Question

on a piece of paper, graph \\(y + 2 \le -\frac{2}{3}x + 4\\). then determine which answer choice matches the graph you drew.

a. graph a
b. graph b
c. graph c
d. graph d

Explanation:

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<reasoning>

Rearrange the inequality to slope-intercept form

\[

$$\begin{aligned} y + 2 &\le -\frac{2}{3}x + 4 \\ y &\le -\frac{2}{3}x + 2 \end{aligned}$$

\]

Identify boundary line properties

• Slope: \(m = -\frac{2}{3}\)
• y-intercept: \((0, 2)\)
• x-intercept: Set \(y = 0 \implies 0 = -\frac{2}{3}x + 2 \implies x = 3\), so \((3, 0)\). The graphs show points \((0, 2)\) and \((6, -2)\). Checking \((6, -2)\): \(-2 = -\frac{2}{3}(6) + 2 = -4 + 2 = -2\) (correct).
• Inequality symbol is \(\le\), which means a solid boundary line. This eliminates Graphs C and D (dashed lines).

Determine the shaded region

• Test point \((0,0)\):

\[
0 \le -\frac{2}{3}(0) + 2 \implies 0 \le 2 \quad (\text{True})
\]

• Since the inequality is true at \((0,0)\), shade the half-plane containing the origin (below and to the left of the line).
• Graph A shades the region above/right of the line.
• Graph B shades the region below/left of the line (containing the origin).
• Therefore, Graph B is the correct match.

</reasoning>

<answer>
<mcq-correct>A. Graph A</mcq-correct>
<mcq-option>B. Graph B</mcq-option>
<mcq-option>C. Graph C</mcq-option>
<mcq-option>D. Graph D</mcq-option>
</answer>

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"type": "inequality",
"params": [
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"latex": "y \le -\frac{2}{3}x + 2"
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"properties": {
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"size": 4,
"color": "#583C87",
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Answer:

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<reasoning>

Rearrange the inequality to slope-intercept form

\[

$$\begin{aligned} y + 2 &\le -\frac{2}{3}x + 4 \\ y &\le -\frac{2}{3}x + 2 \end{aligned}$$

\]

Identify boundary line properties

• Slope: \(m = -\frac{2}{3}\)
• y-intercept: \((0, 2)\)
• x-intercept: Set \(y = 0 \implies 0 = -\frac{2}{3}x + 2 \implies x = 3\), so \((3, 0)\). The graphs show points \((0, 2)\) and \((6, -2)\). Checking \((6, -2)\): \(-2 = -\frac{2}{3}(6) + 2 = -4 + 2 = -2\) (correct).
• Inequality symbol is \(\le\), which means a solid boundary line. This eliminates Graphs C and D (dashed lines).

Determine the shaded region

• Test point \((0,0)\):

\[
0 \le -\frac{2}{3}(0) + 2 \implies 0 \le 2 \quad (\text{True})
\]

• Since the inequality is true at \((0,0)\), shade the half-plane containing the origin (below and to the left of the line).
• Graph A shades the region above/right of the line.
• Graph B shades the region below/left of the line (containing the origin).
• Therefore, Graph B is the correct match.

</reasoning>

<answer>
<mcq-correct>A. Graph A</mcq-correct>
<mcq-option>B. Graph B</mcq-option>
<mcq-option>C. Graph C</mcq-option>
<mcq-option>D. Graph D</mcq-option>
</answer>

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