Question

determining two - variable linear inequalities with no solution
which linear inequality will not have a shared solution set with the graphed linear inequality?
$y < \frac{5}{3}x - 2$
$y < -\frac{5}{3}x + 1$
$y > \frac{5}{3}x + 2$
$y > -\frac{5}{3}x + 2$

Explanation:

Step1: Identify graphed inequality

The dashed line has slope $\frac{5}{3}$ and y-intercept $1$, shaded below, so it is $y < \frac{5}{3}x + 1$.

Step2: Analyze parallel inequalities

Two inequalities have no shared solutions if they are parallel (same slope) and their regions do not overlap. The graphed inequality has slope $\frac{5}{3}$, so we check options with slope $\frac{5}{3}$.

Step3: Check non-overlapping region

For $y > \frac{5}{3}x + 2$, its line is parallel and above $y = \frac{5}{3}x + 1$. The region $y > \frac{5}{3}x + 2$ is entirely above this upper line, while $y < \frac{5}{3}x + 1$ is entirely below the lower line—no overlap.

Answer:

$y > \frac{5}{3}x + 2$