Problem 57 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(x \geq 5\) and \(y \leq-3\)

Step-by-Step Solution

Answer

The solution set for the compound inequality \(x \geq 5\) and \(y \leq -3\) can be represented graphically as a region in the coordinate plane. Draw a vertical solid line at \(x = 5\) and shade the region to the right of the line. Draw a horizontal solid line at \(y = -3\) and shade the region below the line. The overlapped shaded area represents the solution set for the compound inequality.

Step 1: Analyze the first inequality

In the first inequality, \(x \geq 5\), we can see that it involves just the x-variable. We need to graph the region in the xy-plane where all the x-values are greater than or equal to 5. This inequality represents a vertical line at x = 5, and the solution set includes all points to the right of this line, since x > 5.

Step 2: Graph the first inequality

To graph \(x \geq 5\), start by drawing a xy-plane. Then, draw a vertical line at x = 5. To represent the inequality symbol, we will use a solid line at \(x=5\) since our inequality is greater than or equal to 5. Finally, shade the region to the right of the line, which represents the solution set for this inequality.

Step 3: Analyze the second inequality

In the second inequality, \(y \leq -3\), it involves just the y-variable. We need to graph the region in the xy-plane where all the y-values are less than or equal to -3. This inequality represents a horizontal line at \(y = -3\), and the solution set includes all the points below this line, since \(y \leq -3\).

Step 4: Graph the second inequality

To graph \(y \leq -3\), start by drawing a xy-plane. Then, draw a horizontal line at \(y = -3\). To represent the inequality symbol, we will use a solid line because our inequality is less than or equal to -3. Finally, shade the region below the line, which represents the solution set for this inequality.

Step 5: Combine both inequalities

Now that we have graphed both inequalities separately, we need to combine them both, as the problem asks for the regions where both inequalities are true. In the xy-plane, find the overlapping region of the shaded areas from both inequalities. This overlapping region is the solution set for the given compound inequality, which satisfies both \(x \geq 5\) and \(y \leq -3\). After completing these steps, your graph should have a solid vertical line at \(x = 5\) and a solid horizontal line at \(y = -3\), and the overlapped shaded area that satisfies both inequalities. This region is the graphical representation of the solution set for the compound inequality.