Question

graph the inequality on the axes below.
$y \geq -\frac{5}{4}x - 2$

Explanation:

Step1: Identify the boundary line

The inequality is \( y \geq -\frac{5}{4}x - 2 \). First, we graph the boundary line \( y = -\frac{5}{4}x - 2 \). This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m=-\frac{5}{4} \) and the y - intercept \( b=- 2 \).

• To find the y - intercept, we set \( x = 0 \). Then \( y=-\frac{5}{4}(0)-2=-2 \). So the line passes through the point \( (0,-2) \).
• To find another point, we use the slope. The slope \( m = \frac{\text{rise}}{\text{run}}=-\frac{5}{4} \). From the point \( (0,-2) \), we can go down 5 units (since the rise is - 5) and then 4 units to the right (run is 4) to get the point \( (4,-2 - 5)=(4,-7) \), or we can go up 5 units and 4 units to the left to get the point \( (-4,-2 + 5)=(-4,3) \).

Since the inequality is \( y\geq-\frac{5}{4}x - 2 \) (the "greater than or equal to" symbol), the boundary line should be a solid line (because the inequality includes the equal sign, so the points on the line are part of the solution set).

Step2: Determine the region to shade

To determine which side of the line to shade, we can test a point that is not on the line. A common test point is the origin \( (0,0) \).
Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y\geq-\frac{5}{4}x - 2 \):
\( 0\geq-\frac{5}{4}(0)-2 \)
\( 0\geq - 2 \)
This statement is true. So we shade the region that contains the origin \( (0,0) \), which is the region above the line \( y = -\frac{5}{4}x-2 \) (since the test point \( (0,0) \) is above the line \( y = -\frac{5}{4}x - 2 \) as \( 0>-2 \)).

(Note: Since the problem asks to graph the inequality, the final answer is the graph with a solid line \( y = -\frac{5}{4}x - 2 \) and the region above the line shaded. But since we can't draw the graph here, we can describe the steps to graph it as above.)

Answer:

To graph \( y\geq-\frac{5}{4}x - 2 \):

1. Draw a solid line with slope \( -\frac{5}{4} \) and y - intercept \( - 2 \) (passing through \( (0,-2) \) and other points like \( (4,-7) \) or \( (-4,3) \)).
2. Shade the region above the line (the region containing the origin \( (0,0) \)).