which is the graph of the linear inequality 2x - 3y < 12?

Explanation:

Step1: Rewrite the inequality in slope - intercept form

First, solve $2x−3y < 12$ for $y$. Subtract $2x$ from both sides: $-3y< - 2x + 12$. Then divide by $-3$. When dividing an inequality by a negative number, the direction of the inequality sign changes. So we get $y>\frac{2}{3}x - 4$.

Step2: Analyze the boundary line

The boundary line of the inequality $y>\frac{2}{3}x - 4$ is the equation $y = \frac{2}{3}x - 4$. The slope of the line is $\frac{2}{3}$ and the $y$-intercept is $-4$. Since the inequality is $y>\frac{2}{3}x - 4$, the boundary line is a dashed line (because the points on the line are not included in the solution set).

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 $2x−3y<12$. We have $2(0)-3(0)=0<12$, which is true. So the region that contains the point $(0,0)$ is part of the solution set.

Answer:

The graph with a dashed line having a slope of $\frac{2}{3}$, $y$-intercept of $-4$, and the region above the line and containing the origin is the correct graph.