##### Document Text Contents

Page 1

9.1

Focus on…

After this lesson, you

will be able to…

• represent single-

variable linear

inequalities verbally,

algebraically, and

graphically

• determine if a given

number is a possible

solution of a linear

inequality

Representing Inequalities

The official rule book of the NHL states limits for the equipment players

can use. One of the rules states that no hockey stick can exceed 160 cm.

What different ways can you use to represent the allowable lengths of

hockey sticks?

Explore Inequalities

1. a) Show how you can use a number line to graph lengths of hockey

sticks in centimetres. Use a convenient scale for the range of values

you have chosen to show. Why did you select the scale you chose?

b) Mark the maximum allowable length of stick on your line.

2. a) Consider the NHL’s rule about stick length. Identify three different

allowable stick lengths that are whole numbers. Identify three that

are not whole numbers. Mark each value on your number line.

b) Think about all the possible values for lengths of sticks that are

allowable. Describe where all of these values are located on the

number line. How could you mark all of these values on the

number line?

Did You Know?

Zdeno Chara is the

tallest person who

has ever played in the

NHL. He is 206 cm tall

and is allowed to use

a stick that is longer

than the NHL’s

maximum allowable

length.

340 MHR • Chapter 9

Page 2

3. a) Give three examples of stick lengths that are too long. Where are

these values located on the number line?

b) Discuss with a partner how to state the possible length of the

shortest illegal stick. Is it reasonable to have a minimum length

for the shortest illegal stick? Why or why not?

Refl ect and Check

4. The value of 160 cm could be called a boundary point for the

allowable length of hockey sticks.

a) Look at the number line and explain what you think the term

boundary point means.

b) In this situation, is the boundary point included as an allowable

length of stick? Explain.

5. The allowable length of hockey sticks can be expressed mathematically

as an inequality . Since sticks must be less than or equal to 160 cm in

length, the linear inequality is l ≤ 160, where l, in centimetres,

represents the stick length.

Write an inequality to represent the lengths of illegal sticks.

Discuss your answer with a classmate.

Did You Know?

Most adult hockey

sticks range from

142 cm to 157.5 cm

in length.

inequality

• a mathematical

statement comparing

expressions that may

not be equal

• can be written using

the symbol <, >, ≤, ≥,

or ≠

Did You Know?

The world’s largest hockey stick

and puck are in Duncan, British

Columbia. The stick is over 62 m

in length and weighs almost

28 000 kg.

9.1 Representing Inequalities • MHR 341

Page 16

Example 3: Model and Solve a Problem

A games store is offering games on sale for $12.50, including tax. Sean has

set his spending limit at $80. How many games can Sean buy and stay

within his limit?

a) Write an inequality to model the problem.

b) Solve the inequality and interpret the solution.

Solution

a) If n represents the number of games that Sean can buy, the cost

of n games is 12.5 times n. Sean must spend no more than $80.

The situation can be modelled with the inequality 12.5n ≤ 80.

b) 12.5n ≤ 80

12.5n ______

12.5

≤ 80 _____

12.5

n ≤ 6.4

Sean can buy up to and including six games and stay

within his spending limit.

Yvonne is planting trees as a summer job. She gets paid $0.10 per tree

planted. She wants to earn at least $20/h. How many trees must she

plant per hour in order to achieve her goal?

a) Write an inequality to model the number of trees Yvonne must plant

to reach her goal.

b) Will the solution be a set of whole numbers or a set of integers? Explain.

c) Solve the inequality and interpret the solution.

Show You Know

12 × 6 = 72 12 × 7 = 84

The number of games Sean can buy

is between 6 and 7.

Did You Know?

Piecework is work paid

by the amount done,

not by the time it

takes. For example,

tree planters are paid

by the number of

trees they plant.

Since it is not possible to buy part of a

game, 6.4 is not a solution to the

original problem, even though it is a

solution to the inequality 12.5n ≤ 80.

Only a whole number is a possible

solution for this situation.

9.2 Solving Single-Step Inequalities • MHR 355

Page 17

Key Ideas

• The solution to an inequality is the value or values that makes the

inequality true.

5x > 10

A specifi c solution is any value greater than 2. For example, 2.1, 3, or 22.84.

The set of all solutions is x > 2.

1 2 3

• You can solve an inequality involving addition, subtraction,

multiplication, and division by isolating the variable.

x - 3 ≤ 5 8x ≤ 24

x

___

-2

> 6

x - 3 + 3 ≤ 5 + 3

8x

___

8

≤

24

___

8

x

___

-2

× -2 < 6 × -2

x ≤ 8 x ≤ 3 x < -12

Reverse the inequality

symbol when

multiplying or

dividing both sides by

a negative number.

Check Your Understanding

Communicate the Ideas

1. Maria and Ryan are discussing the inequality 2x > 10.

Maria: Ryan:

What does Ryan mean?

2. Explain how the process for verifying a solution is different for a linear

inequality than for a linear equation. Discuss your answer with a classmate.

The solution to the

inequality is 6. When I

substitute 6 for x, a true

statement results.

I agree that 6 is a

solution but it is not

the whole solution.

• To verify the solution to an inequality, substitute possible values into

the inequality:

� Substitute the value of the boundary point to check if both sides

are equal.

� Substitute specifi c value(s) from the solution to check that the

inequality symbol is correct.

Check if x ≥ -3 is the solution to -8x ≤ 24.

Substitute the boundary point -3.

-8x = 24

-8(-3) = 24

24 = 24

The two sides are equal. Therefore, -3 is

the correct boundary point.

Substitute a value greater than the boundary point -3.

-8x ≤ 24

-8(0) ≤ 24

0 ≤ 24

Substituting a value greater than -3 results in a true

statement. Therefore, the inequality symbol is correct.

356 MHR • Chapter 9

Page 31

Chapter 9 Practice Test

For #1 to #5, select the best answer.

1. Karen told her mother that she would be

out for no more than 4 h. If t represents the

time in hours, which inequality represents

this situation?

A t < 4 B t ≤ 4

C t > 4 D t ≥ 4

2. Which inequality does the number line

represent?

-3 -2 -1 10

A x < -1 B x ≤ -1

C x > -1 D x ≥ -1

3. Which number is not a specifi c solution

for the inequality y - 2 ≥ -4?

A -6 B -2

C 2 D 6

4. Solve: 5 - x < 2

A x < 3 B x > 3

C x < 7 D x > 7

5. What is the solution of 5(x - 3) ≤ 2x + 3?

A x ≤ -6 B x ≥ -6

C x ≤ 6 D x ≥ 6

Complete the statements in #6 and #7.

6. The number line representing the inequality

x < 5 would have a(n)�circle at 5 and

an arrow pointing to the�.

7. The solution to -4x < 16 is x � than �.

Short Answer

8. Represent each inequality on a number line.

a) -3 < x

b) x ≤ 6.8

9. Verify whether x > -3 is the correct

solution to the inequality 8 - 5x < 23.

Show your thinking. If the solution is

incorrect, explain why.

10. Christine is researching a career as an airline

pilot. One airline includes the following

criteria for pilots. Express each of the

criteria algebraically as an inequality.

a) Pilots must be shorter than 185 cm.

b) Pilots must be at least 21 years old.

11. Solve and graph each inequality.

a) -6 + x ≥ 10

b) 2.4x - 11 > 4.6

c) 12 - 8x < 17 - 6x

12. Represent each situation algebraically as an

inequality.

a) Luke earns $4.75 per item sold and must

earn over $50.

b) It takes Nicole 3 h to sew beads on a

pair of mitts. She has no more than 40 h

of time to sew beads on all the mitts she

plans to give to her relatives as presents.

370 MHR • Chapter 9

Page 32

You are an amusement park manager who has been off ered a job planning a new

park in a diff erent location.

a) Give your park a name and choose a location. Explain how you made your choice.

State the population of the area around the park that you chose.

b) Choose a reasonable number of rides for your park. Assume that the fi xed costs

include $5000 in addition to maintenance and wages. Assume maintenance and

repairs cost $400 per ride and that it takes eight employees to operate and

supervise each ride. Conduct research and then decide:

• the number of hours that rides will be open

• the average hourly wage for employees

c) Organize your estimates about operating expenses and revenues for the park.

You can use the table in the Math Link on page 367 as a reference.

d) Write an expression to represent each of the following for the number of rides

you chose:

• expenses per visitor

• revenue per visitor

e) For the number of rides you chose, how many visitors will be needed for the park

to make a profi t? Show all your work. Justify your solution mathematically.

f) Assume that you have now opened your park. You fi nd that 0.1% of the people in

the area come to the park per day, on average. Using this information, will your

park earn a profi t? If not, explain what changes you could make. Show all your

work and justify your solution.

Extended Response

13. Consider the inequality 6x - 4 > 9x + 20.

a) Solve the inequality algebraically.

b) Represent the solution graphically.

c) Give one value that is a specifi c solution

and one that is a non-solution.

d) To solve the inequality, Min fi rst

subtracted 6x from both sides. Alan

fi rst subtracted 9x from both sides.

Which method do you prefer? Explain

why.

14. The Lightning Soccer Club plans to buy

shirts for team members and supporters.

Pro-V Graphics charges a $75 set-up fee

plus $7 per shirt. BT Designs has no set-up

fee but charges $10.50 per shirt. How many

shirts does the team need to order for Pro-V

Graphics to be the better option?

15. Dylan is organizing a curling tournament. The

sports complex charges $115/h for the ice

rental. Dylan has booked it for 6 h. He will

charge each of the 14 teams in the tournament

an entrance fee. How much must he charge

each team in order to make a profi t?

What might be the

problem if you

choose too few or

too many rides?

Chapter 9 Practice Test • MHR 371

9.1

Focus on…

After this lesson, you

will be able to…

• represent single-

variable linear

inequalities verbally,

algebraically, and

graphically

• determine if a given

number is a possible

solution of a linear

inequality

Representing Inequalities

The official rule book of the NHL states limits for the equipment players

can use. One of the rules states that no hockey stick can exceed 160 cm.

What different ways can you use to represent the allowable lengths of

hockey sticks?

Explore Inequalities

1. a) Show how you can use a number line to graph lengths of hockey

sticks in centimetres. Use a convenient scale for the range of values

you have chosen to show. Why did you select the scale you chose?

b) Mark the maximum allowable length of stick on your line.

2. a) Consider the NHL’s rule about stick length. Identify three different

allowable stick lengths that are whole numbers. Identify three that

are not whole numbers. Mark each value on your number line.

b) Think about all the possible values for lengths of sticks that are

allowable. Describe where all of these values are located on the

number line. How could you mark all of these values on the

number line?

Did You Know?

Zdeno Chara is the

tallest person who

has ever played in the

NHL. He is 206 cm tall

and is allowed to use

a stick that is longer

than the NHL’s

maximum allowable

length.

340 MHR • Chapter 9

Page 2

3. a) Give three examples of stick lengths that are too long. Where are

these values located on the number line?

b) Discuss with a partner how to state the possible length of the

shortest illegal stick. Is it reasonable to have a minimum length

for the shortest illegal stick? Why or why not?

Refl ect and Check

4. The value of 160 cm could be called a boundary point for the

allowable length of hockey sticks.

a) Look at the number line and explain what you think the term

boundary point means.

b) In this situation, is the boundary point included as an allowable

length of stick? Explain.

5. The allowable length of hockey sticks can be expressed mathematically

as an inequality . Since sticks must be less than or equal to 160 cm in

length, the linear inequality is l ≤ 160, where l, in centimetres,

represents the stick length.

Write an inequality to represent the lengths of illegal sticks.

Discuss your answer with a classmate.

Did You Know?

Most adult hockey

sticks range from

142 cm to 157.5 cm

in length.

inequality

• a mathematical

statement comparing

expressions that may

not be equal

• can be written using

the symbol <, >, ≤, ≥,

or ≠

Did You Know?

The world’s largest hockey stick

and puck are in Duncan, British

Columbia. The stick is over 62 m

in length and weighs almost

28 000 kg.

9.1 Representing Inequalities • MHR 341

Page 16

Example 3: Model and Solve a Problem

A games store is offering games on sale for $12.50, including tax. Sean has

set his spending limit at $80. How many games can Sean buy and stay

within his limit?

a) Write an inequality to model the problem.

b) Solve the inequality and interpret the solution.

Solution

a) If n represents the number of games that Sean can buy, the cost

of n games is 12.5 times n. Sean must spend no more than $80.

The situation can be modelled with the inequality 12.5n ≤ 80.

b) 12.5n ≤ 80

12.5n ______

12.5

≤ 80 _____

12.5

n ≤ 6.4

Sean can buy up to and including six games and stay

within his spending limit.

Yvonne is planting trees as a summer job. She gets paid $0.10 per tree

planted. She wants to earn at least $20/h. How many trees must she

plant per hour in order to achieve her goal?

a) Write an inequality to model the number of trees Yvonne must plant

to reach her goal.

b) Will the solution be a set of whole numbers or a set of integers? Explain.

c) Solve the inequality and interpret the solution.

Show You Know

12 × 6 = 72 12 × 7 = 84

The number of games Sean can buy

is between 6 and 7.

Did You Know?

Piecework is work paid

by the amount done,

not by the time it

takes. For example,

tree planters are paid

by the number of

trees they plant.

Since it is not possible to buy part of a

game, 6.4 is not a solution to the

original problem, even though it is a

solution to the inequality 12.5n ≤ 80.

Only a whole number is a possible

solution for this situation.

9.2 Solving Single-Step Inequalities • MHR 355

Page 17

Key Ideas

• The solution to an inequality is the value or values that makes the

inequality true.

5x > 10

A specifi c solution is any value greater than 2. For example, 2.1, 3, or 22.84.

The set of all solutions is x > 2.

1 2 3

• You can solve an inequality involving addition, subtraction,

multiplication, and division by isolating the variable.

x - 3 ≤ 5 8x ≤ 24

x

___

-2

> 6

x - 3 + 3 ≤ 5 + 3

8x

___

8

≤

24

___

8

x

___

-2

× -2 < 6 × -2

x ≤ 8 x ≤ 3 x < -12

Reverse the inequality

symbol when

multiplying or

dividing both sides by

a negative number.

Check Your Understanding

Communicate the Ideas

1. Maria and Ryan are discussing the inequality 2x > 10.

Maria: Ryan:

What does Ryan mean?

2. Explain how the process for verifying a solution is different for a linear

inequality than for a linear equation. Discuss your answer with a classmate.

The solution to the

inequality is 6. When I

substitute 6 for x, a true

statement results.

I agree that 6 is a

solution but it is not

the whole solution.

• To verify the solution to an inequality, substitute possible values into

the inequality:

� Substitute the value of the boundary point to check if both sides

are equal.

� Substitute specifi c value(s) from the solution to check that the

inequality symbol is correct.

Check if x ≥ -3 is the solution to -8x ≤ 24.

Substitute the boundary point -3.

-8x = 24

-8(-3) = 24

24 = 24

The two sides are equal. Therefore, -3 is

the correct boundary point.

Substitute a value greater than the boundary point -3.

-8x ≤ 24

-8(0) ≤ 24

0 ≤ 24

Substituting a value greater than -3 results in a true

statement. Therefore, the inequality symbol is correct.

356 MHR • Chapter 9

Page 31

Chapter 9 Practice Test

For #1 to #5, select the best answer.

1. Karen told her mother that she would be

out for no more than 4 h. If t represents the

time in hours, which inequality represents

this situation?

A t < 4 B t ≤ 4

C t > 4 D t ≥ 4

2. Which inequality does the number line

represent?

-3 -2 -1 10

A x < -1 B x ≤ -1

C x > -1 D x ≥ -1

3. Which number is not a specifi c solution

for the inequality y - 2 ≥ -4?

A -6 B -2

C 2 D 6

4. Solve: 5 - x < 2

A x < 3 B x > 3

C x < 7 D x > 7

5. What is the solution of 5(x - 3) ≤ 2x + 3?

A x ≤ -6 B x ≥ -6

C x ≤ 6 D x ≥ 6

Complete the statements in #6 and #7.

6. The number line representing the inequality

x < 5 would have a(n)�circle at 5 and

an arrow pointing to the�.

7. The solution to -4x < 16 is x � than �.

Short Answer

8. Represent each inequality on a number line.

a) -3 < x

b) x ≤ 6.8

9. Verify whether x > -3 is the correct

solution to the inequality 8 - 5x < 23.

Show your thinking. If the solution is

incorrect, explain why.

10. Christine is researching a career as an airline

pilot. One airline includes the following

criteria for pilots. Express each of the

criteria algebraically as an inequality.

a) Pilots must be shorter than 185 cm.

b) Pilots must be at least 21 years old.

11. Solve and graph each inequality.

a) -6 + x ≥ 10

b) 2.4x - 11 > 4.6

c) 12 - 8x < 17 - 6x

12. Represent each situation algebraically as an

inequality.

a) Luke earns $4.75 per item sold and must

earn over $50.

b) It takes Nicole 3 h to sew beads on a

pair of mitts. She has no more than 40 h

of time to sew beads on all the mitts she

plans to give to her relatives as presents.

370 MHR • Chapter 9

Page 32

You are an amusement park manager who has been off ered a job planning a new

park in a diff erent location.

a) Give your park a name and choose a location. Explain how you made your choice.

State the population of the area around the park that you chose.

b) Choose a reasonable number of rides for your park. Assume that the fi xed costs

include $5000 in addition to maintenance and wages. Assume maintenance and

repairs cost $400 per ride and that it takes eight employees to operate and

supervise each ride. Conduct research and then decide:

• the number of hours that rides will be open

• the average hourly wage for employees

c) Organize your estimates about operating expenses and revenues for the park.

You can use the table in the Math Link on page 367 as a reference.

d) Write an expression to represent each of the following for the number of rides

you chose:

• expenses per visitor

• revenue per visitor

e) For the number of rides you chose, how many visitors will be needed for the park

to make a profi t? Show all your work. Justify your solution mathematically.

f) Assume that you have now opened your park. You fi nd that 0.1% of the people in

the area come to the park per day, on average. Using this information, will your

park earn a profi t? If not, explain what changes you could make. Show all your

work and justify your solution.

Extended Response

13. Consider the inequality 6x - 4 > 9x + 20.

a) Solve the inequality algebraically.

b) Represent the solution graphically.

c) Give one value that is a specifi c solution

and one that is a non-solution.

d) To solve the inequality, Min fi rst

subtracted 6x from both sides. Alan

fi rst subtracted 9x from both sides.

Which method do you prefer? Explain

why.

14. The Lightning Soccer Club plans to buy

shirts for team members and supporters.

Pro-V Graphics charges a $75 set-up fee

plus $7 per shirt. BT Designs has no set-up

fee but charges $10.50 per shirt. How many

shirts does the team need to order for Pro-V

Graphics to be the better option?

15. Dylan is organizing a curling tournament. The

sports complex charges $115/h for the ice

rental. Dylan has booked it for 6 h. He will

charge each of the 14 teams in the tournament

an entrance fee. How much must he charge

each team in order to make a profi t?

What might be the

problem if you

choose too few or

too many rides?

Chapter 9 Practice Test • MHR 371