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Chapter 5: Problem 95

Explain how to graph the solution set of a system of inequalities.

Short Answer

Expert verified
The graph of the system of inequalities typically involves graphing each inequality on the same set of axes. One would shade above the line for inequalities showing y is greater, or beneath the line when y is less. The solution set where these shadings overlap is the pairs of \(x, y\) that satisfy both inequalities simultaneously.

Step by step solution

01

Identify the Inequalities

In order to begin, you need to know the inequalities being dealt with. Let's consider a system of inequalities, for example, \(y \geq x + 1\) and \(y < 3x - 1\). These will be graphed on the same coordinate system.
02

Graph the First Inequality

The first step is to graph the first inequality, \(y \geq x + 1\), as if it were a linear equation. Plot the line \(y = x + 1\). Since the inequality symbol is '\(\geq\)', this requires a solid line because the line is included in the solution set. Then, because \(y\) is greater or equal to \(x + 1\), the area above the line is shaded.
03

Graph the Second Inequality

Next graph the second inequality, \(y < 3x - 1\), like a linear equation. Plot the line \(y = 3x - 1\). Since the inequality symbol is '<', a dashed line is required because the line is not included in the solution set. Since \(y\) is less than \(3x - 1\), the area below the line is shaded.
04

Identify the Solution Set

The solution set is the area where the shadings from both inequalities overlap. This area represents all the pairs of \(x, y\) that satisfy both inequalities simultaneously.

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Most popular questions from this chapter

In Exercises 5鈥�14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$\begin{aligned}&z=3 x+2 y\\\&\left\\{\begin{array}{c}x \geq 0, y \geq 0 \\\2 x+y \leq 8 \\\x+y \geq 4\end{array}\right.\end{aligned}$$

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises \(71-72,\) you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\).

In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(2,-1,3)\\\ &\left\\{\begin{array}{c} x+y+z=4 \\ x-2 y-z=1 \\ 2 x-y-2 z=-1 \end{array}\right. \end{aligned} $$

Many elevators have a capacity of 2000 pounds. a. If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when \(x\) children and \(y\) adults will cause the elevator to be overloaded. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 2 x^{2}+y^{2}-18 \\ x y-4 \end{array}\right.$$
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