10. solve $|-2x| - 16 = 2x - 20$.

* no solutions
* $\{1\}$
* $\{-1\}$
* all real numbers

Question

10. solve $|-2x| - 16 = 2x - 20$.

* no solutions
* $\{1\}$
* $\{-1\}$
* all real numbers

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Absolute Value Equations", "Solving Absolute Value Equations", "Extraneous Solutions" ], "new_concepts": [], "current_concepts": [ "Absolute Value Equations", "Solving Absolute Value Equations", "Extraneous Solutions" ] } ## 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.

no solutions

{1}

{-1}

all real numbers

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Absolute Value Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Isolate the absolute value expression

Set up and solve the two cases

Check for extraneous solutions

Answer: