Question

consider this system of inequalities.
x + y ≤ -3
y < \frac{x}{2}
which graph shows the solution for this system?

Explanation:

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

Rewrite \( x + y \leq -3 \) as \( y \leq -x - 3 \). The boundary line \( y = -x - 3 \) has a slope of \(-1\) and a y - intercept of \(-3\). Since the inequality is \( \leq \), the line should be solid, and we shade below the line.

Step2: Analyze \( y < \frac{x}{2} \)

The boundary line \( y=\frac{x}{2} \) has a slope of \( \frac{1}{2} \) and a y - intercept of \( 0 \). Since the inequality is \( < \), the line should be dashed, and we shade below the line (because for a linear inequality \( y < mx + b \), we shade below the line).

Step3: Find the intersection of the two regions

We need to find the region that is shaded for both inequalities. The first inequality (\( y \leq -x - 3 \)) has a solid line with negative slope and the second (\( y < \frac{x}{2} \)) has a dashed line with positive slope. By looking at the three graphs, we check the line types (solid/dashed) and the shading regions. The correct graph should have a solid line for \( y=-x - 3 \) (from \( x + y=-3 \)) and a dashed line for \( y = \frac{x}{2} \), with the overlapping shaded region matching the solution of both inequalities.

Answer:

The graph (assuming the third graph in the options, but based on the analysis: the graph with a solid line \( y=-x - 3 \) (slope - 1, y - intercept - 3) and a dashed line \( y=\frac{x}{2} \) (slope \( \frac{1}{2} \), y - intercept 0) with the correct overlapping shaded region)