Question

this graph shows the solution to which inequality?

a. \\(y < \frac{1}{3}x - 2\\)
b. \\(y > \frac{1}{3}x - 2\\)
c. \\(y \le \frac{1}{3}x - 2\\)
d. \\(y \ge \frac{1}{3}x - 2\\)

Explanation:

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<reasoning>

Find the boundary line equation

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} &m = \frac{-1 - (-3)}{3 - (-3)} = \frac{2}{6} = \frac{1}{3}\\ &y - (-1) = \frac{1}{3}(x - 3) \implies y = \frac{1}{3}x - 2 \end{aligned}$$

\]

Determine the inequality sign

Using the Shading Half-Planes knowledge point
\[

$$\begin{aligned} &\text{Boundary line is dashed} \implies < \text{ or } >\\ &\text{Test point } (0,0) \text{ is in shaded region} \implies 0 > \frac{1}{3}(0) - 2 \implies y > \frac{1}{3}x - 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>A. \(y < \frac{1}{3}x - 2\)</mcq-option>
<mcq-correct>B. \(y > \frac{1}{3}x - 2\)</mcq-correct>
<mcq-option>C. \(y \le \frac{1}{3}x - 2\)</mcq-option>
<mcq-option>D. \(y \ge \frac{1}{3}x - 2\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Find the boundary line equation

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} &m = \frac{-1 - (-3)}{3 - (-3)} = \frac{2}{6} = \frac{1}{3}\\ &y - (-1) = \frac{1}{3}(x - 3) \implies y = \frac{1}{3}x - 2 \end{aligned}$$

\]

Determine the inequality sign

Using the Shading Half-Planes knowledge point
\[

$$\begin{aligned} &\text{Boundary line is dashed} \implies < \text{ or } >\\ &\text{Test point } (0,0) \text{ is in shaded region} \implies 0 > \frac{1}{3}(0) - 2 \implies y > \frac{1}{3}x - 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>A. \(y < \frac{1}{3}x - 2\)</mcq-option>
<mcq-correct>B. \(y > \frac{1}{3}x - 2\)</mcq-correct>
<mcq-option>C. \(y \le \frac{1}{3}x - 2\)</mcq-option>
<mcq-option>D. \(y \ge \frac{1}{3}x - 2\)</mcq-option>
</answer>

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