Question

1. which graph represents the linear inequality $y \geq -\frac{5}{2}x + 3$

Explanation:

Step1: Analyze the inequality form

The linear inequality is \( y \geq -\frac{5}{2}x + 3 \). For a linear inequality \( y \geq mx + b \), the line \( y = mx + b \) should be solid (since the inequality is "greater than or equal to") and the region above the line should be shaded.

Step2: Check the y-intercept

The y-intercept \( b = 3 \), so the line should cross the y-axis at \( (0, 3) \). The given graph has a solid line (or in this case, a red line with a slope) crossing the y-axis at \( (0, 3) \), which matches.

Step3: Check the slope

The slope \( m = -\frac{5}{2} \), which is a negative slope (going down from left to right). The graph's line has a negative slope, consistent with \( -\frac{5}{2} \).

Step4: Check the shaded region

Since the inequality is \( y \geq -\frac{5}{2}x + 3 \), we shade above the line. The given graph has the region above the red line shaded, which matches the inequality.

Answer:

The graph shown (with the red line and the shaded region above it, crossing the y-axis at (0,3) and having a negative slope) represents the linear inequality \( y \geq -\frac{5}{2}x + 3 \).