Answer :
To solve the problem of finding the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex] is equal to 15, follow these steps:
1. Set the function equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides:
[tex]\[
4|x-5| = 12
\][/tex]
3. Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Solve the absolute value equation:
The equation [tex]\( |x-5| = 3 \)[/tex] tells us that the expression inside the absolute value can be either 3 or -3. So, we set up two separate equations to solve for [tex]\( x \)[/tex]:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
[tex]\[
x - 5 = 3
\][/tex]
Adding 5 to both sides:
[tex]\[
x = 8
\][/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
[tex]\[
x - 5 = -3
\][/tex]
Adding 5 to both sides:
[tex]\[
x = 2
\][/tex]
5. Conclusion:
The values of [tex]\( x \)[/tex] that make [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 the function equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides:
[tex]\[
4|x-5| = 12
\][/tex]
3. Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Solve the absolute value equation:
The equation [tex]\( |x-5| = 3 \)[/tex] tells us that the expression inside the absolute value can be either 3 or -3. So, we set up two separate equations to solve for [tex]\( x \)[/tex]:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
[tex]\[
x - 5 = 3
\][/tex]
Adding 5 to both sides:
[tex]\[
x = 8
\][/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
[tex]\[
x - 5 = -3
\][/tex]
Adding 5 to both sides:
[tex]\[
x = 2
\][/tex]
5. Conclusion:
The values of [tex]\( x \)[/tex] that make [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].
