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 (since $>$) with slope $-\frac{1}{3}$, y-intercept $-1$, shaded 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 (since $<$) with slope $\frac{1}{2}$, y-intercept $1$, shaded below the line.

Step3: Find overlapping shaded region

The correct graph has:

1. Dashed line $y = -\frac{1}{3}x - 1$, shaded above it
2. Dashed line $y = \frac{1}{2}x + 1$, shaded below it
3. The overlap is the region that satisfies both inequalities.

Answer:

The third graph (left to right, the one with the overlapping shaded region between the two dashed lines, bounded above by $y=\frac{1}{2}x+1$ and below by $y=-\frac{1}{3}x-1$)