﻿ Graphing Inequalities in Two Variables Worksheet | Problems & Solutions Graphing Inequalities in Two Variables Worksheet

Graphing Inequalities in Two Variables Worksheet
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1.
The quantity to be maximized or minimized is represented by a linear function called as _________. a. The restriction b. The objective function c. The feasible function d. The constraint

Solution:

The quantity to be maximized or minimized is represented by a linear function called the objective function.
[Definition of the objective function.]

2.
Draw the graph of 5$x$ + 3$y$ ≤ 15. State the boundary of the region.

Solution:

5x + 3y ≤ 15
[Original in equality.]

The graph of the line 5x + 3y = 15 is part of the graph of the inequality and should be drawn as a solid line.
[Replace ≤ by =.] The pair (1, 1) is a solution of the equation because 15 ≥ 5(1) + 3(1) = 8. Thus the set of all points on and below the line 5x + 3y = 15 are the solutions to the given inequality as shown.
[Draw the inequality.]

The line 5x + 3y = 15 is the boundary of the region.

3.
State whether the ordered pair (4, 6) is a solution of the inequality $\frac{x}{2}$ - $\frac{5y}{6}$ > 7. a. No b. Yes

Solution:

x2 - 5y6 > 7
[Original inequality.]

4 / 2 - 5(6) / 6 > 7
[Substitute 4 for x and 6 for y.]

2 - 5 > 7
[Simplify.]

-3 > 7
[Subtract.]

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

4.
The graphical tool which identifies conditions that make as large as possible or as small as possible is known as________. a. Linear programming b. Optimization technique c. Graph theory d. Curve sketching

Solution:

The graphical tool which identifies conditions that make as large as possible or as small as possible is known as linear programming.
[Definition of linear programming.]

5.
Which of the following is the graph of a linear system of constraints? a. The linear graph b. The linear region c. The feasible region d. The constraint region

Solution:

The graph of the linear system of constraints is called the feasible region.
[Definition of the feasible region.]

6.
Limits on the variables in the objective function are called _______. a. Extremes b. Vertices c. Restrictions d. Objectives

Solution:

Limits on the variables in the objective function are called restrictions.
[Definition of the restrictions.]

7.
Write the system of linear inequalities that defines the shaded region.  a. - $x$ + $y$ > 0 and $x$ + $y$ > 0 b. - $y$ > 0 and $x$ > 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.]

Since 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.

8.
Write a system of linear inequalities that defines the shaded region.  a. $x$ ≥ - 3, $y$ < 3, $x$ < 2 and $y$ ≥ - 2 b. $x$ ≥ - 3 and $y$ ≥ - 2 c. $y$ ≤ 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.

9.
Write a system of linear inequalities that define the shaded region.  a. $x$ + $y$ = -1 and $y$ = 1 b. $x$ + $y$ ≥ -1 and $x$ + $y$ < 1 c. $x$ ≥ -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.

10.
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 the 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.