Question

choose all that give the correct inequality for the graph shown.
a
$y\leq -\frac{1}{3}x + 1$
b
$y < -\frac{1}{3}x - 2$
c
$y \geq \frac{1}{3}x + 1$
d
$y > \frac{1}{3}x - 2$

Explanation:

Step1: Analyze Graph A

First, identify the line equation: the line has slope $-\frac{1}{3}$ and y-intercept 1, so $y = -\frac{1}{3}x + 1$. The line is solid (includes equality), and the shaded region is below the line, so the inequality is $y \leq -\frac{1}{3}x + 1$. This matches the given inequality for A, so A is correct.

Step2: Analyze Graph B

The line has slope $-\frac{1}{3}$ and y-intercept -2, so $y = -\frac{1}{3}x - 2$. The line is dashed (no equality), but the given inequality is $y \leq -\frac{1}{3}x - 2$ (includes equality), which does not match. So B is incorrect.

Step3: Analyze Graph C

The line has slope $\frac{1}{3}$ and y-intercept 1, so $y = \frac{1}{3}x + 1$. The line is dashed (no equality), and the shaded region is above the line, so the inequality is $y > \frac{1}{3}x + 1$. This matches the given inequality for C, so C is correct.

Step4: Analyze Graph D

The line has slope $-\frac{1}{3}$ and y-intercept -2, so $y = -\frac{1}{3}x - 2$, but the given inequality is $y \geq \frac{1}{3}x - 2$ (wrong slope). The shaded region is below the line, which also contradicts the "greater than or equal to" inequality. So D is incorrect.

Answer:

A. $y \leq -\frac{1}{3}x + 1$
C. $y > \frac{1}{3}x + 1$