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Chapter 7: Problem 25

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{r}2 x+y<3 \\ x-y>2\end{array}\right.\)

Short Answer

Expert verified
The solution set for the system of inequalities is the region where the shading from the two inequalities overlaps. If there is no overlap between the two shaded regions, then there is no solution for the system.

Step by step solution

01

Graph the First Inequality

Start by graphing the first inequality, \(2x + y < 3\). First, rewrite this inequality in slope-intercept form, \(y = mx + b\), which leads to \(y < -2x + 3\). The graph of the equation \(y = -2x + 3\) is a line. Because the inequality symbol is '<', the region below the line will be shaded.
02

Graph the Second Inequality

Next, graph the second inequality, \(x - y > 2\). This can be rewritten in slope-intercept form as \(y < x - 2\). Because the inequality symbol is '>', the region above the line will be shaded.
03

Identify the Solution Set

The solution set is the region where the shading from the two inequalities overlaps. Points in this region satisfy both of the original inequalities. If there is no overlap between the two shaded regions, then the system of inequalities has no solution.

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