Problem 60 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(x \geq 2\) or \(y \geq-6\)

Step-by-Step Solution

Answer

Graph the compound inequality by separately graphing the two inequalities. For \(x \geq 2\), draw a vertical line at \(x=2\) and shade the region to the right of the line. For \(y \geq -6\), draw a horizontal line at \(y=-6\) and shade the region above the line. Finally, combine the graphs and shade the overlapping regions, as it represents an "or" situation. The final graph shows all points that satisfy either inequality or both.

Step 1: Graph the first inequality

We are given \(x \geq 2\). To graph this inequality: 1. Create a coordinate plane. 2. Draw a vertical line at \(x=2\). The line passes through all the points having \(x=2\). 3. Shade the region to the right of the line, including the line itself. This represents all the points having \(x\) greater than or equal to 2.

Step 2: Graph the second inequality

We are given \(y \geq -6\). To graph this inequality: 1. Use the same coordinate plane created in step 1. 2. Draw a horizontal line at \(y=-6\). The line passes through all the points having \(y=-6\). 3. Shade the region above the line, including the line itself. This represents all the points having \(y\) greater than or equal to -6.

Step 3: Combine both graphs

Now that we have graphed the individual inequalities, we will combine them. 1. Use the same coordinate plane. 2. Overlay the graph of the first inequality (\(x \geq 2\)) with the graph of the second inequality (\(y \geq -6\)). 3. Since it is an "or" situation, the regions of both the inequalities must be colored/shaded, including the regions where the two graphs intersect. The final graph represents all the points that satisfy either the first inequality (\(x \geq 2\)) or the second inequality (\(y \geq -6\)) or both.