Problem 64 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(y<5 x+2\) or \(x+4 y<12\)

Step-by-Step Solution

Answer

To graph the compound inequality \(y<5x+2\) or \(x+4y<12\), first graph each inequality separately: 1. Graph the boundary line \(y=5x+2\) as a dashed line and shade the region below the line. 2. Graph the boundary line \(x+4y=12\) as a dashed line and shade the region below the line. Combine the shaded regions from both graphs. The combined shaded region represents the graph of the compound inequality \(y<5x+2\) or \(x+4y<12\).

Step 1: Graph the first inequality \(y<5x+2\)

First, we need to graph the boundary line of the inequality \(y=5x+2\). This is a linear function with a slope of 5 and a y-intercept of 2. To graph this line: 1. Plot the y-intercept (0, 2). 2. From this point, use the slope 5 to find another point on the line: rise 5 and run 1. 3. Draw a dashed line through the two points, as the inequality is strictly less than, meaning the points on the line are not part of the solution. Now, we need to determine which side of the boundary line the inequality represents. 1. Choose a test point that is not on the line, such as (0, 0). 2. Plug the test point into the inequality: \(0<5(0)+2\). 3. Since the inequality is true, shade the region containing the test point.

Step 2: Graph the second inequality \(x+4y<12\)

Now, let's graph the boundary line of the inequality \(x+4y=12\). This is also a linear function that can be written in the slope-intercept form as \(y=-\frac{1}{4}x+3\). To graph this line: 1. Plot the y-intercept (0, 3). 2. From this point, use the slope -1/4 to find another point on the line: drop 1 and run 4. 3. Draw a dashed line through the two points, as the inequality is strictly less than, meaning the points on the line are not part of the solution. Next, we need to determine which side of the boundary line the inequality represents. 1. Choose a test point that is not on the line, such as (0, 0). 2. Plug the test point into the inequality: \(0+4(0)<12\). 3. Since the inequality is true, shade the region containing the test point.

Step 3: Combine the graphs

Since this is an "or" compound inequality, the solution will include any region where either of the inequalities is true. This means that any point in the shaded region of either inequality is part of the solution. To graph the compound inequality: 1. Overlay the two shaded regions from steps 1 and 2. 2. The combined shaded region represents the graph of the compound inequality \(y<5x+2\) or \(x+4y<12\).