Problem 63 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(y \geq \frac{2}{3} x-4\) and \(4 x+y \leq 3\)

Step-by-Step Solution

Answer

First, rewrite the second inequality as \(y \leq -4x + 3\). Then, identify the slope and y-intercept for both inequalities: \(\frac{2}{3}\) and -4 for the first; -4 and 3 for the second. Plot the y-intercepts and graph both lines using their respective slopes. Shade the region above the first line and below the second line. The overlapping region represents the solution to the compound inequality.

Step 1: Rewrite equations in slope-intercept form

To begin, the inequalities are already in slope-intercept form, which is \(y = mx + b\). The given inequalities are \(y \geq \frac{2}{3} x - 4\) and \(4x+y \leq 3\), and the second inequality can be rewritten as \(y \leq -4x + 3\).

Step 2: Identify the slope and y-intercept

For the first inequality, \(y \geq \frac{2}{3} x - 4\), the slope (m) is \(\frac{2}{3}\) and the y-intercept (b) is -4. For the second inequality, \(y \leq -4x + 3\), the slope (m) is -4 and the y-intercept (b) is 3.

Step 3: Plot y-intercepts and graph lines

Firstly, plot the y-intercepts, (-4) for the first inequality and (3) for the second inequality, on the y-axis. Then, graph the first line using the slope of \(\frac{2}{3}\) (rise 2, run 3) from the y-intercept, and the second line using the slope of -4 (rise -4, run 1) from the y-intercept. Remember to draw solid lines since the inequalities include equal values.

Step 4: Shade the region that satisfies both inequalities

For \(y \geq \frac{2}{3} x - 4\), shade the region above the first line. For the second inequality, \(y \leq -4x + 3\), shade the region below the second line. The area where both regions overlap is the solution to the compound inequality. To check, you can choose a point within the overlapping region and see if it satisfies both inequalities.