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Chapter 7: Problem 57

Describe how to solve a system of linear inequalities.

Short Answer

Expert verified
The solution to a system of linear inequalities is the intersecting region that satisfies all inequalities simultaneously, if it exists. The solution is graphically represented as a shaded overlapping region.

Step by step solution

01

Identify the Inequalities

First, identify all the inequalities in the system. The inequalities are in the form \(ax + by \leq c\), \(ax + by \geq c\), \(ax + by < c\) or \(ax + by > c\).
02

Graph Each Inequality

Next, each inequality should be graphed on the same coordinate axis. An easy way to do this is to treat each inequality as if it was an equation and graph the resulting line. After that, the area above or below the line should be shaded according to the inequality. If the inequality sign is \(>\) or \(<\), then a dashed line is used. If the inequality sign is \(\leq\) or \(\geq\), then a solid line is used.
03

Find the Intersection of Inequalities

Finally, inspect the graph and identify the area that satisfies all inequalities simultaneously. This is called the feasible region. Each point in this area is a solution to the system of inequalities. In case there's no overlapping region, it means that the system of inequalities has no solution.

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