Question

solve the system of inequalities by graphing.
y > 8x - 7
y ≤ -\frac{1}{7}x - 3
select a line to change it between solid and dotted. select a region to shade it.

Explanation:

Step1: Analyze the first inequality \( y > 8x - 7 \)

The inequality \( y > 8x - 7 \) is a linear inequality. The boundary line is \( y = 8x - 7 \), which has a slope of \( 8 \) and a \( y \)-intercept of \( -7 \). Since the inequality is \( y > 8x - 7 \) (not \( \geq \)), the boundary line should be dashed (to indicate that the points on the line are not included in the solution set). To determine the region to shade, we can test a point not on the line, such as the origin \( (0,0) \). Substituting \( x = 0 \) and \( y = 0 \) into the inequality: \( 0 > 8(0) - 7 \) simplifies to \( 0 > -7 \), which is true. So we shade the region above the dashed line \( y = 8x - 7 \).

Step2: Analyze the second inequality \( y \leq -\frac{1}{7}x - 3 \)

The inequality \( y \leq -\frac{1}{7}x - 3 \) has a boundary line \( y = -\frac{1}{7}x - 3 \) with a slope of \( -\frac{1}{7} \) and a \( y \)-intercept of \( -3 \). Since the inequality is \( \leq \) (not \( < \)), the boundary line should be solid (to indicate that the points on the line are included in the solution set). Testing the origin \( (0,0) \) in this inequality: \( 0 \leq -\frac{1}{7}(0) - 3 \) simplifies to \( 0 \leq -3 \), which is false. So we shade the region below the solid line \( y = -\frac{1}{7}x - 3 \).

Step3: Find the intersection of the two shaded regions

The solution to the system of inequalities is the region where the shaded regions from both inequalities overlap. This is the region that is above the dashed line \( y = 8x - 7 \) and below the solid line \( y = -\frac{1}{7}x - 3 \).

Answer:

To solve the system:

1. Graph \( y = 8x - 7 \) as a dashed line (since \( y > 8x - 7 \)) and shade above it.
2. Graph \( y = -\frac{1}{7}x - 3 \) as a solid line (since \( y \leq -\frac{1}{7}x - 3 \)) and shade below it.
3. The solution is the overlapping shaded region.