# Graph Linear Inequalities Worksheet - Page 3

Graph Linear Inequalities Worksheet
• Page 3
21.
Write a system of linear inequalities to describe the graph.

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

#### Solution:

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

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

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

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

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

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

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

22.
Which of the inequalities represents 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:

The y-intercept of the line is 4 and the slope is -2.

The equation of the line in slope-intercept form is y = -2x + 4.
[Substitute m = -2 and b = 4 in the equation y = mx + b.]

As the boundary line is a solid line and the region below the line is shaded, the inequality should be y ≤ -2x + 4.

3y ≤ -6x + 12
[Multiply by 3 on both sides of inequality.]

6x + 3y ≤ 12
[Rearrange the inequality.]

So, 6x + 3y ≤ 12 represents the graph.

23.
Which of the inequalities represents 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:

From the garph y-intercept is 4 and the slope is 2.

The equation of the boundary line in the slope-intercept form is y = 2x + 4.
[Substitute m = 2 and b = 4 in the equation y = mx + b.]

As the boundary line is a dotted line and the side not containing the origin is shaded the inequality should be y > 2x + 4.

3y > 6x + 12
[Multiply the above inequality by 3.]

-6x + 3y > 12
[Rearrange the above inequality.]

So, the inequality -6x + 3y > 12 represents the graph.

24.
Write the equation of the boundary line -4$x$ + 7$y$ ≤ 28 in slope-intercept form.
 a. y = -($\frac{4}{7}$)x - 4 b. y = ($\frac{4}{7}$)x + 4 c. y = -($\frac{4}{7}$)x + 4 d. y = ($\frac{4}{7}$)x - 4

#### Solution:

-4x + 7y ≤ 28
[Original equation.]

-4x + 7y = 28
[Write the inequality in form of equality.]

7y = 4x + 28

y = (47)x + 4
[Divide by 7 on each side.]

25.
Write the equation of the boundary line 3$x$ + 7$y$ ≥ 14 in slope-intercept form.
 a. y = $\frac{-3}{7}$x + 2 b. $y$ = $\frac{3}{7}$$x$ - 2 c. y = $\frac{-3}{7}$x - 2 d. y = $\frac{3}{7}$x + 2

#### Solution:

3x + 7y ≥ 14
[Original equation.]

3x + 7y = 14
[Write the inequality in the form of equality.]

7y = -3x + 14

y = -37x + 2
[To write in slope-intercept form divide both sides by 7.]

The equation of the boundary line in slope - intercept form is y = -3 / 7x + 2.

26.
Write the equation of the boundary line 8$x$ - 9$y$ ≥ 45 in slope-intercept form.
 a. y = ($\frac{8}{9}$)x + 5 b. y = -($\frac{8}{9}$)x + 5 c. y = -($\frac{8}{9}$)x - 5 d. y = ($\frac{8}{9}$)x - 5

#### Solution:

8x - 9y ≥ 45
[Original equation.]

8x - 9y = 45
[Write the inequality in the form of equality.]

9y = 8x - 45

y = (89)x - 5
[Divide by 9 on each side.]

27.
Tell whether the boundary line 7$x$ + $y$ ≥ 9 is solid or dashed.
 a. Dashed b. Solid

#### Solution:

7x + y ≥ 9
[Original Equation.]

If the points on the boundary line make 7x + y ≥ 9 true, then the boundary line will be solid or else dashed.

As the inequality involves greater than or equal to(≥) (7x+y) can be equal to 9. So, there is a possibility that the points that lie on the boundary line make the inequality true.

So, the boundary line is a solid line.

28.
Tell whether the boundary line of the inequality 9$x$ + $y$ > 2 is solid or dashed.
 a. Solid b. Dashed

#### Solution:

9x + y > 2
[Original Equation.]

If the points on the boundary line make 9x + y > 2 true, then the boundary line will be solid or else dashed.

As the inequality involves '>', (9x + y) is always greater than but not equal to 2. So, the points that lie on the boundary line does not satisfy the inequality.

So, the boundary line is a dashed line.