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Middle Years

10.01 Linear Inequalities

Worksheet
Linear inequalities
1

State whether the following inequalities are represented using a solid line or a dashed line on a graph:

a

y - 3 x > 5

b

x \geq 4

c

- 2 y + x \leq - 10

d

y + 3 x > 7

e

- y - 6 x > 0

f

y + 3 x \leq - 10

g

y \geq - 2

h

3 y + 3 x > 4

2

Write the inequality that describes the following shaded regions:

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
b
-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
6
7
y
c
-4
-3
-2
-1
1
2
3
4
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
y
d
-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
6
7
y
e
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
f
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
y
3

Is \left(6, 20\right) a solution of y\geq 4 x + 5?

4

Is \left(3, 2\right) a solution of 3 x + 2 y \geq 12 ?

5

Determine which point satisfies each of the following inequalities:

a
x - y > 7
A
\left(-1, - 7 \right)
B
\left(8, - 5 \right)
C
\left(13, 6\right)
D
\left(0, 0\right)
b
y > 2 x
A
\left(9, 15\right)
B
\left( - 1 , - 2 \right)
C
\left(1, 4\right)
D
\left(0, 0\right)
c
y\gt - 10 x - 2
A
\left(5, - 52 \right)
B
\left( - 1 , 3\right)
C
\left( - 3 , 26\right)
D
\left(1, - 11 \right)
Graphing linear inequalities
6

Describe the three steps required to graph the inequality y + 7 x \geq 5.

7

Consider the inequality y \lt- 3 x - 6.

a

Solve for the x and y-intercepts of the line y = - 3 x - 6.

b

Determine which of the following points satisfy the inequality y \lt- 3 x - 6:

A
\left( - 7 , 2\right)
B
\left( - 6 , 3\right)
C
\left(3, - 2 \right)
D
\left( - 6 , - 3 \right)
c

Hence sketch the graph of y \lt - 3 x - 6.

d

Do the points on the line satisfy the inequality?

8

Consider the inequality y \leq - 2 x + 2.

a

Solve for the x and y intercepts of the line y = - 2 x + 2.

b

Determine which of the following points satisfies the inequality y \leq - 2 x + 2:

A
\left(2, 3\right)
B
\left(3, - 6 \right)
C
\left(4, - 2 \right)
D
\left(1, 2\right)
c

Hence sketch the graph of y \leq - 2 x + 2.

d

Do the points on the line satisfy the inequality?

9

The solutions to the equation y = x - 2 are plotted on the following graph:

a

Graph the solutions to y \gt x - 2.

b

Graph the solutions to y \leq x - 2.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
10

Sketch the graphs of the following inequalities:

a
y \leq x - 2
b
5 x + 2 y \geq 20
c
4 x + 3 y \gt 12
d
6 x + 3 y \gt 12
e

y \gt 2x - 4

f

y \leq 3 x - 4

g

y \lt 3x - 9

h

y \geq - 2x + 4

i
2 x + 3 y \lt 12
j

y \gt 2x - 4

k

y \leq x + 5

11

State the linear inequality that satisfies the following:

a

A linear inequality has solutions of \left(0, - 5 \right) and \left( - 1 , - 9 \right) on its boundary line and is not satisifed by \left(1, 3\right).

b

A linear inequality has solutions of \left(0, - 6 \right) and \left( - 6 , - 24 \right) on its boundary line and is satisifed by \left( - 3 , - 10 \right).

Applications
12

Luigi has several bottles of lemonade and Pepsi that he wants to take to a picnic, but he only has enough room in his backpack for 9 bottles.

Let x represent the number of bottles of lemonade that he takes to the picnic.

Let y represent the number of bottles of Pepsi that he takes to the picnic.

a

Write an inequality to represent the situation.

b

Graph the inequality on a cartesian plane.

c

Use the graph to determine whether Luigi can take the following combinations of bottles to the picnic:

i

1 bottle of lemonade and 11 bottles of Pepsi

ii

3 bottles of lemonade and 11 bottles of Pepsi

iii

3 bottles of lemonade and 4 bottles of Pepsi

iv

5 bottles of lemonade and 3 bottles of Pepsi

13

Airline passengers are told to be at the gate no less than 15 minutes before a flight's departure. It takes Gwen 30 minutes to get to the airport, and y minutes to go through check in and security, depending on how busy the airport is.

a

Form an inequality relating x and y, where x represents the number of minutes left until Gwen's flight departs.

b

Graph the inequality on a cartesian plane.

c

If the check-in and security process takes 20 minutes, how many minutes before her flight's departure can Gwen leave her house at the latest?

14

Roxanne has set aside \$14.00 in her shopping budget for fruit this week. Currently, oranges are on sale for \$1.40 each, while apples are on sale for \$1.75 each.

Let x represent the number of oranges that Roxanne buys.

Let y represent the number of apples that Roxanne buys.

a

Write an inequality to represent the situation.

b

Graph the inequality on a cartesian plane.

c

Use your graph to determine whether Roxanne can buy the following combination of fruits:

i

6 oranges and 7 apples

ii

3 oranges and 4 apples

iii

7 oranges and 4 apples

iv

8 oranges and 1 apples

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