To solve for the values [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 7 \)[/tex], where the function is given by [tex]\( f(x) = 0.5|x - 4| - 3 \)[/tex], we'll follow these steps:
1. Set up the equation:
[tex]\[
0.5 |x - 4| - 3 = 7
\][/tex]
2. Add 3 to both sides to isolate the absolute value expression:
[tex]\[
0.5 |x - 4| = 10
\][/tex]
3. Multiply both sides by 2 to further isolate the absolute value expression:
[tex]\[
|x - 4| = 20
\][/tex]
4. The definition of the absolute value tells us that:
[tex]\[
x - 4 = 20 \quad \text{or} \quad x - 4 = -20
\][/tex]
5. Solve each of these linear equations separately:
[tex]\[
x - 4 = 20 \quad \Rightarrow \quad x = 24
\][/tex]
[tex]\[
x - 4 = -20 \quad \Rightarrow \quad x = -16
\][/tex]
Thus, the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 7 \)[/tex] are [tex]\( x = 24 \)[/tex] and [tex]\( x = -16 \)[/tex]. Therefore, the correct answer is:
[tex]\[
\boxed{x = -16, x = 24}
\][/tex]
