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Chapter 7: Problem 14

Graph each linear inequality. \(y \leq 2 x-1\)

Short Answer

Expert verified
First, the line of equation \(y = 2x - 1\) is plotted on the graph, with a solid line to include points lying on it. After testing the origin, it is found that the solution of the inequality does not include the origin, thus shading the area that does not contain the origin (i.e., the area below the line).

Step by step solution

01

Plot the Line

First, the line of the equation \(y = 2x - 1\) should be plotted on the graph. It has a slope of 2 and a y-intercept of -1. This can be done by plotting the y-intercept (0, -1) and then by using the slope to find another point. The slope of 2 means that for every 1 unit increase in x, y increases by 2 units. So, another point on this line, for instance, could be \(x = 1\), \(y = 1\).
02

Draw the Line

Next, the line passing through these points is drawn. Since the inequality includes an equal sign (\(\leq\), meaning 鈥渓ess than or equal to鈥�), the line should be solid, indicating that the points on the line are included within the solution set.
03

Determine the Shading Side

To determine which side of the line to shade, a test point not on the line is selected. The origin (0,0) is commonly chosen as the test point if the line does not pass through it. Substituting (0,0) into the inequality \(y \leq 2x - 1\) gives \(0 \leq -1\), which is not true. Therefore, the side of the line containing (0,0) isn't part of the solution set. Thence, the area on the other side of the line is shaded.

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Most popular questions from this chapter

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 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\).

Graph each linear inequality. \(x \leq-4\)

Graph each linear inequality. \(y<-\frac{1}{3} x\)

Compare the graphs of \(3 x-2 y>6\) and \(3 x-2 y \leq 6\). Discuss similarities and differences between the graphs.

In Exercises 1-22, graph each linear inequality. \(x+y \geq 2\)
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