9. solve $|t + 15| = 2t + 25$.

no solutions

$\{-\frac{40}{3}\}$

$\{-\frac{40}{3}, -10\}$

$\{-10\}$

Question

9. solve $|t + 15| = 2t + 25$.

no solutions

$\{-\frac{40}{3}\}$

$\{-\frac{40}{3}, -10\}$

$\{-10\}$

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" ], "new_concepts": [], "current_concepts": [ "Absolute Value Equations", "Solving Absolute Value Equations" ] } ## Set up the two cases for the absolute value equation $$ |t + 15| = 2t + 25 $$ $$ \begin{aligned} \text{Case 1: } & t + 15 = 2t + 25 \ \text{Case 2: } & t + 15 = -(2t + 25) \end{aligned} $$ ## Solve both cases for t $$ \begin{aligned} \text{Case 1: } & t - 2t = 25 - 15 \implies -t = 10 \implies t = -10 \ \text{Case 2: } & t + 15 = -2t - 25 \implies 3t = -40 \implies t = -\frac{40}{3} \end{aligned} $$ ## Check for extraneous solutions $$ \begin{aligned} \text{For } t = -10: & \quad |-10 + 15| = 2(-10) + 25 \implies |5| = 5 \quad (\text{True}) \ \text{For } t = -\frac{40}{3}: & \quad |-\frac{40}{3} + 15| = 2(-\frac{40}{3}) + 25 \implies |\frac{5}{3}| = -\frac{80}{3} + \frac{75}{3} = -\frac{5}{3} \quad (\text{False}) \end{aligned} $$

no solutions

$\{-\frac{40}{3}\}$

$\{-\frac{40}{3}, -10\}$

$\{-10\}$

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

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QUESTION IMAGE

Question

Explanation:

Set up the two cases for the absolute value equation

Solve both cases for t

Check for extraneous solutions

Answer: