Question

graph the solution of the following system. -8x + y ≥ -9; y ≤ (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 \(-8x + y \geq -9\). We can rewrite it in slope - intercept form (\(y=mx + b\)) by solving for \(y\).
Adding \(8x\) to both sides, we get \(y\geq8x - 9\). The slope of this line is \(m = 8\) and the \(y\) - intercept is \(b=-9\). Since the inequality is \(\geq\), we will draw a solid line (because the inequality includes equality) and shade the region above the line.

Step2: Analyze the second inequality

The second inequality is \(y\leq\frac{8}{9}x\). The slope of this line is \(m=\frac{8}{9}\) and the \(y\) - intercept is \(b = 0\) (since the equation is in the form \(y=mx\) with \(b = 0\)). Since the inequality is \(\leq\), we will draw a solid line and shade the region below the line.

Step3: Graph the lines and find the solution region

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

• When \(x = 0\), \(y=-9\). When \(y = 0\), \(0=8x - 9\), so \(8x=9\) and \(x=\frac{9}{8}=1.125\). Plot the points \((0,-9)\) and \((1.125,0)\) and draw a solid line through them. Shade the region above this line.

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

• When \(x = 0\), \(y = 0\). When \(x = 9\), \(y = 8\). Plot the points \((0,0)\) and \((9,8)\) and draw a solid line through them. 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 the problem asks to graph the solution, the final answer is the graph obtained by following the above steps. If we were to describe the graph: It consists of two solid lines, \(y = 8x-9\) (with slope 8, \(y\) - intercept - 9) and \(y=\frac{8}{9}x\) (with slope \(\frac{8}{9}\), \(y\) - intercept 0), and the solution region is the area that is above \(y = 8x - 9\) and below \(y=\frac{8}{9}x\).)

Answer:

The graph is composed of two solid lines: \(y = 8x-9\) (slope 8, \(y\) - intercept - 9, shade above) and \(y=\frac{8}{9}x\) (slope \(\frac{8}{9}\), \(y\) - intercept 0, shade below). The solution region is the intersection of the two shaded regions.