Answer :
To solve for the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 15 \)[/tex] for the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex], follow these steps:
1. Set up the equation:
We want to find the values of [tex]\( x \)[/tex] such that:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression:
Subtract 3 from both sides:
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value expression:
Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Solve for [tex]\( x \)[/tex] using the definition of absolute value:
The equation [tex]\( |x-5| = 3 \)[/tex] can be split into two separate equations:
- Case 1: [tex]\( x-5 = 3 \)[/tex]
Solving this gives:
[tex]\[ x = 3 + 5 = 8 \][/tex]
- Case 2: [tex]\( x-5 = -3 \)[/tex]
Solving this gives:
[tex]\[ x = -3 + 5 = 2 \][/tex]
5. Check the solutions in the context of the problem:
Both [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex] are solutions that satisfy [tex]\( f(x) = 15 \)[/tex].
Thus, the values of [tex]\( x \)[/tex] for which [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]
1. Set up the equation:
We want to find the values of [tex]\( x \)[/tex] such that:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression:
Subtract 3 from both sides:
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value expression:
Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Solve for [tex]\( x \)[/tex] using the definition of absolute value:
The equation [tex]\( |x-5| = 3 \)[/tex] can be split into two separate equations:
- Case 1: [tex]\( x-5 = 3 \)[/tex]
Solving this gives:
[tex]\[ x = 3 + 5 = 8 \][/tex]
- Case 2: [tex]\( x-5 = -3 \)[/tex]
Solving this gives:
[tex]\[ x = -3 + 5 = 2 \][/tex]
5. Check the solutions in the context of the problem:
Both [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex] are solutions that satisfy [tex]\( f(x) = 15 \)[/tex].
Thus, the values of [tex]\( x \)[/tex] for which [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]
