Question

graph the solution of the following system.
-8x + y geq -9
( y leq \frac{8}{9}x )
use the graphing tool to graph the system.
click to enlarge graph

Explanation:

Step1: Analyze the first inequality

The first inequality is \(-8x + y \geq -9\). Let's rewrite it in slope - intercept form (\(y=mx + b\)):
Add \(8x\) to both sides of the inequality: \(y\geq8x - 9\). The slope \(m = 8\) and the \(y\) - intercept \(b=-9\). Since the inequality is \(\geq\), we will draw a solid line (because the points on the line are included in the solution set) and shade the region above the line.

Step2: Analyze the second inequality

The second inequality is \(y\leq\frac{8}{9}x\). The slope \(m=\frac{8}{9}\) and the \(y\) - intercept \(b = 0\) (since when \(x = 0\), \(y = 0\)). Since the inequality is \(\leq\), we will draw a solid line (because the points on the line are included in the solution set) and shade the region below the line.

Step3: Graph the lines and find the solution region

1. For the line \(y = 8x-9\):

• To find two points on the line, when \(x = 0\), \(y=-9\). When \(x = 1\), \(y=8(1)-9=-1\). Plot the points \((0,-9)\) and \((1, - 1)\) and draw a solid line through them. Then shade the region above this line.

1. For the line \(y=\frac{8}{9}x\):

• When \(x = 0\), \(y = 0\). When \(x = 9\), \(y=\frac{8}{9}\times9 = 8\). Plot the points \((0,0)\) and \((9,8)\) and draw a solid line through them. Then shade the region below this line.

The solution to the system of inequalities is the region that is shaded by both inequalities (the intersection of the two shaded regions).

(Note: Since this is a graphing problem, the final answer would be the graph with the two lines \(y = 8x-9\) (solid, shaded above) and \(y=\frac{8}{9}x\) (solid, shaded below) and the overlapping shaded region. However, if we were to describe the steps to graph it, the above steps are the way to go.)

Answer:

The graph should have a solid line \(y = 8x - 9\) (shaded above) and a solid line \(y=\frac{8}{9}x\) (shaded below), with the overlapping region as the solution.