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
