Problem 52 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(y \leq 2\) or \(y \leq \frac{4}{5} x+2\)

Step-by-Step Solution

Answer

Graph the boundary lines \(y=2\) and \(y=\frac{4}{5}x+2\). Shade the region below each line for the respective inequalities \(y \leq 2\) and \(y \leq \frac{4}{5}x+2\). The solution set of the compound inequality is the union of both shaded regions, which is the area that is shaded by both inequalities below the lines \(y=2\) and \(y=\frac{4}{5}x+2\).

Step 1: Graph the boundary lines

The boundary lines for the inequalities are: 1. \(y=2\) 2. \(y=\frac{4}{5}x+2\) Plot these lines on a coordinate plane.

Step 2: Determine the solution set for each inequality

The given inequalities are 1. \(y \leq 2\) 2. \(y \leq \frac{4}{5}x+2\) For the first inequality, the solution set includes all points on the line \(y=2\) and below it. So, shade the region below the line \(y=2\). For the second inequality, the solution set includes all points on the line \(y=\frac{4}{5}x+2\) and below it. So, shade the region below the line \(y=\frac{4}{5}x+2\).

Step 3: Combine the solution sets

The problem says the solution set is to be the union of both inequalities, which means we will find the region that is shaded by both inequalities. In other words, every point in the shaded region below both the line \(y=2\) and the line \(y=\frac{4}{5}x+2\) corresponds to a solution in the compound inequality.