College

Given the function [tex]f(x)=4|x-5|+3[/tex], for what values of [tex]x[/tex] is [tex]f(x)=15[/tex]?

A. [tex]x=2, x=8[/tex]
B. [tex]x=1.5, x=8[/tex]
C. [tex]x=2, x=7.5[/tex]
D. [tex]x=0.5, x=7.5[/tex]

Answer :

To solve the problem of finding the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] for the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex], follow these steps:

1. Set the Equation:
We start with the equation given by the problem when [tex]\( f(x) = 15 \)[/tex]:
[tex]\[
4|x-5| + 3 = 15
\][/tex]

2. Isolate the Absolute Value:
Subtract 3 from both sides to isolate the absolute value term:
[tex]\[
4|x-5| = 12
\][/tex]

3. Solve for the Absolute Value:
Divide both sides by 4 to further isolate the absolute value:
[tex]\[
|x-5| = 3
\][/tex]

4. Remove the Absolute Value:
The equation [tex]\( |x-5| = 3 \)[/tex] implies there are two possible cases to consider:

- Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]

5. Solve for [tex]\( x \)[/tex] in Each Case:

- Case 1:
[tex]\[
x - 5 = 3
\][/tex]
Add 5 to both sides:
[tex]\[
x = 8
\][/tex]

- Case 2:
[tex]\[
x - 5 = -3
\][/tex]
Add 5 to both sides:
[tex]\[
x = 2
\][/tex]

6. Conclusion:
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex].

The correct option is:
- [tex]\( x = 2, x = 8 \)[/tex]

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