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Chapter 6: 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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Inequalities
A system of inequalities consists of two or more inequalities that are considered simultaneously. It is similar to a system of equations, but instead of solving for a single point, a system of inequalities seeks regions on a graph where the inequalities overlap. When dealing with such systems, each inequality represents a condition that needs to be satisfied by the variables involved. For example, in the original exercise, we are given two inequalities:
  • \(3x - 2y > -13\)

  • \(-x + 2y > 5\)
The solution to the system of inequalities is the region on the coordinate plane where the solutions to each individual inequality coincide.Visualizing systems of inequalities graphically can make it easier to see all possible solutions that satisfy each condition at the same time. The overlapping region of the shaded areas for each inequality shows all possible solutions.
Bounded and Unbounded Regions
Identifying whether a solution set is bounded or unbounded is a crucial part of solving a system of inequalities. A bounded region is closed and finite. It does not stretch out to infinity and is typically contained within a geometric shape such as a triangle or rectangle. For example, if the intersection of shaded areas on a graph creates a shape with defined edges on all sides, it is considered bounded. On the other hand, an unbounded region extends infinitely in at least one direction. It represents solutions that satisfy the system of inequalities but do not reside within a confined space. An unbounded region might appear when there are fewer constraints or the constraints open up to infinity. In practice, determining if the solution set is bounded or unbounded involves looking at the graph and checking whether the overlap offers a complete enclosure or stretches endlessly.
Coordinate Plane Graphing
Graphing on a coordinate plane is the process of visually representing equations and inequalities by plotting points, lines, and shaded areas. A coordinate plane consists of two perpendicular number lines called axes: the horizontal line (x-axis) and the vertical one (y-axis). Coordinate plane graphing is essential when solving systems of inequalities. The steps for graphing involve:
  • Rewriting each inequality as an equation to find the boundary line.

  • Determining the intercepts (where the lines cross the axes) to plot the lines.

  • Shading the region of the plane that satisfies the inequality. Often, the test point (like the origin) helps decide which side of the line to shade.
Graphical solutions provide a clear interpretation of the relationships between various inequalities by visualizing the overlap and intersections of shaded regions. This method illustrates how multiple conditions interact on the coordinate plane, highlighting the solution set for the system.

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Most popular questions from this chapter

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{cccccc|c} x & y & z & u & v & P & \text { Constant } \\ \hline 3 & 0 & 5 & 1 & 1 & 0 & 28 \\ 2 & 1 & 3 & 0 & 1 & 0 & 16 \\ \hline 2 & 0 & 8 & 0 & 3 & 1 & 48 \end{array} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=10 x+12 y \\ \text { subject to } & x+2 y \leq 12 \\ & 3 x+2 y \leq 24 \\ & x \geq 0, y \geq 0 \end{array} $$

A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $$\$ 45$$, and the profit for each chair is $$\$ 20 .$$ In a certain week, the company has 3200 board feet of wood available and 520 labor-hours available. How many tables and chairs should Winston manufacture in order to maximize its profit? What is the maximum profit?

Use the technique developed in this section to solve the minimization problem. $$ \begin{aligned} \text { Minimize } & C=x-2 y+z \\ \text { subject to } & x-2 y+3 z \leq 10 \\ & 2 x+y-2 z \leq 15 \\ & 2 x+y+3 z \leq 20 \\ & x \geq 0, y \geq 0, z & \geq 0 \end{aligned} $$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Choosing the pivot column by requiring that it be the column associated with the most negative entry to the left of the vertical line in the last row of the simplex tableau ensures that the iteration will result in the greatest increase or, at worse, no decrease in the objective function.
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