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 \( m = -3 \) and the y - intercept \( b = 4 \). 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, to determine which side of the line to shade, we can test a point not on the line, for example, the origin \((0,0)\). Substitute \( x = 0\) and \( y = 0\) into the inequality: \( 0 < -3(0)+4\), which simplifies to \( 0 < 4 \), a true statement. So we shade the region below the dashed line \( y=-3x + 4\).

Step2: Graph \( y\geq3x - 2 \)

Next, graph the line \( y = 3x-2 \). The slope \( m = 3\) and the y - intercept \( b=-2\). Since the inequality is \( y\geq3x - 2\), the line should be solid (because the inequality includes equality, \( y\) can be equal to \( 3x - 2\)). To determine the shading, we test the origin \((0,0)\) again. Substitute \( x = 0\) and \( y = 0\) into the inequality: \( 0\geq3(0)-2\), which simplifies to \( 0\geq - 2\), a true statement. So we shade the region above the solid line \( y = 3x-2\).

Step3: Find the Intersection Region

The solution to the system of inequalities is the region where the two shaded regions overlap. We can also find the point of intersection of the two lines \( y=-3x + 4\) and \( y = 3x-2\) by setting 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 could use either line), we get \( y=3(1)-2=1\). So the lines intersect at the point \((1,1)\). The overlapping region is bounded by the dashed line \( y=-3x + 4\) (above the intersection point, since we shade below it) and the solid line \( y = 3x-2\) (above the intersection point, since we shade above it) and between the two lines.

(Note: Since the problem asks to graph the solution, the final answer is the graph with the overlapping shaded region of the two inequalities, with the appropriate lines (dashed for \( y < -3x + 4\) and solid for \( y\geq3x - 2\)) and the intersection point \((1,1)\) identified.)

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 lines \( y=-3x + 4\) (dashed) and \( y = 3x-2\) (solid) graphed, and the intersection point \((1,1)\) marked. The graph should show this overlapping shaded area.