Let's solve the given equation [tex]\( f(x) = 7 \)[/tex] for the function [tex]\( f(x) = 0.5|x-4|-3 \)[/tex].
1. Start with the equation:
[tex]\[
f(x) = 7
\][/tex]
Substitute [tex]\( f(x) \)[/tex] with [tex]\( 0.5|x-4|-3 \)[/tex]:
[tex]\[
0.5|x-4| - 3 = 7
\][/tex]
2. Add 3 to both sides of the equation to isolate the absolute value term:
[tex]\[
0.5|x-4| = 10
\][/tex]
3. Multiply both sides by 2 to further isolate the absolute value term:
[tex]\[
|x-4| = 20
\][/tex]
4. The equation [tex]\( |x-4| = 20 \)[/tex] gives us two possible cases:
[tex]\[
x - 4 = 20 \quad \text{or} \quad x - 4 = -20
\][/tex]
5. Solve each case separately:
- Case 1: [tex]\( x - 4 = 20 \)[/tex]
[tex]\[
x = 20 + 4
\][/tex]
[tex]\[
x = 24
\][/tex]
- Case 2: [tex]\( x - 4 = -20 \)[/tex]
[tex]\[
x = -20 + 4
\][/tex]
[tex]\[
x = -16
\][/tex]
Thus, the solutions for the equation [tex]\( f(x) = 7 \)[/tex] are [tex]\( x = 24 \)[/tex] and [tex]\( x = -16 \)[/tex].
The correct answer is:
[tex]\[
x = -16, x = 24
\][/tex]
