Question

which is the graph of the linear inequality \\(\frac{1}{2}x - 2y > -6\\)?

Explanation:

Step1: Rearrange inequality to slope-intercept form

Start with $\frac{1}{2}x - 2y > -6$. Subtract $\frac{1}{2}x$ from both sides:
$-2y > -\frac{1}{2}x - 6$

Step2: Solve for y (reverse inequality sign)

Divide all terms by $-2$:
$y < \frac{1}{4}x + 3$

Step3: Identify line type and shading

The inequality uses $<$, so the boundary line $y=\frac{1}{4}x + 3$ is dashed, and we shade below the line.

Step4: Verify intercepts

For x-intercept: set $y=0$, $0=\frac{1}{4}x + 3 \implies x=-12$ (matches the dashed line's left intercept). For y-intercept: set $x=0$, $y=3$ (matches the dashed line's top intercept).

Answer:

The correct graph is the bottom-right option (dashed line $y=\frac{1}{4}x + 3$, shaded region below the line).