Question

1. which region of the graph represents the solution to the system of inequalities?

y < -x + 2
y > 2x - 4

Explanation:

Step1: Analyze \( y < -x + 2 \)

The line \( y=-x + 2 \) has a slope of \(-1\) and a \( y \)-intercept of \( 2 \). Since the inequality is \( y < -x + 2 \), the region is below this dashed line (dashed because the inequality is strict, \( < \), not \( \leq \)).

Step2: Analyze \( y > 2x - 4 \)

The line \( y = 2x-4 \) has a slope of \( 2 \) and a \( y \)-intercept of \(-4\). Since the inequality is \( y > 2x - 4 \), the region is above this dashed line (dashed because the inequality is strict, \( > \), not \( \geq \)).

Step3: Find the Intersection Region

The solution to the system of inequalities is the region that is both below \( y=-x + 2 \) and above \( y = 2x-4 \). To confirm, we can pick a test point in the intersection region, say \( (0,0) \). For \( y < -x + 2 \): \( 0 < 0 + 2 \) (true). For \( y > 2x - 4 \): \( 0 > 0 - 4 \) (true). So the region satisfying both inequalities is where these two half - planes overlap.

Answer:

The region that is below the line \( y=-x + 2 \) (dashed) and above the line \( y = 2x-4 \) (dashed) is the solution region for the system \(

$$\begin{cases}y < -x + 2\\y > 2x - 4\end{cases}$$

\).