Question

the point \\((0, 0)\\) is a solution to which of these inequalities?

a. \\(y + 7 < 2x + 6\\)
b. \\(y - 7 < 2x - 6\\)
c. \\(y - 6 < 2x - 7\\)
d. \\(y + 7 < 2x - 6\\)

Explanation:

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<reasoning>

Substitute point into Option A

\[

$$\begin{aligned} y + 7 &< 2x + 6 \\ 0 + 7 &< 2(0) + 6 \\ 7 &< 6 \quad (\text{False}) \end{aligned}$$

\]

Substitute point into Option B

\[

$$\begin{aligned} y - 7 &< 2x - 6 \\ 0 - 7 &< 2(0) - 6 \\ -7 &< -6 \quad (\text{True}) \end{aligned}$$

\]

Substitute point into Options C and D

\[

$$\begin{aligned} \text{Option C: } 0 - 6 &< 2(0) - 7 \implies -6 < -7 \quad (\text{False}) \\ \text{Option D: } 0 + 7 &< 2(0) - 6 \implies 7 < -6 \quad (\text{False}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>A. \(y+7 < 2x+6\)</mcq-option>
<mcq-correct>B. \(y-7 < 2x-6\)</mcq-correct>
<mcq-option>C. \(y-6 < 2x-7\)</mcq-option>
<mcq-option>D. \(y+7 < 2x-6\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Substitute point into Option A

\[

$$\begin{aligned} y + 7 &< 2x + 6 \\ 0 + 7 &< 2(0) + 6 \\ 7 &< 6 \quad (\text{False}) \end{aligned}$$

\]

Substitute point into Option B

\[

$$\begin{aligned} y - 7 &< 2x - 6 \\ 0 - 7 &< 2(0) - 6 \\ -7 &< -6 \quad (\text{True}) \end{aligned}$$

\]

Substitute point into Options C and D

\[

$$\begin{aligned} \text{Option C: } 0 - 6 &< 2(0) - 7 \implies -6 < -7 \quad (\text{False}) \\ \text{Option D: } 0 + 7 &< 2(0) - 6 \implies 7 < -6 \quad (\text{False}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>A. \(y+7 < 2x+6\)</mcq-option>
<mcq-correct>B. \(y-7 < 2x-6\)</mcq-correct>
<mcq-option>C. \(y-6 < 2x-7\)</mcq-option>
<mcq-option>D. \(y+7 < 2x-6\)</mcq-option>
</answer>

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