Question

which graph matches the system of inequalities shown below?
y ≥ -x + 3
y < 3x - 1

Explanation:

Step1: Analyze \( y \geq -x + 3 \)

The inequality \( y \geq -x + 3 \) has a linear equation \( y = -x + 3 \). The slope is \(-1\) and the y - intercept is \( 3 \). Since the inequality is \( \geq \), the line should be solid, and we shade above the line.

Step2: Analyze \( y < 3x - 1 \)

The inequality \( y < 3x - 1 \) has a linear equation \( y = 3x - 1 \). The slope is \( 3 \) and the y - intercept is \(-1\). Since the inequality is \( < \), the line should be dashed, and we shade below the line.

Step3: Find the intersection

The solution to the system of inequalities is the region that is shaded by both inequalities. We need to check which graph has a solid line for \( y=-x + 3\) (shaded above) and a dashed line for \( y = 3x-1\) (shaded below). Looking at the given graph, the blue - shaded region (from \( y\geq - x + 3\)) and the region that is also below \( y = 3x-1\) (dashed line) should be the overlapping region. The graph shown has a solid line with slope - 1 (for \( y=-x + 3\)) and a dashed line with slope 3 (for \( y = 3x - 1\)), and the overlapping region (the area that is both above the solid line and below the dashed line) is correctly represented.

Answer:

The graph shown (the one with the solid line \( y=-x + 3\) and dashed line \( y = 3x-1\) with the overlapping shaded region)