﻿ Graphing Linear Inequalities and Equations Worksheet | Problems & Solutions
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Graphing Linear Inequalities and Equations Worksheet
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1.
Which of the following is a linear inequality with two variables? a. 2$x$ + 8$y$ ≥ 9 b. 8$x$ + 9$y$ = 0 c. 10$x$ + 7 ≤ 2 d. None of the above

#### Solution:

A linear inequality in two variables will contain only two variables and an inequality symbol like ≠, ≤, <, ≥ or >.

Among the choices, 2x + 8y ≥ 9 is a linear inequality in two variables x and y.

Correct answer : (1)
2.
State whether the ordered pair (15,5) is a solution of the inequality $\frac{x}{5}$ - $\frac{4y}{5}$ > 7. a. No b. Yes

#### Solution:

x / 5 - 4y / 5 > 7
[Original inequality.]

15 / 5 - 4(5) / 5 > 7
[Substitute 15 for x and 5 for y.]

3 - 4 > 7
[Simplify.]

-1 > 7
[Subtract.]

The above statement is false. So, the ordered pair is not a solution of the inequality.

Correct answer : (1)
3.
Write a system of linear inequalities that represents the shaded region of the figure.  a. $x$ + 5$y$ < 5 and $x$ > -1 b. $x$ + 5$y$ ≤ 5 and $x$ ≥ -1 c. $x$ + 5$y$ ≥ 5 and $x$ ≤ -1 d. None of the above

#### Solution:

As the shaded region is bounded by two lines, the system must have two linear inequalities.

From the graph, the first inequality is bounded by the line that passes through the points (0, 1) and (5, 0).

m = (y2 - y1) / (x2 - x1)
[Use slope formula to find slope of line (1).]

m = (0-1)(5-0)
[Substitute coordinates in the formula.]

m = -15
[Simplify.]

As line (1) crosses y-axis at the point (0, 1), its equation can be found using the slope-intercept form.

y = mx + b
[Write slope-intercept form.]

y = -15x + 1
[Substitute -1 / 5 for m and 1 for b.]

x + 5y = 5
[Simplify.]

As the shaded region is below the solid boundary line (1), the first inequality is x + 5y ≤ 5.

From the graph, the second inequality crosses the x-axis at (-1, 0). So, it is bounded by the line x = -1.

As the shaded region is to the right of the solid boundary line (2), the second inequality is x ≥ -1.

The system of inequalities that represent the shaded region is x + 5y ≤ 5 and x ≥ -1.

Correct answer : (2)
4.
Write a system of linear inequalities that defines the shaded region in the figure.  a. 2$y$ < 2 and -$x$ + $y$ < -1 b. -$x$ ≤ 2 and -$x$ + $y$ = -1 c. -$x$ + 2$y$ < 2 and -$x$ + $y$ > -1 d. None of the above

#### Solution:

There are two lines that bound the shaded region. So, the system must have two linear inequalities.

Line (1) passes through the points (-2, 0) and (0, 1).

m = (y2 - y1) / (x2 - x1)
[Use slope formula to find slope of line (1).]

m = (1-0)(0-(-2)) = 12
[Substitute coordinates in the formula and simplify.]

Since, line (1) crosses y-axis at (0, 1), its equation can be found using the slope-intercept form where b = 1.

y = mx + b
[Write slope-intercept form.]

-x + 2y = 2
[Substitute values of m and b and simplify.]

Since the shaded region is below the dashed boundary line (1), the inequality is -x + 2y < 2.

Line (2) passes through the points (0, -1) and (1, 0).

m = (y2 - y1) / (x2 - x1)
[Use slope formula to find slope of line (2).]

m = 0-(-1)1-0 = 1
[Substitute cordinates in the formula and simplify.]

Since, line (2) passes through the point (0, -1), its equation can be found using the slope-intercept form.

y = mx + b
[Write slope-intercept form.]

-x + y = -1
[Substitute 1 for m and -1 for b and simplify.]

Since the shaded region is above the dashed boundary line (2), the inequality is -x + y > -1.

The system of inequalities that describes the shaded region is -x + 2y < 2 and -x + y > -1.

Correct answer : (3)
5.
Write a system of linear inequalities that define the shaded region.  a. $x$ ≥ -1 and $x$ + $y$ = 1 b. $x$ + $y$ = -1 and $y$ = 1 c. $x$ + $y$ ≥ -1 and $x$ + $y$ < 1 d. None of the above

#### Solution:

The shaded region is between the two lines. So, the system must have two linear inequalities.

Line (1) passes through the points (0, -1) and (-1, 0).

m = (y2 - y1) / (x2 - x1)
[Use slope formula to find slope of line (1).]

m = (0-(-1))(-1-0) = -1
[Substitute coordinates in the formula and simplify.]

Since line (1) crosses y-axis at the point (0, -1), its equation can be found using the slope-intercept form.
[The y intercept of the line is -1]

y = mx + b
[Write slope-intercept form.]

x + y = -1
[Substitute -1 for m and -1 for b and simplify.]

Since the shaded region is on and above the solid boundary line (1), the inequality is x + y ≥ -1.

Line (2) passes through the points (1, 0) and (0, 1).

m = (1-0)(0-1) = -1
[Substitute coordinates in the slope formula and simplify.]

Since line (2) crosses y-axis at the point (0, 1), its equation can be found using the slope-intercept form.
[The y-intercept of the line is 1.]

x + y = 1
[Substitute -1 for m and 1 for b in the slope intercept form, y = mx + b.]

Since the shaded region is below the dashed boundary line (2), the inequality is x + y < 1.

The system of inequalities that describes the shaded region is x + y ≥ -1 and x + y < 1.

Correct answer : (3)
6.
Write a system of linear inequalities that defines the shaded region.  a. $y$ ≤ 3, $x$ ≤ 2 and $y$ > -2 b. $x$ ≥ -3 and $y$ ≥ -2 c. $x$ ≥ -3, $y$ < 3, $x$ < 2 and $y$ ≥ -2 d. None of the above

#### Solution:

There are four lines that bound the shaded region. So, the system must have four linear inequalities.

Equation of line (1) that passes through the point (-3, 0) is x = -3. The first inequality is bounded by this line.

Since the shaded region is on and to the right of the solid boundary line (1), the inequality is x ≥ -3.

Equation of line (2) that passes through the point (0, 3) is y = 3. The second inequality is bounded by this line.

Since the shaded region is below the dashed boundary line (2), the inequality is y < 3.

Equation of line (3) that passes through the point (2, 0) is x = 2. The third inequality is bounded by this line.

Since the shaded region is to the left of the dashed boundary line (3), the inequality is x < 2.

Equation of line (4) that passes through the point (0, -2) is y = -2. The fourth inequality is bounded by this line.

Since the shaded region is on and above the solid boundary line (4), the inequality is y ≥ -2.

The system of inequalities that describes the shaded region is x ≥ -3, y < 3, x < 2 and y ≥ -2.

Correct answer : (3)
7.
Write the system of linear inequalities that defines the shaded region.  a. -$y$ > 0 and $x$ > 0 b. -$x$ + $y$ > 0 and $x$ + $y$ > 0 c. -$x$ > 0 and $y$ > 0 d. None of the above

#### Solution:

There are two lines that bound the shaded region. So, the system must have two linear inequalities.

Line (1) passes through the points (-1, -1) and (1, 1).

Slope of a line m = (y2 - y1) / (x2 - x1)
[Formula for slope.]

m = (1-(-1)) / (1-(-1)) = 1
[Substitute coordinates in the formula.]

Line (1) crosses y-axis at the point (0, 0), its equation can be found using the slope-intercept form.

y = mx + b
[Write slope-intercept form.]

-x + y = 0
[Substitute 1 for m and 0 for b and simplify.]

Since the shaded region is above the dashed boundary line (1), the inequality is -x + y > 0.

Line (2) passes through the points (1, -1) and (-1, 1).

m = (1-(-1)) / (-1-1) = -1
[Substitute the coordinates in the slope formula.]

Since line (2) crosses y-axis at the point (0, 0), its equation can be found using the slope-intercept form.

y = mx + b
[Write slope-intercept form.]

x + y = 0
[Substitute -1 for m and 0 for b and simplify.]

Since the shaded region is above the dashed boundary line (2), the inequality is x + y > 0.

The system of inequalities that describes the shaded region is -x + y > 0 and x + y > 0.

Correct answer : (2)
8.
Write a system of linear inequalities that describes the shaded region.  a. $x$ ≥ -2 and $x$ < 0 b. $x$ ≥ -2 and $x$ > 0 c. $x$ ≤ -2 and $x$ < 0 d. $x$ ≤ -2 and $x$ > 0

#### Solution:

There are two lines that bound the shaded region.

So, the system must have two linear inequalities.

The first inequality is bounded by the vertical line that passes through the point (-2, 0).

Then the equation of this line is x = -2.

The shaded region is on and to right of the solid boundary line.

So, the first inequality is x ≥ -2.

The second inequality is bounded by the vertical line that passes through the point (0, 0).

Then the equation of this line is x = 0.

The shaded region is to the left of the dashed boundary line.

So, the second inequality is x < 0.

The system of inequalities that describes the shaded region is x ≥ -2 and x < 0.

Correct answer : (1)
9.
The number of solutions for the system of linear inequalities $y$ ≥ 2 , $y$ ≤ -2, $x$ ≤ -2 and $x$ ≥ 2 is ____. a. 1 b. finite c. infinite

#### Solution:

The graph of y ≥ 2 is the half-plane on and above the solid line y = 2.

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

The graph of x ≤ -2 is the half-plane on and to the left of solid line x = -2.

The graph of x ≥ 2 is the half-plane on and to the right of solid line x = 2.

Graph all the inequalities in the same coordinate plane. The graph of the system is the overlap, or intersection, of the four half-planes. There is no intersection or overlap of four half-planes. So, there is no solution for the given system of linear inequalities.

Correct answer : (1)
10.
Which of the graphs defines the system of inequalities, $x$ + $y$ < 3, -5$x$ + 2$y$ ≤ 10, $y$ ≥ -3?  a. Graph-1 b. Graph-2 c. Graph-3 d. Graph-4

#### 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.

Correct answer : (1)

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