Question

1. solve \\(|-2x| - 16 = 2x - 20\\).

• no solutions
• \\(\\{1\\}\\)
• \\(\\{-1\\}\\)
• all real numbers

Explanation:

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<reasoning>

Isolate the absolute value expression

\[

$$\begin{aligned} |-2x| - 16 &= 2x - 20 \\ |-2x| &= 2x - 4 \end{aligned}$$

\]

Set up and solve the two cases

\[

$$\begin{aligned} \text{Case 1: } -2x &= 2x - 4 \\ -4x &= -4 \\ x &= 1 \\\\ \text{Case 2: } -(-2x) &= 2x - 4 \\ 2x &= 2x - 4 \\ 0 &= -4 \quad (\text{No solution}) \end{aligned}$$

\]

Check for extraneous solutions

\[

$$\begin{aligned} \text{Substitute } x = 1: \\ |-2(1)| - 16 &= 2(1) - 20 \\ |-2| - 16 &= 2 - 20 \\ 2 - 16 &= -18 \\ -14 & eq -18 \end{aligned}$$

\]
Since \(x = 1\) is extraneous, there are no solutions.
</reasoning>

<answer>
<mcq-correct>no solutions</mcq-correct>
<mcq-option>{1}</mcq-option>
<mcq-option>{-1}</mcq-option>
<mcq-option>all real numbers</mcq-option>
</answer>

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Answer:

<pre_analysis>
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</pre_analysis>

<reasoning>

Isolate the absolute value expression

\[

$$\begin{aligned} |-2x| - 16 &= 2x - 20 \\ |-2x| &= 2x - 4 \end{aligned}$$

\]

Set up and solve the two cases

\[

$$\begin{aligned} \text{Case 1: } -2x &= 2x - 4 \\ -4x &= -4 \\ x &= 1 \\\\ \text{Case 2: } -(-2x) &= 2x - 4 \\ 2x &= 2x - 4 \\ 0 &= -4 \quad (\text{No solution}) \end{aligned}$$

\]

Check for extraneous solutions

\[

$$\begin{aligned} \text{Substitute } x = 1: \\ |-2(1)| - 16 &= 2(1) - 20 \\ |-2| - 16 &= 2 - 20 \\ 2 - 16 &= -18 \\ -14 & eq -18 \end{aligned}$$

\]
Since \(x = 1\) is extraneous, there are no solutions.
</reasoning>

<answer>
<mcq-correct>no solutions</mcq-correct>
<mcq-option>{1}</mcq-option>
<mcq-option>{-1}</mcq-option>
<mcq-option>all real numbers</mcq-option>
</answer>

<post_analysis>
{
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Absolute Value Equations"
]
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