Problem 50 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(5 x-3 y>9\) and \(2 x+3 y \leq 12\)

Step-by-Step Solution

Answer

To graph the compound inequality, first convert each inequality to slope-intercept form: 1. \(y < \frac{5}{3}x - 3\) 2. \(y \leq -\frac{2}{3}x + 4\) Next, graph the lines with the first as a dashed line and the second as a solid line. Then, shade the appropriate half-planes for each inequality (below the dashed line for the first inequality, below the solid line for the second). The overlapping shaded region represents the solution for the compound inequality.

Step 1: Convert each inequality to the slope-intercept form

We will first convert each inequality into the slope-intercept form (y = mx + b). For the first inequality, \(5x - 3y > 9\), we'll solve for y: 1. Add \(3y\) to both sides: \(5x - 9 > 3y\) 2. Divide both sides by 3: \(y < \frac{5}{3}x - 3\) For the second inequality, \(2x + 3y \leq 12\), we'll also solve for y: 1. Subtract \(2x\) from both sides: \(3y \leq -2x + 12\) 2. Divide both sides by 3: \(y \leq -\frac{2}{3}x + 4\) Now, we have the two inequalities in slope-intercept forms: 1. \(y < \frac{5}{3}x - 3\) 2. \(y \leq -\frac{2}{3}x + 4\)

Step 2: Graph the lines

Now, let's graph the lines for each inequality as if they were equalities: 1. \(y = \frac{5}{3}x - 3\) (Dashed line because the original inequality is a greater-than ">", not an equal) 2. \(y = -\frac{2}{3}x + 4\) (Solid line because the original inequality is a less-than-or-equal-to "≤")

Step 3: Shade the appropriate half-planes for each inequality

Let's now identify and shade the half-planes that represent the solutions for each inequality: 1. For \(y < \frac{5}{3}x - 3\), we'll shade the region below the dashed line 2. For \(y \leq -\frac{2}{3}x + 4\), we'll shade the region below the solid line

Step 4: Identify the overlapping shaded region

As we have used "and" in the original compound inequality, we need both inequalities to be true simultaneously for a solution. So, we must find the region where the shading for both inequalities intersects or overlaps. This overlapping region is the solution to the compound inequality. Finally, the graph will display two lines with shaded regions, and the overlapping shaded region represents the solution for the compound inequality. If you need to indicate the solutions on the graph, you can use a different shade or add extra hatched lines to highlight the overlapping area.