Let's find the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 15 \)[/tex] for the given function [tex]\( f(x) = 4|x - 5| + 3 \)[/tex].
1. Set up the equation:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the absolute value term:
Subtract 3 from both sides to isolate the term with the absolute value:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Solve for the absolute value:
Divide both sides by 4:
[tex]\[
|x - 5| = 3
\][/tex]
4. Rewrite the absolute value equation as two separate equations:
The absolute value equation [tex]\( |x - 5| = 3 \)[/tex] can be rewritten as two separate linear equations:
[tex]\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\][/tex]
5. Solve each equation for [tex]\( x \)[/tex]:
- For [tex]\( x - 5 = 3 \)[/tex]:
[tex]\[
x = 3 + 5
\][/tex]
[tex]\[
x = 8
\][/tex]
- For [tex]\( x - 5 = -3 \)[/tex]:
[tex]\[
x = -3 + 5
\][/tex]
[tex]\[
x = 2
\][/tex]
So, the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the correct answer is:
[tex]\[ x = 2, x = 8 \][/tex]
