Problem 58 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(x \leq 6\) and \(y \geq 1\)

Step-by-Step Solution

Answer

To graph the compound inequality \(x \leq 6\) and \(y \geq 1\), draw a solid vertical line at \(x=6\) and a solid horizontal line at \(y=1\). Shade the region to the left of the vertical line and above the horizontal line, representing the intersection of the two inequalities. The graph represents all points \((x, y)\) that satisfy the compound inequality.

Step 1: Graph the first inequality\(x \leq 6\)

To graph \(x \leq 6\), we will first draw the vertical line \(x=6\). Since the inequality is less than or equal to, we will draw a solid line (not dashed) to represent the equality portion. Next, shade the region to the left of the line as it represents all the points where \(x\) is less than or equal to 6.

Step 2: Graph the second inequality \(y \geq 1\)

For the inequality \(y \geq 1\), draw a horizontal line at \(y=1\). This inequality is greater than or equal to, hence we use a solid line to represent the equality part. Shade the region above the line, as it represents all the points where \(y\) is greater than or equal to 1.

Step 3: Find the intersection of both inequalities

The intersection of both inequalities can be found by overlapping the graphs of the individual inequalities. The region where both shaded regions intersect is the graph of the compound inequality.

Step 4: Interpret the graph

The graph represents the set of all points \((x, y)\) that satisfy both \(x \leq 6\) and \(y \geq 1\). This set of points is the solution to the compound inequality.