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 the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex] equals 15, we can break it down into a series of steps:

1. Set up the equation:
We are given the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex] and need to find when [tex]\( f(x) = 15 \)[/tex].
So, set the equation:
[tex]\[
4|x-5| + 3 = 15
\][/tex]

2. Isolate the absolute value term:
First, subtract 3 from both sides to isolate the term with the absolute value:
[tex]\[
4|x-5| = 15 - 3
\][/tex]
[tex]\[
4|x-5| = 12
\][/tex]

3. Solve for the absolute value:
Divide both sides by 4 to solve for the absolute value:
[tex]\[
|x-5| = \frac{12}{4}
\][/tex]
[tex]\[
|x-5| = 3
\][/tex]

4. Consider the two cases for the absolute value equation:
The equation [tex]\( |x-5| = 3 \)[/tex] gives us two possible scenarios:
- 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: [tex]\( x - 5 = 3 \)[/tex]
[tex]\[
x = 5 + 3
\][/tex]
[tex]\[
x = 8
\][/tex]

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

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

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