Question

graph the solution of the following system.
-9x + y ≥ -10
( y leq \frac{9}{10}x )
use the graphing tool to graph the system.
click to enlarge graph

Explanation:

Step1: Rewrite the first inequality

Rewrite \(-9x + y \geq -10\) in slope - intercept form (\(y=mx + b\)). Add \(9x\) to both sides: \(y\geq9x - 10\). The slope \(m = 9\) and the \(y\) - intercept \(b=-10\). Since the inequality is \(\geq\), we draw a solid line for \(y = 9x-10\) and shade the region above the line (because \(y\) is greater than or equal to \(9x - 10\)).

Step2: Analyze the second inequality

The second inequality is \(y\leq\frac{9}{10}x\). The slope \(m=\frac{9}{10}\) and the \(y\) - intercept \(b = 0\). Since the inequality is \(\leq\), we draw a solid line for \(y=\frac{9}{10}x\) and shade the region below the line (because \(y\) is less than or equal to \(\frac{9}{10}x\)).

Step3: Find the solution region

The solution to the system of inequalities is the region that is shaded by both inequalities. To graph this, first plot the line \(y = 9x-10\) (solid line, slope \(9\), \(y\) - intercept \(- 10\)) and shade above it. Then plot the line \(y=\frac{9}{10}x\) (solid line, slope \(\frac{9}{10}\), \(y\) - intercept \(0\)) and shade below it. The overlapping region of the two shaded areas is the solution to the system.

(Note: Since this is a graphing problem, the final answer is the graph of the overlapping region of the two inequalities. The steps above describe how to construct the graph.)

Answer:

The solution is the region that is above (or on) the line \(y = 9x-10\) and below (or on) the line \(y=\frac{9}{10}x\). To graph it, plot \(y = 9x - 10\) (solid, shade above) and \(y=\frac{9}{10}x\) (solid, shade below), and the intersection of the shaded regions is the solution.