Problem 55 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(y>-\frac{2}{3} x+1\) or \(-2 x+5 y \leq 0\)

Step-by-Step Solution

Answer

The short answer is: To graph the compound inequality, plot the lines for each inequality, using a dotted line for \(y > -\frac{2}{3}x + 1\) and a solid line for \(y \leq \frac{2}{5}x\). Then, shade the area above the dotted line and the area below the solid line. The combined solution is the union of both shaded areas.

Step 1: Rewrite Inequalities in Slope-Intercept Form

The first inequality is already in slope-intercept form: \(y > -\frac{2}{3}x + 1\). To rewrite the second inequality in slope-intercept form, we need to isolate the y variable. \(-2x + 5y \leq 0\) Add 2x to both sides of the inequality: \(5y \leq 2x\) Divide both sides by 5: \(y \leq \frac{2}{5}x\) Now, we have both inequalities in slope-intercept form: 1) \(y > -\frac{2}{3}x + 1\) 2) \(y \leq \frac{2}{5}x\)

Step 2: Create Lines for Each Inequality

Next, we need to draw a line for each inequality, depending on its sign. If the inequality is strict (greater than or less than), we will draw the line using a dotted line. If the inequality is non-strict (greater than or equal to, or less than or equal to), we will draw the line using a solid line. In this case, we have: - A dotted line for the first inequality: \(y > -\frac{2}{3}x + 1\) - A solid line for the second inequality: \(y \leq \frac{2}{5}x\)

Step 3: Shade the Correct Areas

Now, we will shade the area that satisfies each inequality. 1) For \(y > -\frac{2}{3}x + 1\), we will shade the area above the dotted line. 2) For \(y \leq \frac{2}{5}x\), we will shade the area below the solid line.

Step 4: Identify the Combined Solution

As we are dealing with an "or" compound inequality, the combined solution is the union of the shaded areas for both inequalities. Look for the area on the graph where either of the inequalities is satisfied, which is the combination of both shaded areas.