Skills Practice€¦ · SOLVING LINEAR EQUATIONS AND INEQUALITIES III. Modeling Linear Equations...

72 • MODULE 2: Exploring Constant Change

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SOLVING LINEAR EQUATIONS AND INEQUALITIES

I. Modeling Rates of Change

A. Write a linear equation to model each problem situation and use the equation to answer the question.

1. Autumn creates custom bracelets as a hobby and is planning to start selling them online for $10 per bracelet. Autumn has already sold 5 custom bracelets. Her bracelets are so popular that she expects to sell every bracelet that she makes. Write an equation for the amount of money Autumn makes. If Autumn makes an additional 24 bracelets, how much money will she make?

2. Antonio works at the circus making balloon animals, charging $3 for a balloon animal. Before he took a lunch break at noon, he sold 14 balloon animals. After lunch, he goes back to selling balloon animals for the rest of the day. Write an equation for the amount of money Antonio makes. How many balloon animals would Antonio need to sell after lunch to make $117 for the day?

3. Violet is trying to start an Intramural Club at her school. The principal tells her she must get signatures from students to show support. Each fi lled sheet contains 25 signatures. By Monday, she and her friend already have 6 sheets fi lled with signatures. The principal tells Violet she must have 7 more sheets fi lled with signatures. Write an equation for the number of signatures Violet will get. If she fi lls all of these, how many signatures will she get in all?

4. Tremaine thought it would be okay to check his email, text, listen to music, and eat free food for 1 hour of each of his shifts at Slow Food to Go. He lasted for 6 shifts, and then (to put it nicely) he was let go. Write an equation for the number of hours Tremaine actually worked. If Tremaine actually worked a total of 18 hours during his 6 shifts at Slow Food to Go, how many hours was he scheduled to work each shift?

Skills PracticeName Date

IM1_SP_M02_T02.indd 72IM1_SP_M02_T02.indd 72 24/05/18 2:40 PM24/05/18 2:40 PM

SOLVING LINEAR EQUATIONS AND INEQUALITIES: Skills Practice • 73

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5. Dr. Betz, a vet, is running a free rabies clinic. He estimates that it will take him 12 minutes for each animal he treats. Dr. Betz has already seen 20 animals, the last of which was a shaggy dog. Write an equation for how many minutes Dr. Betz worked. How long did Dr. Betz work at the rabies clinic if he saw 20 more animals after the shaggy dog?

6. Nakida cleans the bird cages at an animal shelter. She doesn’t know how many birds were at the shelter this morning, but 4 adorable birds were adopted today. Each remaining bird has its own cage, each of which takes Nakida 3 minutes to clean. Write an equation for the amount of time Nakida spends cleaning cages. How long will it take to clean the cages if there were nine birds at the shelter this morning?

II. Modeling Linear Equations Given Two Points

A. Determine the slope-intercept equation of each line given two points on the line.

1. (1, 23) and (22, 6) 2. (22, 4) and (4, 22)

3. (3, 9) and (1, 10) 4. (4, 27) and (211, 27)

5. (26, 1 __ 2 ) and (5, 3 __ 2 ) 6. (15, 0) and (21, 21)



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III. Modeling Linear Equations Given an Initial PointA. Write a linear equation to model each problem situation and use the equation to answer the question.

1. An airplane prepares for landing. When it begins its descent towards the airport, the plane's altitude is 27,000 feet. After descending at a steady rate for 6 minutes, its altitude is 19,000 feet. Write an equation for the altitude of the plane. What will the altitude of the plane be after descending for 12 minutes?

2. A college freshman starts the year with $4400 in spending money and after 2 months, he has $3600 left. Assume that he continues to spend his money at this same rate. Write an equation for the amount of spending money he has. How much money will he have left at the end of 7 1 __ 2 months?

3. Taki and Connor invest $20 in drink mix, sugar, and paper cups for a lemonade stand. After 5 hours, they have made $50 in profi ts. Assume that they sell the drinks at the same rate all day. Write an equation for their profi t for the day from the lemonade stand. If their stand continues to sell from 1:00 P.M. to 8:00 P.M. that day, what is their profi t for the day?

4. An experiment is underway to test the eff ect of extreme temperatures on a newly developed liquid. Two hours into the experiment, the temperature of the liquid is 220°C. After 6 hours of the experiment, the temperature of the liquid is 240°C. Assume that the temperature has been changing at a constant rate throughout the experiment and will continue to do so. Write an equation for the temperature of the liquid. How many hours into the experiment will the temperature of the liquid be 2100°C?



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5. A baseball player starts the season with 77 career home runs. Ten games into the season, he has hit 2 home runs. Assume he continues at this pace. Write an equation for the number of career home runs he has hit. How many career home runs would you expect him to have at the end of the season (162 games)?

6. A man retires at age 50 with $605,000 in savings. He spends his savings at a steady rate, and after 6 years of retirement, he has spent $300,000. Write an equation for the amount the man has in savings. When will he have $100,000 in savings?

IV. Solving Linear EquationsA. Solve each equation. Write the properties that justify each step.

1. 23(x 2 4) 5 29(x 2 1) 2. 8x 2 2(x 1 3) 5 4x 1 2

3. 22x 1 1 __________ 2 1 6 5 3x ___ 2 2 10 4. 12x 2 4( 1 __ 2 x 2 5) 5

1 __ 3 (6x 2 15)

5. 7(x 2 1) ________ 4 2 3 __ 4 5 28x 1

3 __ 4 6. 24(2x 2 9) 1 6(2x 1 1) 5 28x 2 5(3x 2 6 __ 5 )



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B. Determine if the equation has one solution, no solution, or infi nite solutions. Show your work.

1. 22(x 2 3) 1 5 5 26(x 1 1) 1 4x

3. 20x 2 2(x 1 10) 5 2(5 2 2x)

2. 3x 1 1 ________ 2 1 6 5 1 __ 2 (3x 2 4) 1

17 ___ 2

4. 3 __ 5 (x 2 12) 5 24(x 1 9) 1 1

5. 27(x 2 1) 5 215x 1 8(x 1 2) 6. 8(x 2 3) ________ 2 1 5x 5 9(x 2 1) 2 3

V. Converting Literal EquationsA. Convert between degrees Fahrenheit and degrees Celsius using the literal equation given. If necessary, round the answer to the nearest hundredth.

C 5 5 __ 9 (F 2 32)

1. 72°F 2. 211°F

3. 102.6°F 4. 25°C

5. 42°C 6. 23.4°C



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B. Convert each equation from standard form to slope-intercept form.

1. 4x 1 6y 5 48 2. 3x 2 5y 5 25

3. 24x 1 9y 5 45 4. 6x 2 2y 5 252

5. 2x 2 8y 5 96 6. 12x 1 28y 5 284

C. Convert each equation from slope-intercept form to standard form.

1. y 5 5x 1 8 2. y 5 24x 1 2

3. y 5 2 __ 3 x 2 6 4. y 5 2 1 __ 2 x 2 3

5. y 5 25x 2 13 6. y 5 3 __ 4 x 1 10



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D. Solve each equation for the variable indicated.

1. The formula for the area of a triangle is A 5 1 __ 2 bh. Solve the equation for h.

2. The formula for the area of a trapezoid is A 5 1 __ 2 (b1 1 b2)h. Solve the equation for b1.

3. The formula for the volume of a cylinder is V 5 pr2h. Solve the equation for h.

4. The formula for the volume of a pyramid is V 5 1 __ 3 lwh. Solve the equation for w.

5. The Ideal Gas Law is pV 5 nRT. Solve the equation for T.

6. Solve the literal equation 1 __ R 5 1 ___ R1

1 1 ___ R2 for R1.

7. Solve the literal equation a1 ___ a0

5 2 b1 ___ b0

for b0.

8. Solve the literal equation Z 5 4X ___ Y 2 1 3W for X.



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VI. Solving and Graphing InequalitiesA. Carlos works at an electronics store selling computer equipment. He can earn a bonus if he sells $10,000 worth of computer equipment this month. So far this month, he has sold $4000 worth of computer equipment. He hopes to sell additional laptop computers for $800 each to reach his goal. The function f(x) 5 800x 1 4000 represents Carlos's total sales as a function of the number of laptop computers he sells.

3. less than $6000 4. at least $9000

5. more than $12,000 6. exactly $8000

Number of Laptop Computers Sold

Tota

l Sal

es (d

olla

rs)

x

y

10,000

8000

1 2 3 40

4000

12,000

2000

5 6 7 8 9

14,000

18,000

16,000

6000

1. at least $10,000 2. less than $7000

Use the graph to write an equation or inequality to determine the number of laptop computers Carlos would need to sell to earn each amount.



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C. Leon plays on the varsity basketball team. So far this season he has scored a total of 52 points. He scores an average of 13 points per game. The function f(x) 5 13x 1 52 represents the total number of points Leon will score this season. Write and solve an inequality to answer each question.

1. How many more games must Leon play in order to score at least 117 points?

2. How many more games must Leon play in order to score fewer than 182 points?

3. How many more games must Leon play in order to score more than 143 points?



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D. Draw an oval on the graph to represent the solution to each question. Write the corresponding inequality statement.

1. A hot air balloon at 4000 feet begins its descent. It descends at a rate of 200 feet per minute. The function f(x) 5 2200x 1 4000 represents the height of the balloon as it descends. How many minutes have passed if the balloon is below 3000 feet?

Time (minutes)

Hei

ght (

feet

)

x

y

3000

50

2000

4000

1000

10 15

5000

6000

2. A bathtub fi lled with 55 gallons of water is drained. The water drains at a rate of 5 gallons per minute. The function f(x) 5 25x 1 55 represents the volume of water in the tub as it drains. How many minutes have passed if the tub still has more than 20 gallons of water remaining in it?

Time (minutes)

Volu

me

(gal

lons

)

x

y

30

50

20

40

10

10 15

90

80

70

60

50



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E. Solve each inequality and then graph the solution on the number line.

1. 4x 1 3 # 3x 2 5

2. 22x . 6

3. 1 __ 8 (3x 2 16) , 4

4. x 2 3 ______ 2 $ 25



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4. All numbers greater than 10 and less than 1000

5. All numbers less than or equal to 87 and greater than or equal to 83

6. All numbers greater than 21 and less than or equal to 39

B. Write an inequality for each graph.

1. 20 4 6 8 10–10 –8 –6 –4 –2

2. 210 3 4 5 76 8 9 10 11

3. 1 5 13 17 21 25 29 339 37

4. 20 4 6 8 10–10 –8 –6 –4 –2

5. 0 40 8–16 –12 –8 –4

6. 1612 20–4 0 4 8



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D. Write a compound inequality for each situation.

1. The fl owers in the garden are 6 inches or taller or shorter than 3 inches.

2. People with a driver's license are at least 16 years old and no older than 85 years old.

3. Kyle's car gets more than 31 miles per gallon on the highway or 26 miles or less per gallon in the city.

4. The number of houses that will be built in the new neighborhood must be at least 14 and no more than 28.

5. At the High and Low Store they sell high-end items that sell for over $1000 and low-end items that sell for less than $10.

6. The heights of the twenty tallest buildings in New York City range from 229 meters to 381 meters.

E. Represent the solution to each part of the compound inequality on the number line. Then write the fi nal solution that is represented by each graph.

1. x . 2 and x # 7



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F. Solve each compound inequality. Then graph and describe the solution.

1. 23 , x 1 7 # 17

2. 4 # 2x 1 2 , 12

3. x 1 5 . 14 or 3x , 9

4. 25x 1 1 $ 16 or x 2 6 # 28

5. 28 # 7 __ 8 x , 42

6. 22x 1 5 # 9 or 2x 2 13 . 231


Skills Practice€¦ · SOLVING LINEAR EQUATIONS AND INEQUALITIES III. Modeling Linear Equations...

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