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 given function [tex]\( f(x) = 4|x-5| + 3 \)[/tex], we can follow these steps:

1. Set the function equal to 15:
Start by setting the equation [tex]\( f(x) = 15 \)[/tex]:
[tex]\[
4|x-5| + 3 = 15
\][/tex]

2. Isolate the absolute value expression:
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 solve for the absolute value:
[tex]\[
|x-5| = 3
\][/tex]

4. Consider the two cases for the absolute value:
Remember, if [tex]\( |A| = B \)[/tex], then [tex]\( A = B \)[/tex] or [tex]\( A = -B \)[/tex]. Here we apply this to our expression:
- Case 1: [tex]\( x-5 = 3 \)[/tex]
- Case 2: [tex]\( x-5 = -3 \)[/tex]

5. Solve each case for [tex]\( x \)[/tex]:

- For Case 1, solve [tex]\( x-5 = 3 \)[/tex]:
[tex]\[
x = 3 + 5
\][/tex]
[tex]\[
x = 8
\][/tex]

- For Case 2, solve [tex]\( x-5 = -3 \)[/tex]:
[tex]\[
x = -3 + 5
\][/tex]
[tex]\[
x = 2
\][/tex]

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