Let's solve the given equation step by step: [tex]\( 4|x + 16| = 52 \)[/tex].
1. Isolate the absolute value expression:
Divide both sides of the equation by 4:
[tex]\[
|x + 16| = \frac{52}{4}
\][/tex]
Simplify the right side:
[tex]\[
|x + 16| = 13
\][/tex]
2. Solve the absolute value equation:
The equation [tex]\( |x + 16| = 13 \)[/tex] means that the expression [tex]\(x + 16\)[/tex] can be either 13 or -13. We need to consider both cases:
- Case 1: [tex]\( x + 16 = 13 \)[/tex]
- Case 2: [tex]\( x + 16 = -13 \)[/tex]
3. Solve for [tex]\( x \)[/tex] in both cases:
- Case 1:
[tex]\[
x + 16 = 13
\][/tex]
Subtract 16 from both sides:
[tex]\[
x = 13 - 16
\][/tex]
Simplify:
[tex]\[
x = -3
\][/tex]
- Case 2:
[tex]\[
x + 16 = -13
\][/tex]
Subtract 16 from both sides:
[tex]\[
x = -13 - 16
\][/tex]
Simplify:
[tex]\[
x = -29
\][/tex]
4. Solutions:
The solutions to the equation [tex]\( 4|x + 16| = 52 \)[/tex] are:
[tex]\[
x = -3 \quad \text{and} \quad x = -29
\][/tex]
Therefore, the final solutions are [tex]\( x = -3 \)[/tex] and [tex]\( x = -29 \)[/tex].
