Question

choose all that give the correct inequality for the graph shown.
a
$y\leq -\frac{1}{3}x + 1$

b
$y < -\frac{3}{5}x - 2$

c
$y > \frac{1}{2}x + 1$

d
$y \geq \frac{2}{5}x - 2$

Explanation:

To solve this, we analyze each option by checking the line type (solid/dashed) and the shaded region relative to the line.

Option A

• Line Type: Solid (so inequality includes equality, $\leq$ or $\geq$).
• Slope/Intercept: The line has slope $-\frac{1}{3}$ and y - intercept $1$ (equation $y = -\frac{1}{3}x + 1$).
• Shaded Region: Below the line (since shaded area is where $y$ is less than or equal to the line’s $y$ - value).
• Conclusion: $y\leq-\frac{1}{3}x + 1$ matches (solid line, shaded below).

Option B

• Line Type: Dashed (so inequality is strict, $<$ or $>$).
• Slope/Intercept: The line has slope $-\frac{3}{5}$ and y - intercept $-2$ (equation $y = -\frac{3}{5}x - 2$).
• Shaded Region: Below the dashed line (shaded area is where $y$ is less than the line’s $y$ - value).
• Conclusion: $y<-\frac{3}{5}x - 2$ matches (dashed line, shaded below).

Option C

• Line Type: Dashed (strict inequality).
• Slope/Intercept: The line has slope $\frac{1}{2}$ and y - intercept $1$ (equation $y=\frac{1}{2}x + 1$).
• Shaded Region: Above the line (shaded area is where $y$ is greater than the line’s $y$ - value).
• Conclusion: $y>\frac{1}{2}x + 1$ matches (dashed line, shaded above).

Option D

• Line Type: Solid (includes equality).
• Slope/Intercept: The line has slope $\frac{2}{5}$ and y - intercept $-2$ (equation $y=\frac{2}{5}x - 2$).
• Shaded Region: Below the line (shaded area is where $y$ is less than the line’s $y$ - value), but the inequality is $y\geq\frac{2}{5}x - 2$ (shaded above would be for $\geq$).
• Conclusion: Does not match (shaded region contradicts the inequality).

Answer:

A. $y\leq-\frac{1}{3}x + 1$, B. $y<-\frac{3}{5}x - 2$, C. $y>\frac{1}{2}x + 1$