To solve the equation [tex]\( |x - 6| = 13 \)[/tex], we need to consider the definition of the absolute value function. The absolute value of a number [tex]\( a \)[/tex] is defined as:
[tex]\[ |a| =
\begin{cases}
a & \text{if } a \geq 0 \\
-a & \text{if } a < 0
\end{cases}
\][/tex]
Given the equation [tex]\( |x - 6| = 13 \)[/tex], this tells us two possible scenarios:
1. [tex]\( x - 6 = 13 \)[/tex]
2. [tex]\( x - 6 = -13 \)[/tex]
Let's solve these two cases one by one:
### Case 1: [tex]\( x - 6 = 13 \)[/tex]
[tex]\[ x - 6 = 13 \][/tex]
To isolate [tex]\( x \)[/tex], add 6 to both sides:
[tex]\[ x = 13 + 6 \][/tex]
[tex]\[ x = 19 \][/tex]
### Case 2: [tex]\( x - 6 = -13 \)[/tex]
[tex]\[ x - 6 = -13 \][/tex]
To isolate [tex]\( x \)[/tex], add 6 to both sides:
[tex]\[ x = -13 + 6 \][/tex]
[tex]\[ x = -7 \][/tex]
Therefore, the solution to the equation [tex]\( |x - 6| = 13 \)[/tex] is:
[tex]\[ x = 19 \quad \text{or} \quad x = -7 \][/tex]
So, the final answer is:
[tex]\[
\{19, -7\}
\][/tex]
