Question

which graph best represents the solution set to the system of inequalities?
y ≤ (1/2)x + 5
y > 3x - 1
a
graph a
c
graph c
b
graph b
d
graph d

Explanation:

Step1: Analyze \( y \leq \frac{1}{2}x + 5 \)

The inequality \( y \leq \frac{1}{2}x + 5 \) has a slope of \( \frac{1}{2} \) and y - intercept at \( (0,5) \). The line should be solid (since \( \leq \)) and the shading is below the line.

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

The inequality \( y > 3x - 1 \) has a slope of \( 3 \) and y - intercept at \( (0, - 1) \). The line should be dashed (since \( > \)) and the shading is above the line.

Step3: Match with graphs

• For \( y \leq \frac{1}{2}x + 5 \), the solid line with slope \( \frac{1}{2} \) and y - intercept 5. For \( y > 3x - 1 \), dashed line with slope 3 and y - intercept - 1.
• Graph D: The solid line has slope \( \frac{1}{2} \) (matches \( y=\frac{1}{2}x + 5 \) as it goes through (0,5) and has a gentle slope), dashed line has slope 3 (steeper, matches \( y = 3x-1 \) as it goes through (0, - 1) when extended), and the shading is below the solid line and above the dashed line.

Answer:

D