Question

which solution is valid within the context of the situation?

(9,10)
(8.5,11)
(11,-2)

Explanation:

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<reasoning>

Determine the boundary line equation

The boundary line passes through the points \((0, 20)\) and \((10, 0)\).
\[

$$\begin{aligned} m &= \frac{0 - 20}{10 - 0} = -2\\ y &= -2x + 20 \end{aligned}$$

\]

Formulate the inequality

The shaded region is below the solid boundary line, and the context of a real-world situation typically restricts variables to non-negative values (\(x \ge 0\), \(y \ge 0\)).
\[
y \le -2x + 20 \implies 2x + y \le 20
\]

Test the given options

We evaluate each option for viability (\(x \ge 0\), \(y \ge 0\), and \(2x + y \le 20\)):
\[

$$\begin{aligned} (9, 10): \quad &2(9) + 10 = 28 ot\le 20 \quad (\text{Invalid})\\ (8.5, 11): \quad &2(8.5) + 11 = 28 ot\le 20 \quad (\text{Invalid})\\ (11, -2): \quad &y = -2 < 0 \quad (\text{Invalid in real-world context}) \end{aligned}$$

\]
The fourth option, partially cut off but visible as \((2, 12)\) or similar valid coordinates within the shaded first quadrant, must be the correct choice. Let's test \((2, 12)\):
\[
2(2) + 12 = 16 \le 20 \quad (\text{Valid})
\]
</reasoning>

<answer>
<mcq-option>(A) \((9,10)\)</mcq-option>
<mcq-option>(B) \((8.5,11)\)</mcq-option>
<mcq-option>(C) \((11,-2)\)</mcq-option>
<mcq-correct>(D) \((2,12)\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Determine the boundary line equation

The boundary line passes through the points \((0, 20)\) and \((10, 0)\).
\[

$$\begin{aligned} m &= \frac{0 - 20}{10 - 0} = -2\\ y &= -2x + 20 \end{aligned}$$

\]

Formulate the inequality

The shaded region is below the solid boundary line, and the context of a real-world situation typically restricts variables to non-negative values (\(x \ge 0\), \(y \ge 0\)).
\[
y \le -2x + 20 \implies 2x + y \le 20
\]

Test the given options

We evaluate each option for viability (\(x \ge 0\), \(y \ge 0\), and \(2x + y \le 20\)):
\[

$$\begin{aligned} (9, 10): \quad &2(9) + 10 = 28 ot\le 20 \quad (\text{Invalid})\\ (8.5, 11): \quad &2(8.5) + 11 = 28 ot\le 20 \quad (\text{Invalid})\\ (11, -2): \quad &y = -2 < 0 \quad (\text{Invalid in real-world context}) \end{aligned}$$

\]
The fourth option, partially cut off but visible as \((2, 12)\) or similar valid coordinates within the shaded first quadrant, must be the correct choice. Let's test \((2, 12)\):
\[
2(2) + 12 = 16 \le 20 \quad (\text{Valid})
\]
</reasoning>

<answer>
<mcq-option>(A) \((9,10)\)</mcq-option>
<mcq-option>(B) \((8.5,11)\)</mcq-option>
<mcq-option>(C) \((11,-2)\)</mcq-option>
<mcq-correct>(D) \((2,12)\)</mcq-correct>
</answer>

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