Question

1. solve \\(|t + 15| = 2t + 25\\).

no solutions

\\(\\{-\frac{40}{3}\\}\\)

\\(\\{-\frac{40}{3}, -10\\}\\)

\\(\\{-10\\}\\)

Explanation:

<pre_analysis>
{
"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"
]
}
</pre_analysis>

<reasoning>

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

\[

$$\begin{aligned} \text{For } t = -10: & \quad |-10 + 15| = 2(-10) + 25 \implies |5| = 5 \quad (\text{True}) \\ \text{For } t = -\frac{40}{3}: & \quad |-\frac{40}{3} + 15| = 2(-\frac{40}{3}) + 25 \implies |\frac{5}{3}| = -\frac{80}{3} + \frac{75}{3} = -\frac{5}{3} \quad (\text{False}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>no solutions</mcq-option>
<mcq-option>\(\{-\frac{40}{3}\}\)</mcq-option>
<mcq-option>\(\{-\frac{40}{3}, -10\}\)</mcq-option>
<mcq-correct>\(\{-10\}\)</mcq-correct>
</answer>

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

Answer:

<pre_analysis>
{
"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"
]
}
</pre_analysis>

<reasoning>

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

\[

$$\begin{aligned} \text{For } t = -10: & \quad |-10 + 15| = 2(-10) + 25 \implies |5| = 5 \quad (\text{True}) \\ \text{For } t = -\frac{40}{3}: & \quad |-\frac{40}{3} + 15| = 2(-\frac{40}{3}) + 25 \implies |\frac{5}{3}| = -\frac{80}{3} + \frac{75}{3} = -\frac{5}{3} \quad (\text{False}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>no solutions</mcq-option>
<mcq-option>\(\{-\frac{40}{3}\}\)</mcq-option>
<mcq-option>\(\{-\frac{40}{3}, -10\}\)</mcq-option>
<mcq-correct>\(\{-10\}\)</mcq-correct>
</answer>

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