Question

identifying the graph of a system of two - variable linear inequalities
which is the graph of the system (x + 3y > - 3) and (y < \frac{1}{2}x + 1)?

Explanation:

Step1: Rewrite first inequality to slope-intercept

Rearrange $x + 3y > -3$ to solve for $y$:
$3y > -x - 3$
$y > -\frac{1}{3}x - 1$
This is a dashed line with slope $-\frac{1}{3}$, y-intercept $-1$, shading above the line.

Step2: Analyze second inequality

The inequality $y < \frac{1}{2}x + 1$ is already in slope-intercept form:
It is a dashed line with slope $\frac{1}{2}$, y-intercept $1$, shading below the line.

Step3: Match to correct graph

Find the graph with:

1. Dashed line $y = -\frac{1}{3}x - 1$ (shaded above)
2. Dashed line $y = \frac{1}{2}x + 1$ (shaded below)

The overlapping shaded region is the solution set, which corresponds to the third graph (left-to-right).

Answer:

The third graph (left-to-right: the one with overlapping shading in the upper-right/center area, bounded by the two dashed lines)