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Chapter 8: Problem 95

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

Short Answer

Expert verified
Graphing the solution set of a system of inequalities involves expressing the inequalities in slope-intercept form, graphing these lines using the slope and y-intercept, shading the appropriate side of each line according to the inequality sign, and identifying the area of overlap as the solution set.

Step by step solution

01

Express Inequalities in Slope-Intercept Form

Write each inequality in the system in slope-intercept form, y = mx + b, where m is the slope, and b is the y-intercept. This form will make the inequalities simple to graph as lines on a coordinate plane.
02

Graph the Lines

Using the slope and y-intercept from Step 1, graph each line on the same coordinate plane. If the inequality is strict (meaning less than '<' or greater than '>'), draw a dashed line, which indicates that the points on the line are not included in the solution. If the inequality is not strict (meaning less than or equal to '鈮�' or greater than or equal to '鈮�'), draw a solid line, which indicates that the points on the line are included in the solution.
03

Shade the Appropriate Areas

Then, determine which side of each line to shade. For a 'greater than' inequality, shade above the line, and for a 'less than' inequality, shade below the line. Use a different texture or color for each line.
04

Identify Solution Set

The solution to the system of inequalities will be the area where the shadings for all the inequalities overlap. This area represents all possible solutions to the system of inequalities.

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Most popular questions from this chapter

If \(x=3, y=2,\) and \(z=-3,\) does the ordered triple \((x, y, z)\) satisfy the equation \(2 x-y+4 z=-8 ?\)

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. In 1978 , a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon \(20 .\) To aid in their decision, executives at Federal Express analyzed the following data: $$\begin{array}{ll} {} & {\text { Boeing } 727 \quad \text { Falcon } 20} \\ {\text { Direct Operating cost }} & {\$ 1400 \text { per hour } \$ 500 \text { per hour }} \\ {\text { Payload }} & {42,000 \text { pounds } \quad 6000 \text { pounds }} \end{array}$$ Federal Express was faced with the following constraints: \(\cdot\) Hourly operating cost was limited to 35,000. \(\cdot\) Total payload had to be at least 672,000 pounds. \(\cdot\) Only twenty 727 s were available. Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?

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\(g(x)=\frac{x-6}{x^{2}-36}\) (Section 3.5, Example 1)
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