Problem 56 - Introductory Algebra for College Students

Question

Explain how to graph \(2 x-3 y<6\)

Step-by-Step Solution

Answer

The graph of the inequality \(2 x-3 y<6\) is a dashed line with slope -2/3 and y-intercept 2, and the region below the line is shaded.

Step 1: Rearrange the inequality into slope-intercept form

Divide each term in the inequality by -3 to isolate y. Because the divisor is negative, the inequality sign flips. The result is \(y > -\frac{2}{3}x + 2\). Thus, the slope of the graph is -2/3 and the y-intercept is 2.

Step 2: Draw the graph of the line

First, plot the y-intercept, which is 2. Then, use the slope to plot another point on the line. Since the slope is -2/3, this means to go down 2 units and right 3 units from the y-intercept. After plotting the second point, draw a dashed line through the two points. The line is dashed because the original inequality was '<', which does not include equality.

Step 3: Choose a test point and shade the correct side of the line

Most commonly, the test point is (0,0) if it's not on the line. In this case, (0,0) satisfies the inequality \(y > -\frac{2}{3}x + 2\), so the region containing the point (0,0) is shaded.