Question

inequalities by graphing.
select a line to change it between solid and dotted. select a region to shade it.
$y > \frac{6}{7}x - 3$
$-9x + y < -5$

Explanation:

Step1: Analyze the first inequality \( y > \frac{6}{7}x - 3 \)

The inequality is in slope - intercept form \( y=mx + b \), where the slope \( m=\frac{6}{7} \) and the y - intercept \( b=- 3 \). Since the inequality is \( y>\frac{6}{7}x - 3 \) (not \( y\geq\frac{6}{7}x - 3 \)), the line should be dotted. To graph the line \( y=\frac{6}{7}x - 3 \), we start at the y - intercept \( (0,-3) \). Then, using the slope \( \frac{6}{7} \) (rise 6, run 7), we can find another point. For example, from \( (0,-3) \), moving 6 units up and 7 units to the right gives us the point \( (7,3) \). After drawing the dotted line, we shade the region above the line because \( y \) is greater than \( \frac{6}{7}x - 3 \).

Step2: Analyze the second inequality \( - 9x + y<-5 \)

First, we rewrite it in slope - intercept form. Solve for \( y \): \( y<9x - 5 \). The slope \( m = 9 \) and the y - intercept \( b=-5 \). Since the inequality is \( y < 9x-5 \) (not \( y\leq9x - 5 \)), the line should be dotted. To graph the line \( y = 9x-5 \), we start at the y - intercept \( (0,-5) \). Using the slope \( 9=\frac{9}{1} \), from \( (0,-5) \), moving 9 units up and 1 unit to the right gives us the point \( (1,4) \). After drawing the dotted line, we shade the region below the line because \( y \) is less than \( 9x - 5 \).

Step3: Find the overlapping region

The solution to the system of inequalities is the region that is shaded for both inequalities. We need to check the intersection of the region above \( y=\frac{6}{7}x - 3 \) and below \( y = 9x-5 \).

Answer:

To graph the system:

1. For \( y>\frac{6}{7}x - 3 \): Draw a dotted line with slope \( \frac{6}{7} \) and y - intercept \( - 3 \), shade above the line.
2. For \( -9x + y<-5\) (or \( y < 9x-5 \)): Draw a dotted line with slope \( 9 \) and y - intercept \( - 5 \), shade below the line.

The solution is the region where the two shaded regions overlap.