Question

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

Explanation:

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

First, graph the line \( y = -3x + 4 \). The slope is \( -3 \) and the y - intercept is \( 4 \). Since the inequality is \( y < -3x + 4 \), the line should be dashed (because the inequality is strict, \( y\) is not equal to \( -3x + 4 \)). Then, we test a point not on the line, say \( (0,0) \). Substitute into the inequality: \( 0 < -3(0)+4 \), which is \( 0 < 4 \), a true statement. So we shade the region below the dashed line \( y = -3x + 4 \).

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

Next, graph the line \( y = 3x - 2 \). The slope is \( 3 \) and the y - intercept is \( -2 \). Since the inequality is \( y\geq3x - 2 \), the line should be solid (because \( y\) can be equal to \( 3x - 2 \)). Then, we test a point not on the line, say \( (0,0) \). Substitute into the inequality: \( 0\geq3(0)-2 \), which is \( 0\geq - 2 \), a true statement. So we shade the region above (or on) the solid line \( y = 3x - 2 \).

Step3: Find the Intersection

The solution to the system of inequalities is the region that is shaded for both inequalities. To find the intersection point of the two lines \( y=-3x + 4 \) and \( y = 3x-2 \), we set them equal to each other:
\[

$$\begin{align*} -3x + 4&=3x-2\\ 4 + 2&=3x+3x\\ 6&=6x\\ x&=1 \end{align*}$$

\]
Substitute \( x = 1 \) into \( y = 3x-2 \), we get \( y=3(1)-2=1 \). So the intersection point is \( (1,1) \). The solution region is the area that is below the dashed line \( y=-3x + 4 \) and above (or on) the solid line \( y = 3x - 2 \), bounded by these two lines and including the part of \( y = 3x - 2 \) that is below \( y=-3x + 4 \).

(Note: Since this is a graphing problem, the final answer is the graph of the overlapping region of the two inequalities. The key steps are graphing each inequality correctly (dashed/solid line, correct shading) and identifying the overlapping region.)

Answer:

The solution is the region that is below the dashed line \( y = - 3x+4 \) and above (or on) the solid line \( y=3x - 2 \), with the intersection point at \( (1,1) \). To graph it:

1. Draw a dashed line for \( y=-3x + 4 \) and shade below it.
2. Draw a solid line for \( y = 3x-2 \) and shade above (or on) it.
3. The overlapping shaded region is the solution.