Question

solve the following absolute value equation.

\\4|x - 8| = 36\\

\\x = ?, \quad \\

enter the smallest solution first.

Explanation:

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<reasoning>

Isolate the absolute value expression

Using the One-Step Equations knowledge point
\[

$$\begin{aligned} 4|x - 8| &= 36 \\ |x - 8| &= \frac{36}{4} \\ |x - 8| &= 9 \end{aligned}$$

\]

Set up the two case equations

To solve an absolute value equation of the form \(|u| = c\) (where \(c \ge 0\)), we split it into two separate linear equations:
\[
x - 8 = 9 \quad \text{or} \quad x - 8 = -9
\]

Solve the first case equation

Using the Linear Equations knowledge point
\[

$$\begin{aligned} x - 8 &= 9 \\ x &= 9 + 8 \\ x &= 17 \end{aligned}$$

\]

Solve the second case equation

Using the Linear Equations knowledge point
\[

$$\begin{aligned} x - 8 &= -9 \\ x &= -9 + 8 \\ x &= -1 \end{aligned}$$

\]

Order the solutions

Comparing the two solutions:
\[
-1 < 17
\]
The smallest solution is \(-1\) and the largest solution is \(17\).
</reasoning>

<answer>
Solve the following absolute value equation.
\[4|x - 8| = 36\]
\(x =\) <blank>-1</blank>, <blank>17</blank>

Enter the smallest solution first.
</answer>

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"Mathematics",
"Algebra",
"Absolute Value Equations"
]
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Isolate the absolute value expression

Using the One-Step Equations knowledge point
\[

$$\begin{aligned} 4|x - 8| &= 36 \\ |x - 8| &= \frac{36}{4} \\ |x - 8| &= 9 \end{aligned}$$

\]

Set up the two case equations

To solve an absolute value equation of the form \(|u| = c\) (where \(c \ge 0\)), we split it into two separate linear equations:
\[
x - 8 = 9 \quad \text{or} \quad x - 8 = -9
\]

Solve the first case equation

Using the Linear Equations knowledge point
\[

$$\begin{aligned} x - 8 &= 9 \\ x &= 9 + 8 \\ x &= 17 \end{aligned}$$

\]

Solve the second case equation

Using the Linear Equations knowledge point
\[

$$\begin{aligned} x - 8 &= -9 \\ x &= -9 + 8 \\ x &= -1 \end{aligned}$$

\]

Order the solutions

Comparing the two solutions:
\[
-1 < 17
\]
The smallest solution is \(-1\) and the largest solution is \(17\).
</reasoning>

<answer>
Solve the following absolute value equation.
\[4|x - 8| = 36\]
\(x =\) <blank>-1</blank>, <blank>17</blank>

Enter the smallest solution first.
</answer>

<post_analysis>
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"knowledge_point": [
"Mathematics",
"Algebra",
"Absolute Value Equations"
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