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

Explanation:

Step1: Rewrite the inequality in slope - intercept form

First, start with $\frac{1}{2}x−2y > - 6$. Subtract $\frac{1}{2}x$ from both sides: $-2y>-\frac{1}{2}x - 6$. Then divide each term by $- 2$. Remember to reverse the inequality sign when dividing by a negative number. So we get $y<\frac{1}{4}x + 3$.

Step2: Analyze the boundary line

The boundary line of the inequality $y<\frac{1}{4}x + 3$ is the equation $y = \frac{1}{4}x+3$. Since the inequality is $y<\frac{1}{4}x + 3$, the boundary line is a dashed line (because the points on the line are not included in the solution set).

Step3: Determine the shaded region

To find the shaded region, we can test a point. Let's test the origin $(0,0)$. Substitute $x = 0$ and $y = 0$ into the original inequality $\frac{1}{2}(0)-2(0)>-6$, which simplifies to $0>-6$. This is a true statement. So, the region that contains the origin is the shaded region.

Answer:

The graph with a dashed line $y=\frac{1}{4}x + 3$ and the region below the line shaded.