﻿ Graphing Linear Inequalities in Two Variables Worksheet | Problems & Solutions

# Graphing Linear Inequalities in Two Variables Worksheet

Graphing Linear Inequalities in Two Variables Worksheet
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1.

Which of the graphs best suits the inequality $y$ > 1?

 a. Graph 1 b. Graph 3 c. Graph 2 d. Graph 4

#### Solution:

As the inequality of the line is >, use dashed line to represent y = 1.

The origin (0, 0) is not a solution and it lies below the line.

The solution of the inequality is the other half of the plane, which does not include the origin (0,0).

Therefore, Graph 2 best suits the inequality y > 1.

2.
Which of the graphs best suits the inequality $y$ < 1?

 a. Graph-2 b. Graph-4 c. Graph-3 d. Graph-1

#### Solution:

Graph the corresponding equation y = 1 with dashed line.
[Since the inequality is <.]

The point (0, 0) is a solution to the inequality and it lies below the line.

The graph of y < 1 is the half plane below the graph of y = 1.

Therefore, Graph 2 best suits the inequality y < 1

3.
Which of the graphs best suits the inequality $x$ - $y$ > - 1?

 a. Graph 1 b. Graph 3 c. Graph 4 d. Graph 2

#### Solution:

x - y > - 1
[Original inequality.]

x - y = - 1
[Write corresponding equation].

y = x + 1
[Subtract x from each side].

The corresponding equation in slope-intercept form is y = x + 1.

The graph of the line has a slope of 1 and a y-intercept of 1. Since the inequality is > use a dashed line.

The point (0, 0) satisfies the inequality. So, the solution is the half of the plane that includes the point (0, 0).

Therefore, graph 1 best suits the inequality x - y > - 1.

4.
Which of the following ordered pairs is not a solution of the inequality 3$x$ - 4$y$ > -4?
 a. (4, 3) b. (7, 3) c. (3, - 4) d. (- 3, 3)

#### Solution:

3x - 4y > -4
[Write original inequality.]

3(4) - 4(3) > -4
[Replace x with 4 and y with 3.]

0 > -4, which is true.
[Simplify.]

So, the ordered pair (4, 3) is a solution.

3x - 4y > -4
[Write original inequality.]

3(7) - 4(3) > -4
[Replace x with 7 and y with 3.]

9 > -4, which is true.
[Simplify.]

So, the ordered pair (7, 3) is a solution.

3x - 4y > -4
[Write original inequality.]

3(3) - 4(- 4) > -4
[Replace x with 3 and y with - 4.]

25 > -4, which is true.
[Simplify.]

So, the ordered pair (3, - 4) is a solution.

3x - 4y > -4
[Write original inequality.]

3(- 3) - 4(3) > -4
[Replace x with - 3 and y with 3.]

- 21 > -4, which is not true.
[Simplify.]

So, the ordered pair (- 3, 3) is not a solution.

Therefore, (- 3, 3) is not a solution for the inequality 3x - 4y > -4.

5.
Choose the ordered pair that is not the solution of the inequality whose graph is shown.

 a. (2, - 4) b. (4, - 2) c. (4, - 4) d. (- 2, 2)

#### Solution:

The y-intercept of the line is - 4 and the slope is 1.

The equation of the line in slope-intercept form is y = x - 4.
[Substitute m = 1 and b = - 4 in the equation y = mx + b.]

As the boundary line is a dashed line and the region below the line is shaded, the inequality should be y < x - 4.

y < x - 4
[Write original inequality.]

- 4 < 2 - 4
[Replace x with 2 and y with - 4.]

- 4 < - 2, which is true.
[Simplify.]

So, the ordered pair (2, - 4) is a solution.

y < x - 4
[Write original inequality.]

- 2 < 4 - 4
[Replace x with 4 and y with - 2.]

- 2 < 0, which is true.
[Simplify.]

So, the ordered pair (4, - 2) is a solution.

y < x - 4
[Write original inequality.]

- 4 < 4 - 4
[Replace x with 4 and y with - 4.]

- 4 < 0, which is true.
[Simplify.]

So, the ordered pair (4, - 4) is a solution.

y < x - 4
[Write original inequality.]

2 < - 2 - 4
[Replace x with - 2 and y with 2.]

2 < - 6, which is not true.
[Simplify.]

So, the ordered pair (- 2, 2) is not a solution.

Therefore, (- 2, 2) is not a solution for the inequality of the graph shown.

6.
Which of the following ordered pairs is not a solution of the inequality $x$ - $\frac{y}{5}$ > 6?
 a. (8, - 5) b. (3, 5) c. (0, - 40) d. (11, 0)

#### Solution:

x - y5 > 6
[Original inequality.]

3 - 55 > 6
[Replace x with 3 and y with 5.]

2 > 6, which is not true.
[Simplify.]

So, the ordered pair (3, 5) is not a solution.

x - y5 > 6
[Original inequality.]

8 - (- 55) > 6
[Replace x with 8 and y with - 5.]

9 > 6, which is true.
[Simplify.]

So, the ordered pair (8, - 5) is a solution.

x - y5 > 6
[Original inequality.]

11 - (05) > 6
[Replace x with 11 and y with 0.]

11 > 6, which is true.
[Simplify.]

So, the ordered pair (11, 0) is a solution.

x - y5 > 6
[Original inequality.]

0 - (- 405) > 6
[Replace x with 0 and y with - 40.]

8 > 6, which is true.
[Simplify.]

So, the ordered pair (0, - 40) is a solution.

The ordered pair (3, 5) is not a solution of the inequality x - y5 > 6.

7.
Which of the following is a linear inequality with two variables?
 a. 2$x$ + 6$y$ = 0 b. $x$ + 6 = 0 c. 8$x$ + 1 ≤ 6 d. 6$x$ + 2$y$ ≥ 6

#### Solution:

A linear inequality in two variables will contain only two variables and an inequality symbol like ≠, ≤, <, ≥ or >.

Among the choices, 6x + 2y ≥ 6 is a linear inequality in two variables x and y.

8.
Identify the inequality which has the ordered pair (2, - 3) as a solution.
 a. 5$x$ + 3$y$ ≤ 0 b. 5$x$ - 3$y$ < 0 c. 5$x$ - 3$y$ ≤ 0 d. 5$x$ + 3$y$ ≥ 0

#### Solution:

5x + 3y ≤ 0

5(2) + 3(- 3) ≤ 0
[Replace x with 2 and y with - 3.]

1 ≤ 0, which is false.
[Simplify.]

5x - 3y < 0

5(2) - 3(- 3) < 0
[Replace x with 2 and y with - 3.]

19 < 0, which is false.
[Simplify.]

5x - 3y ≤ 0

5(2) - 3(- 3) ≤ 0
[Replace x with 2 and y with - 3.]

19 ≤ 0, which is false.
[Simplify.]

5x + 3y ≥ 0

5(2) + 3(- 3) ≥ 0
[Replace x with 2 and y with - 3.]

1 ≥ 0, which is true.
[Simplify.]

So, (2, - 3) is a solution of 5x + 3y ≥ 0.

9.
Choose the inequality which has the ordered pair (0, 0) as the solution.
 a. 3$x$ + 3$y$ > 4 b. 3$x$ - 3$y$ > 4 c. - 3$x$ + 3$y$ > 4 d. 3$x$ - 3$y$ < 4

#### Solution:

3x + 3y > 4

3(0) + 3(0) > 4
[Replace x with 0 and y with 0.]

0 > 4, which is false.
[Simplify.]

3x - 3y > 4

3(0) - 3(0) > 4
[Replace x with 0 and y with 0.]

0 > 4, which is false.
[Simplify.]

- 3x + 3y > 4

- 3(0) + 3(0) > 4
[Replace x with 0 and y with 0.]

0 > 4, which is false.
[Simplify.]

3x - 3y < 4

3(0) - 3(0) < 4
[Replace x with 0 and y with 0.]

0 < 4, which is true.
[Simplify.]

So, (0, 0) is a solution of the inequality 3x - 3y < 4.

10.
Choose the inequality which has the ordered pair (0, - $\frac{4}{3}$ ) as a solution.
 a. 3$x$ + 2$y$ ≥ 0 b. 2$x$ - 3$y$ < 0 c. 2$x$ + 3$y$ > 0 d. 2$x$ - 3$y$ > 0

#### Solution:

2x - 3y > 0

2(0) - 3(- 43 ) > 0
[Replace x with 0 and y with - 4 / 3.]

4 > 0, which is true.
[Subtract.]

2x - 3y < 0

2(0) - 3(- 43 ) < 0
[Replace x with 0 and y with - 4 / 3.]

4 < 0, which is false.
[Subtract.]

2x + 3y > 0

2(0) + 3(- 43 ) > 0
[Replace x with 0 and y with - 4 / 3.]

- 4 > 0, which is false.
[Subtract.]

3x + 2y ≥ 0

3(0) + 2(- 43 ) ≥ 0
[Replace x with 0 and y with - 4 / 3.]

- 83 ≥ 0, which is false.
[Subtract.]

So, the ordered pair (0, - 4 / 3 ) is a solution to the inequality 2x - 3y > 0.