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Chapter 3: Problem 20

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{r} 3 x-2 y>-13 \\ -x+2 y>5 \end{array} $$

Short Answer

Expert verified
The solution set is the intersection of the shaded regions, which represents the values of x and y that satisfy both inequalities. In this case, the intersection is an unbounded region extending infinitely to the right. Therefore, the solution set is unbounded.

Step by step solution

01

Rewrite inequalities as equations.

First, let's rewrite the given inequalities as equations: 1. \(3x - 2y = -13\) 2. \(-x + 2y = 5\)
02

Graph the equations.

Next, graph the equations on a coordinate plane. Equation 1: To graph \(3x - 2y = -13\), find the x and y-intercepts: - When x = 0: \(-2y=-13 \implies y=\dfrac{13}{-2}\) - When y = 0: \(3x=-13 \implies x=-\dfrac{13}{3}\) Plot these points (0,-6.5) and (-4.33,0) and draw a line through them. Equation 2: To graph \(-x + 2y = 5\), find the x and y-intercepts: - When x = 0: \(2y=5 \implies y=\dfrac{5}{2}\) - When y = 0: \(-x=5 \implies x=-5\) Plot these points (0,2.5) and (-5,0) and draw a line through them.
03

Shade the Regions.

Now, shade the regions that satisfy each inequality. For inequality \(3x - 2y > -13\), test a point not on the line, usually the origin (0,0) for simplicity. - \(3(0)-2(0)>-13\) - \(0>-13\) which is true, so shade the region containing the origin. For inequality \(-x + 2y > 5\), test a point not on the line, for example, (0,0). - \(-0+2(0)>5\) - \(0>5\) which is false, so shade in the region not containing the origin.
04

Identify the intersection of the shaded regions.

Finally, look at the graph and identify the intersection of the shaded regions. The intersection is the part of the graph where both inequalities are satisfied, and it represents the solution set of the system.
05

Determine if the solution set is bounded or unbounded.

Look at the intersection of the shaded regions. If the intersection forms a closed region (such as a triangle, rectangle, etc.), it is bounded. If it extends infinitely in any direction, it is unbounded. After completing these steps, you will have graphically determined the solution set for the system of inequalities and identified whether it is bounded or unbounded.

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Most popular questions from this chapter

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