91Ó°ÊÓ

Take the inequality: \(3x + 4y \leq -2\) Replace the inequality sign with an equal sign to get the line equation: \(3x + 4y = -2\)

To graph the line equation, we need to find two points that are on the line. We can do this by solving for x and y. When \(x = 0\), solve for \(y\): \(3(0) + 4y = -2\) \(4y = -2\) \(y = -\frac{1}{2}\) When \(y = 0\), solve for \(x\): \(3x + 4(0) = -2\) \(3x = -2\) \(x = -\frac{2}{3}\) These two points are \((- \frac{2}{3}, 0)\) and \((0, -\frac{1}{2})\). Plot these points on the coordinate plane and draw the line through them.

Now, we must determine which region on the coordinate plane, above or below the line, satisfies the inequality. To do this, we can choose a test point that is not on the line and plug it into the original inequality. If the inequality is true, then the region that contains the test point is the solution. A common test point to use is the origin \((0, 0)\). Plug this point into the original inequality: \(3(0) + 4(0) \leq -2\) \(0 \leq -2\) The inequality is false, so the region that contains the origin is not the solution. Therefore, the region opposite the origin (below the line) is the solution. Now, sketch the region below the line on the graph. The graphical solution of the inequality \(3x + 4y \leq -2\) is the region below the line passing through the points \((- \frac{2}{3}, 0)\) and \((0, -\frac{1}{2})\).