﻿ Graph Linear Inequalities Worksheet - Page 2 | Problems & Solutions

# Graph Linear Inequalities Worksheet - Page 2

Graph Linear Inequalities Worksheet
• Page 2
11.
Which of the graphs represents the inequality $y$ < $x$ + 4?

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

#### Solution:

Since the inequality involves less than (<), use dashed line to represent the boundary of y < x + 4.

y < x + 4
0 < 0 + 4
0 < 4 True
Test a point not on the boundary line.
Test (0, 0) in the inequality.
[Substitute.]

Since the inequality is true for (0, 0), shade the region containing (0, 0).

The above graph matches with the graph 2.

12.
Which of the graphs represents the inequality $y$ > $x$ + 4?

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

#### Solution:

Since the inequality involves greater than (>), use dashed line to represent the boundary of y > x + 4.

y > x + 4
0 > 0 + 4
0 > 4
Test a point not on the boundary line.
Test (0, 0) in the inequality.
[Substitute.]
[False.]

Since the inequality is false for (0, 0), shade the region that does not contain (0, 0).

The above graph matches with the graph 3.

13.
Which of the graphs represents the inequality $y$$x$ - 4?

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

#### Solution:

Since the inequality involves greater than or equal to (≥), the boundary line of the inequality y = x - 4 is a solid line.

yx - 4
0 ≥ 0 - 4
0 ≥ -4
Test a point not on the boundary line.
Test (0, 0) in the inequality.
[Substitute.]

[True.]

Since the inequality is true for (0, 0), shade the region that contains (0, 0).

The above graph matches with the graph 4.

14.
Which of the graphs represents the inequality $y$ ≤ 5$x$ + 3?

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

#### Solution:

Since the inequality involves less than or equal to (≤), the boundary line of the inequality y = 5x + 3 is a solid line as shown in the graph.

y ≤ 5x + 3
0 ≤ 0 + 3
0 ≤ 3
Test a point not on the boundary line.
Test (0, 0) in the inequality.
[Substitute.]
[True.]

Since the inequality is true for (0, 0), shade the region that contains (0, 0).

The above graph matches graph 3.

15.
Which of the graphs represent the linear inequality $y$ ≥ 5$x$ - 3?

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

#### Solution:

Since the inequality involves greater than or equal to (≥), the boundary line of the inequality y ≥ 5x - 3 is a solid line as shown in the graph.

y ≥ 5x - 3
0 ≥ 0 - 3
0 ≥ -3
Test a point not on the boundary line.
Test (0, 0) in the inequality.
[Substitute.]

[True.]

Since the inequality is true for (0, 0), shade the region that contains (0, 0).

The above graph matches with Graph 1.

16.
Write the linear inequality for the graph.

 a. 6$x$ - 3$y$ ≤ 12 b. 6$x$ - 3$y$ ≥ 12 c. -6$x$ + 3$y$ ≥ 12 d. 6$x$ + 3$y$ ≤ -12

#### Solution:

Since the boundary line is solid, the linear inequality must be either ≥ or ≤.

Take a point from the shaded region i.e. solution and check which of the equations satisfies the linear inequality.

6(3) - 3(1) ≥ 12

15 ≥ 12
[Replace x with 3 and y with 1 in the equation 6x -3y ≥ 12.]
[True.]

The linear inequality for the graph is 6x - 3y ≥ 12.

17.
Write the equation of the boundary line 5$x$ + 6$y$ ≤ 18, in slope-intercept form.
 a. y = $\frac{-5}{18}$x - 3 b. y = $\frac{5}{6}$x - 3 c. x = $\frac{-5}{6}$y + 3 d. y = $\frac{-5}{6}$x + 3

#### Solution:

5x + 6y ≤ 18
[Original equation.]

5x + 6y = 18
[Write the inequality in the form of equality.]

6y = -5x + 18

y = -56x + 3
[To write in slope-intercept form divide each side by 6.]

The equation of the boundary line in slope - intercept form is y = -5 / 6x + 3.

18.
Write a system of linear inequalities to describe the graph.

 a. y ≤ $\frac{3}{2}$x + 3; y ≥ -x - 3 b. y ≥ $\frac{-3}{2}$x - 3; y ≤ -x + 3 c. y ≥ $\frac{3}{2}$x + 3; y > -x + 3 d. y ≤ $\frac{3}{2}$x + 3; y > -x + 3

#### Solution:

The y-intercept of the Line 1 is 3 and the slope is 3 / 2.
[From the graph.]

The equation of Line 1 in slope-intercept form is y = 3 / 2x + 3
[Substitute m = 3 / 2 and b = 3 in the equation y = mx + b.]

As Line 1 is a solid line and the region above the line is shaded, the equation should be y3 / 2x + 3.
[From the graph.]

The y-intercept of Line 2 is 3 and the slope is -1.
[From the graph.]

The equation of Line 2 in slope-intercept form is y = -x + 3
[Substitute m = -1 and b = 3 in the equation y = mx + b.]

As Line 2 is a dashed line and the region above the line is shaded, the inequality should be y > -x + 3.
[From the graph.]

So, the system of linear inequalities is y3 / 2x + 3 and y > -x + 3.

19.
Find the two numbers whose difference is less than or equal to 5. Show the solution by graphing an inequality.

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

#### Solution:

Let one number be x and the other number be y.

x - y ≤ 5
[The difference of the two numbers is less than or equal to 5.]

Since the inequality involves less than or equal to(≤) the boundary line of the equation x - y ≤ 5 is a solid line as shown

Substitute any point in the equation that does not lie on the boundary line.

x - y ≤ 5

2 - 1 ≤ 5
[Test (2, 1).]

1 ≤ 5
[True.]

The inequality 1 ≤ 5 is true. So, shade the region that contains (2, 1) as shown

The above graph matches with the graph 2.

20.
Write a system of linear inequalities to describe the graph.

 a. y < $\frac{2}{3}$x + 1; y > -x - 3 b. y > $\frac{2}{3}$x + 2; y ≤ -x - 3 c. y < $\frac{2}{3}$x + 2; y ≥ -x - 3 d. y < $\frac{2}{3}$x + 1; y ≤ -x - 3

#### Solution:

The y-intercept of Line 1 is 2 and slope is 2 / 3.
[From the graph.]

The equation of Line 1 in slope-intercept form is y = 2 / 3x + 2
[Substitute m = 2 / 3 and b = 2 in the equation y = mx + b.]

As Line 1 is a dashed line and the region above the line is shaded, the equation should be y > 2 / 3x + 2.
[From the graph.]

The y-intercept of Line 2 is -3 and slope is -1.
[From the graph.]

The equation of Line 2 in the slope-intercept form is y = -x - 3
[Substitute m = -1 and b = -3 in the equation y = mx + b.]

As Line 2 is a solid line and the region below the line is shaded, the inequality should be y ≤ -x - 3.
[From the graph.]

So, the system of linear inequalities is y > 2 / 3x + 2 and y ≤ -x - 3.