Answer :
Let's solve the problem step-by-step to find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] given the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex].
1. Set the function equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression:
[tex]\[
4|x-5| = 15 - 3
\][/tex]
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value:
[tex]\[
|x-5| = \frac{12}{4}
\][/tex]
[tex]\[
|x-5| = 3
\][/tex]
4. Remove the absolute value by setting up two equations:
- Case 1: [tex]\( x-5 = 3 \)[/tex]
- Case 2: [tex]\( x-5 = -3 \)[/tex]
5. Solve each case:
- Case 1:
[tex]\[
x - 5 = 3
\][/tex]
[tex]\[
x = 3 + 5
\][/tex]
[tex]\[
x = 8
\][/tex]
- Case 2:
[tex]\[
x - 5 = -3
\][/tex]
[tex]\[
x = -3 + 5
\][/tex]
[tex]\[
x = 2
\][/tex]
6. Conclusion:
The solutions are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the answer is [tex]\( x = 2, x = 8 \)[/tex].
1. Set the function equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression:
[tex]\[
4|x-5| = 15 - 3
\][/tex]
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value:
[tex]\[
|x-5| = \frac{12}{4}
\][/tex]
[tex]\[
|x-5| = 3
\][/tex]
4. Remove the absolute value by setting up two equations:
- Case 1: [tex]\( x-5 = 3 \)[/tex]
- Case 2: [tex]\( x-5 = -3 \)[/tex]
5. Solve each case:
- Case 1:
[tex]\[
x - 5 = 3
\][/tex]
[tex]\[
x = 3 + 5
\][/tex]
[tex]\[
x = 8
\][/tex]
- Case 2:
[tex]\[
x - 5 = -3
\][/tex]
[tex]\[
x = -3 + 5
\][/tex]
[tex]\[
x = 2
\][/tex]
6. Conclusion:
The solutions are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the answer is [tex]\( x = 2, x = 8 \)[/tex].
