Problem 64 - College Algebra

Question

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

Step-by-Step Solution

Answer

The graph of \(2x - 3y < 6\) is a dashed line drawn through the points where y equals \(2/3x -2\), with shading above the line.

Step 1: Convert Inequality to Slope-Intercept Form

First, rewrite the given inequality \(2x - 3y < 6\) into slope-intercept form (y = mx + b). To do this, subtract 2x from both sides to get \(-3y < -2x + 6\). Then divide by -3 to find \(y > 2/3x - 2\). Note that the inequality sign is reversed when multiplying or dividing by a negative number.

Step 2: Plot the Y-Intercept

Next, graph the line of the inequality, beginning with the y-intercept (-2 in this case). This point is on the y-axis.

Step 3: Use Slope to Find Another Point

The slope of the line is 2/3, which means move up 2 units and to the right 3 units from the y-intercept to plot another point.

Step 4: Draw the Line and Shade the Correct Region

Draw a dashed line through the points. Since the inequality is 'greater than', not 'greater than or equal to', do not include the points on the line in the solution set, indicating this by using a dashed line. Test a point not on the line (like (0,0)) in the inequality and if it makes the inequality true, shade the region that contains that point. Here, shading will be done above the line.