Question

which graph matches the system of inequalities shown below?
y ≥ 3x - 3
y ≤ (1/2)x + 2

Explanation:

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

The inequality \( y \geq 3x - 3 \) has a slope of \( 3 \) (steep positive) and y - intercept \( - 3 \). The "greater than or equal" sign means the line is solid, and we shade above the line.

Step2: Analyze \( y \leq \frac{1}{2}x + 2 \)

The inequality \( y \leq \frac{1}{2}x + 2 \) has a slope of \( \frac{1}{2} \) (gentle positive) and y - intercept \( 2 \). The "less than or equal" sign means the line is solid, and we shade below the line.

Step3: Match with the graph

We need to find a graph where there is a solid line with slope \( 3 \) (steep) and shading above it, and a solid line with slope \( \frac{1}{2} \) (gentle) and shading below it. The overlapping region of these two shadings is the solution. From the given graph, the one with the steep solid line (for \( y \geq 3x - 3 \)) and gentle solid line (for \( y \leq \frac{1}{2}x + 2 \)) with correct shadings (above the steep line and below the gentle line) is the correct graph. Assuming the first graph (the one with the blue and red shadings) has the correct lines and shadings: the steep line (dashed? Wait, no, \( y\geq3x - 3 \) should be solid. Wait, maybe the dashed line is a mistake in the image, but following the slopes: the steep line (slope 3) and gentle line (slope 1/2). The correct graph is the one where the region above \( y = 3x-3 \) and below \( y=\frac{1}{2}x + 2 \) is shaded. So the graph with the steep solid line (or dashed if the image has a typo, but following the slopes) and gentle solid line with correct shading.

Answer:

The first graph (the one with the blue and red shaded regions, assuming the lines are solid with slopes 3 and 1/2 and correct shading above \( y = 3x - 3 \) and below \( y=\frac{1}{2}x + 2 \))