﻿ Graphing Linear Inequalities in Two Variables Worksheet - Page 2 | Problems & Solutions

# Graphing Linear Inequalities in Two Variables Worksheet - Page 2

Graphing Linear Inequalities in Two Variables Worksheet
• Page 2
11.
Choose the ordered pair which is the solution for the inequality 4$x$ + 5$y$ ≤ 0.
 a. (0, $\frac{6}{5}$) b. (0, - $\frac{6}{5}$) c. (6, - $\frac{6}{5}$) d. (- 1, $\frac{6}{5}$)

#### Solution:

4x + 5y ≤ 0
[Original inequality.]

4(0) + 5(- 65) ≤ 0
[Replace x with 0 and y with - 6 / 5.]

- 6 ≤ 0, which is true
[Subtract.]

4(0) + 5(65) ≤ 0
[Replace x with 0 and y with 6 / 5.]

6 ≤ 0, which is false
[Subtract.]

4(6) + 5(- 65) ≤ 0
[Replace x with 6 and y with - 6 / 5.]

18 ≤ 0, which is false
[Subtract.]

4(- 1) + 5(65) ≤ 0
[Replace x with - 1 and y with 6 / 5.]

2 ≤ 0, which is false
[Subtract.]

So, the ordered pair (0, - 6 / 5) is a solution to the inequality 4x + 5y ≤ 0.

12.
Choose 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 that is 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.

13.
Choose the equation of the boundary line 5$x$ + 6$y$ ≤ 18, in slope-intercept form.
 a. $y$ = - $\frac{5}{6}$$x$ + 9 b. $y$ = - $\frac{5}{6}$$x$ + 3 c. $y$ = $\frac{5}{6}$$x$ - 3 d. $x$ = - $\frac{5}{6}$$y$ + 3

#### Solution:

5x + 6y ≤ 18
[Original inequality.]

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.

14.
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.

15.
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.

16.
Write the equation of the boundary line 2$x$ + 5$y$ ≥ 10 in slope-intercept form.
 a. $y$ = $\frac{2}{5}$$x$ - 2 b. $y$ = $\frac{2}{5}$$x$ + 2 c. $y$ = - $\frac{2}{5}$$x$ - 2 d. $y$ = - $\frac{2}{5}$$x$ + 2

#### Solution:

2x + 5y ≥ 10
[Original inequality.]

2x + 5y = 10
[Write the inequality in the form of equality.]

5y = - 2x + 10
[Add - 2x on each side.]

y = - 25x + 2
[To write in slope-intercept form divide both sides by 5.]

The equation of the boundary line in slope - intercept form is y = - 2 / 5x + 2.

17.
Write the equation of the boundary line 2$x$ - 3$y$ ≥ 6 in slope-intercept form.
 a. $y$ = ($\frac{2}{3}$)$x$ - 2 b. $y$ = - ($\frac{2}{3}$)$x$ + 2 c. $y$ = - ($\frac{2}{3}$)$x$ - 2 d. $y$ = ($\frac{2}{3}$)$x$ + 2

#### Solution:

2x - 3y ≥ 6
[Original inequality.]

2x - 3y = 6
[Write the inequality in the form of equality.]

- 3y = - 2x + 6
[Add - 2x to both sides.]

3y = 2x - 6
[Multiply with - 1 on both sides.]

y = (23)x - 2
[Divide by 3 on each side.]

18.
Which of the following inequalities the boundary line in the graph would be solid?6
 a. 4$x$ + 5$y$ > 3 b. 5$x$ - 6$y$ < 2 c. 4$x$ + 5$y$ < 3 d. 5$x$ + 6$y$ ≥ 2

#### Solution:

The boundary line of the half-plane is dashed if the inequality is < or > and solid if the inequality is ≤ or ≥.

For the inequalities 4x + 5y > 3, 5x - 6y < 2 and 4x + 5y < 3 the boundary line of the half-plane is dashed as they contain > and < symbols.

For the inequality 5x + 6y ≥ 2 the boundary line of the half-plane is solid as it contains ≥ symbol.

19.
Which of the following inequalities the boundary line in the graph would be dashed?
 a. 9$x$ + 10$y$ ≤ 3 b. 9$x$ - $y$ ≥ 3 c. $x$ + 9$y$ ≤ 3 d. 9$x$ + $y$ > 3

#### Solution:

The boundary line of the half-plane is dashed if the inequality is < or > and solid if the inequality is ≤ or ≥.

For the inequalities 9x - y ≥ 3, 9x + 10y ≤ 3 and x + 9y ≤ 3 the boundary line of the half-plane is solid as they contain ≥ and ≤ symbols.

For the inequality 9x + y > 3 the boundary line of the half-plane is dashed as it contains > symbol.