Question

which of the following inequalities is graphed on the coordinate plane?

\\(y < -3x + 5\\)
\\(y > -3x + 5\\)
\\(y \le -3x + 5\\)
\\(y \ge -3x + 5\\)

Explanation:

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<reasoning>

Identify the boundary line equation

Using the Linear Functions knowledge point
\[

$$\begin{aligned} &\text{y-intercept: } (0, 5) \implies b = 5\\ &\text{x-intercept: } (1.67, 0) \implies \text{slope } m = \frac{0 - 5}{\frac{5}{3} - 0} = -3\\ &\text{Boundary line: } y = -3x + 5 \end{aligned}$$

\]

Determine the inequality sign

The boundary line is dashed, which means the inequality is strict (\(<\) or \(>\)).
The shaded region is below and to the left of the line.
Testing the origin \((0,0)\):
\[
0 < -3(0) + 5 \implies 0 < 5 \quad (\text{True})
\]
Thus, the correct inequality is \(y < -3x + 5\).
</reasoning>

<answer>
<mcq-correct>(A) \(y < -3x + 5\)</mcq-correct>
<mcq-option>(B) \(y > -3x + 5\)</mcq-option>
<mcq-option>(C) \(y \le -3x + 5\)</mcq-option>
<mcq-option>(D) \(y \ge -3x + 5\)</mcq-option>
</answer>

<plot>
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"elements": [
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"type": "inequality",
"params": [
{
"js": "-3*x + 5",
"latex": "y < -3x + 5"
},
[0, 0]
],
"properties": {
"inverse": false,
"strict": true,
"fillColor": "rgba(140, 85, 242, 0.3)",
"strokeColor": "#8C55F2",
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Answer:

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<reasoning>

Identify the boundary line equation

Using the Linear Functions knowledge point
\[

$$\begin{aligned} &\text{y-intercept: } (0, 5) \implies b = 5\\ &\text{x-intercept: } (1.67, 0) \implies \text{slope } m = \frac{0 - 5}{\frac{5}{3} - 0} = -3\\ &\text{Boundary line: } y = -3x + 5 \end{aligned}$$

\]

Determine the inequality sign

The boundary line is dashed, which means the inequality is strict (\(<\) or \(>\)).
The shaded region is below and to the left of the line.
Testing the origin \((0,0)\):
\[
0 < -3(0) + 5 \implies 0 < 5 \quad (\text{True})
\]
Thus, the correct inequality is \(y < -3x + 5\).
</reasoning>

<answer>
<mcq-correct>(A) \(y < -3x + 5\)</mcq-correct>
<mcq-option>(B) \(y > -3x + 5\)</mcq-option>
<mcq-option>(C) \(y \le -3x + 5\)</mcq-option>
<mcq-option>(D) \(y \ge -3x + 5\)</mcq-option>
</answer>

<plot>
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"type": "inequality",
"params": [
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"latex": "y < -3x + 5"
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[0, 0]
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"strict": true,
"fillColor": "rgba(140, 85, 242, 0.3)",
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