# Graph Linear Inequalities Worksheet

Graph Linear Inequalities Worksheet
• Page 1
1.
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 the values.]

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

The above graph matches with the graph 2.

2.
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 the values.]

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

The above graph matches with the graph 4.

3.
Graph the system of linear inequalities.
$x$ ≥ 0
$y$ ≥ 0
4$x$ + 6$y$ ≤ 12

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

#### Solution:

The graph of x ≥ 0 is the half-plane on and to the right of the solid line x = 0.

The graph of y ≥ 0 is the half-plane on and above the solid line y = 0.

The graph of 4x + 6y ≤ 12 is the half-plane on and below the solid line 4x + 6y = 12.

The shaded portion shown below is the intersection of the three half-planes, which is the graph of the system.

So, Graph 1 represents the system of linear inequalities.

4.
Which of the graphs represents the inequality $x$ - $y$ > - 1?

 a. Graph 2 b. Graph 1 c. Both A and B d. None of the above

#### Solution:

x - y > - 1

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

y = x + 1
[Subtracting x from the two sides of the equation].

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).

5.
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.

6.
Graph the linear inequality: |$x$| ≥ 3

 a. Graph C b. Graph A c. Graph B d. Graph D

#### Solution:

|x| ≥ 3

x ≤ - 3 or x ≥ 3

The graph of |x| ≥ 3 consists of the solid lines x = - 3 and x = 3 and the region between these lines.

7.
Check whether the ordered pair (0, $\frac{-6}{5}$ ) is a solution of the inequality 4$x$ + 5$y$ ≤ 4.
 a. No b. Yes

#### Solution:

4x + 5y ≤ 4

4x + 5y
[Take left hand side of the inequality.]

= 4(0) + 5(-65 )
[Substitute the values.]

= 0 + (-6)
[Simplify.]

-6 ≤ 4
[Compare with the right hand side of the orginal expression.]

The ordered pair is a solution of 4x + 5y ≤ 4.

8.
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 the values.]

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

The above graph matches graph 3.

9.
Graph the system of linear inequalities.
2$x$ - 3$y$ < 6
- $x$ + $y$ ≤ 4
2$x$ + 4$y$ < 8

 a. Graph 1 b. Graph 2

#### Solution:

The graph of 2x - 3y < 6 is the half-plane above the dashed line 2x - 3y = 6.

The graph of - x + y ≤ 4 is the half-plane on and below the solid line - x + y = 4.

The graph of 2x + 4y < 8 is the half-plane below the dashed line 2x + 4y = 8.

Finally, the graph of the system is the intersection of the three half-plane as shown in the graph.

10.
Which of the graphs best represents the system of inequalities, $x$ + $y$ < 3, - 5$x$ + 2$y$ ≤ 10, $y$ ≥ - 3?

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

#### Solution:

Graph all the three inequalitites in the same coordinate plane.

The graph of x + y < 3 is the half-plane below the dashed line x + y = 3.

The graph of - 5x + 2y ≤ 10 is the half-plane on and below the solid line - 5x + 2y = 10.

The graph of y ≥ - 3 is the half-plane on and above the solid line y = - 3.

The graph of the system is the intersection of the three half planes as shown in the graph.

Graph 1 is the correct answer.