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Chapter 3: Problem 9

Graph each inequality. $$x+y \geq 3$$

Short Answer

Expert verified
The graph of the inequality \(x + y \geq 3\) is a solid line along \(x + y = 3\) with all points above that line shaded to represent the solution set for the inequality.

Step by step solution

01

Plot the boundary line

First, change the inequality into an equation to plot the boundary line, here it would be \(x + y = 3\). This is a linear equation in slope-intercept form \(y = -x +3\). So plot this line on the graph. The y-intercept is 3 and the slope is -1. Thus, starting from the y-intercept point (0,3), move down 1 unit and right 1 unit consecutively to plot the line.
02

Shade the appropriate region

Because the inequality is 'greater than or equal to', we draw a solid line to show that the points on the line are part of the solution. Next, choose a test point not on the line to determine which side of the line to shade. A good point is usually the origin (0,0). Substitute x=0 and y=0 into the inequality: \(0 + 0 \geq 3\), which is not true. Therefore, the origin does not satisfy the inequality and we shade the region of the graph that does not include the origin, which is the region above the line.
03

Double Check Your Work

To ensure your shaded region is correct, you can check another point in that region, say (2,2). Substituting x=2 and y=2 into the inequality gives, \(2 + 2 \geq 3\), which is true, confirming that you have shaded the correct region.

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