Question

graph the inequality on the axes below.
$-5x + 2y > -8$

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(-5x + 2y>-8\). We want to solve for \(y\) to get it in the form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept) for easier graphing.
Add \(5x\) to both sides of the inequality: \(2y>5x - 8\).
Then divide each term by \(2\): \(y>\frac{5}{2}x-4\).

Step2: Graph the boundary line

The boundary line for the inequality \(y>\frac{5}{2}x - 4\) is the line \(y=\frac{5}{2}x-4\). Since the inequality is \(y>\) (not \(y\geq\)), the boundary line should be a dashed line (because the points on the line are not included in the solution set).
To graph the line \(y = \frac{5}{2}x-4\):

• The y - intercept \(b=-4\), so we plot the point \((0, - 4)\) on the y - axis.
• The slope \(m=\frac{5}{2}\), which means from the point \((0,-4)\), we move up \(5\) units and then to the right \(2\) units to get another point (for example, \((2,1)\) since \(-4 + 5=1\) and \(0+2 = 2\)). We can also move down \(5\) units and to the left \(2\) units from \((0,-4)\) to get the point \((-2,-9)\). Then we draw a dashed line through these points.

Step3: Shade the solution region

Since the inequality is \(y>\frac{5}{2}x-4\), we need to shade the region above the dashed line. To check, we can pick a test point that is not on the line. A common test point is \((0,0)\).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y>\frac{5}{2}x-4\):
\(0>\frac{5}{2}(0)-4\)
\(0>-4\), which is a true statement. So the region that contains the point \((0,0)\) (the region above the line \(y=\frac{5}{2}x - 4\)) is the solution region.

Answer:

1. Rewrite the inequality as \(y>\frac{5}{2}x - 4\).
2. Graph the dashed line \(y=\frac{5}{2}x-4\) (using the y - intercept \((0,-4)\) and slope \(\frac{5}{2}\)).
3. Shade the region above the dashed line.