 > -7 23. -2h - 7 ≥ 5

24. 10 > a

__ 3 + 2 25. x __ 3 - 8 ≥ -10

26. 5 - 3k < -4 27. 2y + 3

__ 4 > 1

SLesson 7-6


Solve and graph.

-3x - 3 < 9 -3x - 3 < 9

−−−−− 3

−−− 3

-3x < 12

-3x ____ 3 > 12 ___ 3

x > -4

t __ 4 - 15 ≥ 2

t __ 4 - 15 ≥ 2

−−−−

−−−

15 −−−

15

4 ( t __ 4 ) ≥ 4(17)

t ≥ 68


Divide both sides by -3. Change < to >.


Multiply both sides by 4.


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.Name Class Date

Write About It!Write About It!Think and DiscussAnswer these questions to summarize the important concepts from Chapter 7 in your own words.

1. Explain how to solve the equation ax + by = c for y. (Assume b does not equal zero.)

2. Explain how to solve the inequality 3d - 4 ≤ 2d + 1.

3. Explain how to solve the inequality 12 ≤ c ___ -5 .

4. Explain what an open circle and a closed circle means when graphing inequalities on a number line.

Before The TestI need answers to these questions:


LA.8.2.2.3 The student will organize information to show understanding…through…summarizing…


mmulti-step ulti-step florida eequations and quations and

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