Question

which is the graph of $2x - 4y > 6$?

Explanation:

Step1: Rewrite in slope-intercept form

First, isolate $y$ to identify the boundary line and inequality direction.

$$\begin{align*} 2x - 4y &> 6 \\ -4y &> -2x + 6 \\ y &< \frac{1}{2}x - \frac{3}{2} \end{align*}$$

(Note: Inequality flips when dividing by -4)

Step2: Identify boundary line properties

The boundary line is $y = \frac{1}{2}x - \frac{3}{2}$, which has a y-intercept of $-\frac{3}{2}$ (or -1.5) and a slope of $\frac{1}{2}$. Since the inequality is $<$, the line is dashed.

Step3: Determine shaded region

We shade the area below the dashed line (because $y$ is less than the expression). To verify, test the origin $(0,0)$:
$0 < \frac{1}{2}(0) - \frac{3}{2}$ simplifies to $0 < -1.5$, which is false. So the region does not include the origin, confirming we shade below the line.

Answer:

The correct graph is the fourth option (bottom right): dashed line with y-intercept -1.5, slope $\frac{1}{2}$, and shading below the line.