Question

multiple choice 1 point
sketch the solution to this system of inequalities:
$y \geq 2x - 3$
$y > \frac{1}{2}x + 1$
options a, b, c, d with graphs

Explanation:

Step1: Analyze \( y \geq 2x - 3 \)

The line \( y = 2x - 3 \) has a slope of \( 2 \) and a \( y \)-intercept of \( -3 \). The inequality \( y \geq 2x - 3 \) means we shade above (or on) this line, and the line should be solid (since it's "greater than or equal to").

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

The line \( y = \frac{1}{2}x + 1 \) has a slope of \( \frac{1}{2} \) and a \( y \)-intercept of \( 1 \). The inequality \( y > \frac{1}{2}x + 1 \) means we shade above this line, and the line should be dashed (since it's "greater than", not "or equal to").

Step3: Match with Graphs

• Check the slopes and intercepts of the lines in each graph.
• For the solid line (from \( y \geq 2x - 3 \)): slope \( 2 \), \( y \)-intercept \( -3 \). For the dashed line (from \( y > \frac{1}{2}x + 1 \)): slope \( \frac{1}{2} \), \( y \)-intercept \( 1 \).
• Graph A has a solid line with slope \( 2 \) (steep) and a dashed line with slope \( \frac{1}{2} \) (shallow), and the shading is above both, matching the inequalities. Other graphs (B, C, D) have lines with incorrect slopes or intercepts, or wrong shading.

Answer:

A. (The graph labeled A with the solid steep line and dashed shallow line, shaded above both)