Expressions and Equations

My Notes

Up In the Air

Lesson 15-1 Representing Situations with Inequalities

Learning Targets:

•

Write inequalities to represent constraints or conditions within problems.

•

Use substitution to determine whether a given number makes an

inequality true.

•

Graph solution sets of inequalities.

•

Given an inequality, write a corresponding real-world problem.

SUGGESTED LEARNING STRATEGIES: Interactive Word Wall,

Marking the Text, Summarizing, Note Taking, Discussion Group,

Activating Prior Knowledge

Geri wants to become a commercial airline pilot someday. She found the

following information while doing research on this career.

•

The airplane’s captain must be at least 23 years old.

•

The captain must have a minimum of 1500 hours of flying experience.

•

By law, pilots can fly a maximum of 100 hours in a month.

•

By law, pilots may not fly more than 32 hours during any consecutive

7 days.

Each piece of information that Geri found can be modeled by using an

inequality . Phrases like at least, more than, and a maximum express a

quantity that is greater than another.

1. List some phrases that could be used to express a quantity that is less

than another.

The phrases above, and others like them, are clues to help solve the

problem. They help to explain the rules of the situation.

Let’s look at one of the rules about being an airline pilot.

A captain must be at least 23 years old. The words at least tell you that

someone who is 23 years old can be a pilot. If the person is not 23, then

he or she must be older than 23 to be a pilot.

The inequality modeling this situation is written as follows:

x ≥23

, where x represents the age of the person.

An inequality is a mathematical

statement showing that one

quantity is greater than or less

than another. Inequalities use

these symbols:

> is greater than

< is less than

≥ is greater than or equal to

≤ is less than or equal to

MATH TERMS

Activity 15 • Expressions and Equations 187

ACTIVITY 15

My Notes

Lesson 15-1

Representing Situations with Inequalities

Much like the solution to an equation, a solution of an inequality is a

number that makes a statement true when it is substituted for the variable

in the inequality.

18 is one possible solution to the inequality

p <25

because

18 25<

is a

true statement.

Commercial airplanes are required to fly at least 150 feet above the

highest fixed object in a residential area. The highest building in Geri’s

town is 240 feet tall.

2. Define a variable and write an inequality to describe the situation.

3. State three possible solutions to the inequality.

Example A

Linda was told she had to spend less than $15 on flight snacks. Write the

inequality that represents this statement.

Step 1: Define the variable:

Let x = the amount of money Linda can spend.

Step 2: Determine which symbol should be used in the inequality.

Linda needs to spend less than $15, so use the symbol <.

Solution: Write the inequality.

x <15

Try These A

Write inequalities for the following statements.

a. The temperature was less than 20° F on the morning of the test.

b. More than 40 students were in her flight school class.

c. Training uniforms cost at least $50.

d. No more than 25 students in the class will get a job with the airline.

188 Unit 3 • Expressions and Equations

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ACTIVITY 15

My Notes

Lesson 15-1

Representing Situations with Inequalities

The graph of an inequality shows all its possible solutions. Inequalities

with one variable are graphed on a number line.

4. Give three values for the variable a that are not solutions to the

inequality.

Example B

Graph all possible solutions of the inequality, a > 3.

a. Draw a number line.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

b. Draw an open dot at 3.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

c. The solution is all values greater than 3, so draw a line with an arrow

to the right where the values are larger than 3.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

When graphing inequalities use an

open point to show an endpoint

that is not be included in the

graph, < or >.

Use a filled point to show an

endpoint that is included as part

of the solution to the inequality,

≤ or ≥.

MATH TIP

Determine two possible solutions for each of the following inequalities.

x ≤3 5.

6. m>

7. Daria did not want to spend more than $200 for a flight from San

Francisco to San Diego. Write the inequality that models this

situationand determine two possible solutions.

Check Your Understanding

Activity 15 • Expressions and Equations 189

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ACTIVITY 15

My Notes

Lesson 15-1

Representing Situations with Inequalities

In the previous items, inequalities were written from the real-world

scenario about becoming a pilot. Real-world problems can also be written

from an inequality.

Try These B

a. Model with mathematics.

Graph all possible solutions of the inequality y ≤1.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

b. Graph all possible solutions of the inequality 6 ≥ x.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

c. Graph all possible solutions of the inequality x >5.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

8. Look at the inequalities in Try These B above.

a. Name two solutions for each inequality.

b. How many more solutions do you think you can find to each

inequality?

An inequality such as y ≤ 1 or y > 5 has infinitely many solutions.

That means that the number of solutions has no limit.

When drawing a number line, the

scale may be altered to best

describe the situation. It is not

always necessary to scale by ones.

MATH TIP

Example C

Write a real-world problem this inequality could represent.

x ≤ 10

Step 1: Think about what the number 10 could represent.

It could be the number of people going on the same flight.

It could be the height of a room in feet.

It could be the number of seats remaining on a flight.

Step 2: Pick one of your choices.

The height of a room in feet.

Step 3: Make up a problem that could occur in the real world.

Solution: Lenora wants a new dresser and mirror for her

bedroom. Sheneeds to know how high the top of the

dresser could be, so she measures the height of her

room. The height of her room is 10 feet. The inequality

above shows that Lenora’s dresser and mirror can be no

taller than 10 feet.

190 Unit 3 • Expressions and Equations

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ACTIVITY 15

My Notes

Lesson 15-1

Representing Situations with Inequalities

Example D

Write a real-world problem this inequality could represent.

125 < x

Step 1: Think about what the number 125 could represent.

It could be the number of airplanes sitting at the airport.

It could be the number of bags that need to go through security.

It could be the number of songs in a music collection for

flight entertainment.

Step 2: Pick one of your choices.

The number of songs in a music collection for flight

entertainment

Step 3: Make up a problem that could occur in the real world.

Solution: Travel Right Airlines has 30 jazz songs, 45 rock songs, 30

easy listening songs, and 20 rap songs ready to play on a

flight. The rest are movie sound tracks. The inequality above

represents that the number of songs in the music collection

is greater than 125.

Try These C and D

Reason abstractly. Write a real-world problem each inequality could

represent.

a. Use the inequality x < 5.5 to write a problem about the height a

paper airplane can reach.

b. Use the inequality 21 ≤ x to write a problem about the number of

people who want to sign up for a trip.

c. Use the inequality

x ≤105

to write a problem about the number of

snacks served during a flight.

d. Use the inequality

x ≥

to write a problem about the diameter of an

airplane engine.

e. Use the inequality

x ≥ 40

to write a problem about the number of

seats on a Ferris wheel.

f. Use the inequality

x ≤16

to write a problem about the number of

students who sign up for the class trip to Paris.

g. Use the inequality

x ≤12 7.

to write a problem about the cubic feet

ofspace in an overhead compartment on an airplane.

Activity 15 • Expressions and Equations 191

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ACTIVITY 15

My Notes

Lesson 15-1

Representing Situations with Inequalities

Define a variable and write an inequality to represent this situation.

8. No one under 23 is allowed to captain an airplane.

9. Write two possible solutions to the inequality x <

10. Graph all possible solutions to the inequality x ≤ 2.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

11. Write a problem about the number of pieces of luggage for the

inequality 7 < x.

Check Your Understanding

LESSON 15-1 PRACTICE

Define a variable and write an inequality to represent these situations.

12. She finished the license test in no more than 30 minutes.

13. The captain must have a minimum of 1500 hours of flying

experience.

14.

Reason quantitatively. What are two values that are not solutions

for 11.4 > y?

15. What are three possible solutions for x >

16. Graph all possible solutions for

x >

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

17. Graph all possible solutions for x > 9.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

18. Use the inequality x ≤ 20 to write a problem about people waiting

for a flight.

192 Unit 3 • Expressions and Equations

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ACTIVITY 15

My Notes

Lesson 15-2

Solving One-Step Inequalities

Learning Targets:

•

Write one-step inequalities to represent constraints or conditions

within problems.

•

Use substitution to determine whether a given number makes an

inequality true.

•

Solve one-step inequalities.

•

Graph the solution sets of one-step inequalities.

SUGGESTED LEARNING STRATEGIES: Paraphrasing, Marking the

Text, Think Aloud, Create a Plan, Sharing and Responding, Create

Representations, Simplify the Problem

A pilot training class has space for at most 25 students. There are already

12 students who have signed up for the class. How can the number of

spaces remaining in the class be represented?

An inequality can be written similar to an equation, except that instead of

using an equal sign, you use an inequality symbol.

Example E

Find the number of students that can still sign up for pilot training if

there is space for at most 25 students and 12 have already signed up.

Step 1: Define the variable.

The variable x can represent the number of spaces remaining

in the class.

Step 2: The words at most in the statement above mean that the

numbers represented are 25 or less than 25. The symbol used

to write this inequality is ≤.

Step 3: Write the inequality.

Put the maximum number of students who can sign up for

the class on the side indicating that 25 is the greatest amount,

and the number who have signed up and can still sign up, on

the side indicating that these values are less than 25.

x + 12 ≤ 25

Solution: This inequality says that there do not have to be 25 students

in the class. Any number of students below 25 is also

acceptable.

Try These E

Write inequalities to represent the following situations.

a. A pilot training class needs a minimum of 10 students to run. At this

time, 7 students have signed up for the class.

Activity 15 • Expressions and Equations 193

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ACTIVITY 15

My Notes

Lesson 15-2

Solving One-Step Inequalities

Example F

Use substitution to determine if 15 additional people are too many for

the class.

Step 1: Write the inequality using the information that is given.

Let x represent the additional students that can sign up for

the class.

x + 12 ≤ 25

Step 2: Substitute the value 15 for x and solve.

15 + 12 ≤ 25

27 ≤ 25

Solution: You know that 27 is not less than 25, so 15 additional students

cannot sign up for the class.

Try These F

Determine if the given value of x makes the inequality true.

x x− > =5 17 12,

x x+ > =9 21 15,

4 50 13x x≤ =,

Show a question mark over the

inequality symbol to show that

you are not sure if it is true or not.

MATH TIP

To determine the values that make an inequality true, an inequality

can be solved like an equation is solved.

The person registering students for the pilot training class does not

know that there is a maximum number of students that can sign up, or

that some students have already signed up. She signs up 15 more

students. Is there a way to know if she signed up too many people?

b. A hot-air balloon needs to be at least 100 feet in the air to fly safely.

It is already 37 feet in the air.

c. A block is 6 inches high. The tower must be over 100 inches high.

194 Unit 3 • Expressions and Equations

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ACTIVITY 15

My Notes

Lesson 15-2

Solving One-Step Inequalities

Example H

Graph the solution to the inequality x +12 ≤ 25.

a. From Example G you know the solution is x ≤ 13.

b. The inequality includes an equal sign, showing that 13 is included in

the solution, so the point at 13 will be solid.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

c. The solution is all numbers less than 13, so the arrow goes to theleft.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Try These H

Solve and graph each inequality.

x + <5 13

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

− ≥

3 9

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3 24x ≤

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

Example G

You know that 15 students are too many to add to the class, but how

many more can sign up without going over the limit?

Step 1: Write the inequality.

x + 12 ≤ 25

Step 2: Subtract 12 from both sides of the inequality to isolate the

variable.

x + 12 − 12 ≤ 25 − 12

x ≤ 13

Solution: x ≤ 13.

This solution tells you that any number of students, less than or equal

to 13, can sign up for the class without exceeding the limit of 25.

Try These G

Solve each inequality.

x + ≤21 46

2 11x >

x − <1 2 4 8. .

d. x + ≥

Graph the solution of an inequality on a number line.

Activity 15 • Expressions and Equations 195

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ACTIVITY 15

My Notes

Lesson 15-2

Solving One-Step Inequalities

Write an inequality to represent this situation.

1. The parachute needs at least six people to hold it. There are two

people holding it now.

2. Determine if

is a solution for

x + >

3. Determine if 4.7 is a solution for

x − <2 4 5.

4. Solve

2 3 7 7. .+ <x

5. Solve

3 8x <

6. Solve this inequality and graph the solution

x + <23 31

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

Check Your Understanding

LESSON 15-2 PRACTICE

7. A captain can fly a maximum of 100 hours a month. He has flown

52hours.Write the inequality that represents this situation.

Model with mathematics. A paper airplane contest needs at least

65 people to enter. So far, 43 people have entered. Write the

inequality that represents this situation.

9. Determine if 4.4 is a solution for x +1.9 > 7.

10. Determine if

is a solution for

3+ ≤x

11. Solve x −15 < 2.

12. Solve 2.5x ≥12.5.

13. Solve 1.5 + x < 6.5 and graph the solution.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

14. Solve

3+ >x and graph the solution.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

196 Unit 3 • Expressions and Equations

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ACTIVITY 15