Question

graph the solution to the following system of inequalities.
$y < -3x + 4$
$y \geq 3x - 2$

Explanation:

Step1: Analyze \( y < -3x + 4 \)

The inequality \( y < -3x + 4 \) is a linear inequality. The boundary line is \( y=-3x + 4 \), which has a slope of \(-3\) and a \( y \)-intercept of \( 4 \). Since the inequality is \( y < \dots \), the line should be dashed (because the points on the line are not included in the solution), and we shade the region below the line.

Step2: Analyze \( y \geq 3x - 2 \)

The inequality \( y \geq 3x - 2 \) has a boundary line \( y = 3x - 2 \), with a slope of \( 3 \) and a \( y \)-intercept of \(-2\). Since the inequality is \( y \geq \dots \), the line should be solid (because the points on the line are included in the solution), and we shade the region above the line.

Step3: Find the intersection of the two regions

To graph the solution to the system, we find the region that is shaded for both inequalities. First, graph the dashed line \( y=-3x + 4 \) and shade below it. Then, graph the solid line \( y = 3x - 2 \) and shade above it. The overlapping region (where both shadings meet) is the solution to the system.

To find the intersection point of the two boundary lines \( y=-3x + 4 \) and \( y = 3x - 2 \), set them equal:
\[
-3x + 4=3x - 2
\]
Add \( 3x \) to both sides:
\[
4 = 6x - 2
\]
Add \( 2 \) to both sides:
\[
6=6x
\]
Divide both sides by \( 6 \):
\[
x = 1
\]
Substitute \( x = 1 \) into \( y = 3x - 2 \):
\[
y=3(1)-2=1
\]
So the intersection point of the two lines is \( (1,1) \). The solution region is the area that is below the dashed line \( y=-3x + 4 \) and above the solid line \( y = 3x - 2 \), bounded by these two lines and their intersection at \( (1,1) \).

(Note: Since this is a graphing problem, the final answer is the graph with the dashed line \( y = -3x + 4 \) (shaded below), the solid line \( y = 3x - 2 \) (shaded above), and the overlapping region. However, if we were to describe the solution set in terms of the graph, it's the region bounded by the two lines as described.)

Answer:

The solution is the region that is below the dashed line \( y=-3x + 4 \) and above the solid line \( y = 3x - 2 \), with the intersection point of the two lines at \( (1,1) \). The graph should have a dashed line \( y=-3x + 4 \) (slope \(-3\), \( y \)-intercept \( 4 \)) with shading below it, a solid line \( y = 3x - 2 \) (slope \( 3 \), \( y \)-intercept \(-2\)) with shading above it, and the overlapping shaded region is the solution.