Problem 53 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(y \leq 4\) or \(4 y-3 x \geq-8\)

Step-by-Step Solution

Answer

To graph the compound inequality, first graph the line \(y=4\) and shade the region below it, since we need \(y\leq4\). Next, graph the line \(y=\frac{3}{4}x - 2\) and shade the region above it, since we need \(y\geq\frac{3}{4}x - 2\). Then, shade the region that is covered by at least one of the inequalities, as the compound inequality is an "or" inequality, meaning the points must satisfy either of the individual inequalities.

Step 1: Graph the first inequality

To graph \(y \leq 4\), we start by graphing the line \(y = 4\). This line will be a horizontal line that passes through all points whose \(y\)-coordinate is equal to 4. Since \(y \leq 4\), the region we're interested in is below (or equal to) this line, since this is where every point has a \(y\)-value of 4 or less. Shade this region.

Step 2: Graph the second inequality

To graph \(4y - 3x \geq -8\), we first need to rewrite it in slope-intercept form (i.e., \(y = mx + b\)). Add \(3x\) to both sides of the inequality to obtain: \(4y \geq 3x - 8\). Dividing both sides by 4, we get: \(y \geq \frac{3}{4}x - 2\). Now, we can graph the line \(y = \frac{3}{4}x - 2\). This line has a slope of \(\frac{3}{4}\) and a \(y\)-intercept of \(-2\). Since \(y \geq \frac{3}{4}x - 2\), the region we're interested in is above (or equal to) this line, since this is where every point has a \(y\)-value greater than or equal to \(\frac{3}{4}x - 2\). Shade this region.

Step 3: Combine the inequalities

Since we are looking for an "or" compound inequality, we'll look for points that satisfy either of the inequalities, meaning they should be in the shaded regions from Step 1 and Step 2. In other words, the graph of this compound inequality is the union of the respective regions. On your graph, shade the region that is covered by at least one of the inequalities.