﻿ Graph Linear Inequalities Worksheet - Page 2 | Problems & Solutions
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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.

Correct answer : (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.

Correct answer : (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.

Correct answer : (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.

Correct answer : (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.

Correct answer : (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.

Correct answer : (2)
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
[Add -5x on each side.]

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.

Correct answer : (4)
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.

Correct answer : (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.

Correct answer : (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.

Correct answer : (2)

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