To solve the equation [tex]\( |x - 4| = 17 \)[/tex], we need to consider the definition of the absolute value function. The absolute value [tex]\( |a| \)[/tex] is defined as:
[tex]\[ |a| = a \quad \text{if} \quad a \geq 0 \][/tex]
[tex]\[ |a| = -a \quad \text{if} \quad a < 0 \][/tex]
Given [tex]\( |x - 4| = 17 \)[/tex], this means [tex]\( x - 4 \)[/tex] can be either [tex]\( 17 \)[/tex] or [tex]\( -17 \)[/tex].
We will now solve for [tex]\( x \)[/tex] in both cases:
1. Case 1: [tex]\( x - 4 = 17 \)[/tex]
[tex]\[
x - 4 = 17
\][/tex]
To isolate [tex]\( x \)[/tex], add 4 to both sides of the equation:
[tex]\[
x = 17 + 4
\][/tex]
[tex]\[
x = 21
\][/tex]
2. Case 2: [tex]\( x - 4 = -17 \)[/tex]
[tex]\[
x - 4 = -17
\][/tex]
To isolate [tex]\( x \)[/tex], add 4 to both sides of the equation:
[tex]\[
x = -17 + 4
\][/tex]
[tex]\[
x = -13
\][/tex]
So, the solutions to the equation [tex]\( |x - 4| = 17 \)[/tex] are:
[tex]\[
x = 21 \quad \text{or} \quad x = -13
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{B. \ x = -13 \ \text{or} \ x = 21} \][/tex]
