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 $y = \frac{1}{4}x+3$. The slope of the line is $\frac{1}{4}$ and the y - intercept is 3. Since the inequality is $y<\frac{1}{4}x + 3$, the boundary line is a dashed line (because the inequality is strict, i.e., $y$ is strictly less than, not less than or equal to).

Step3: Test a point

We can test a point not on the line, say $(0,0)$. 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 point $(0,0)$ is the solution region.

The graph with a dashed line having a positive slope of $\frac{1}{4}$ and y - intercept of 3 and the region below the line and containing the origin is the correct graph.

Answer:

The graph with a dashed line, slope $\frac{1}{4}$, y - intercept 3, and the region below the line containing the origin.