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 equation [tex]\( f(x) = 4|x - 5| + 3 = 15 \)[/tex], we want to find the values of [tex]\( x \)[/tex] that make [tex]\( f(x) = 15 \)[/tex]. Let's break it down step by step:

1. Set the equation equal to 15:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]

2. Isolate the absolute value:
Start by subtracting 3 from both sides of the equation:
[tex]\[
4|x - 5| = 12
\][/tex]

3. Divide by 4 to isolate the absolute value term:
[tex]\[
|x - 5| = 3
\][/tex]

4. Solve the absolute value equation:
The expression [tex]\(|x - 5| = 3\)[/tex] means that the distance between [tex]\(x\)[/tex] and 5 is 3. This gives us two possible equations:
- [tex]\(x - 5 = 3\)[/tex]
- [tex]\(x - 5 = -3\)[/tex]

5. Solve each equation separately:

- For [tex]\(x - 5 = 3\)[/tex]:
Add 5 to both sides:
[tex]\[
x = 3 + 5 = 8
\][/tex]

- For [tex]\(x - 5 = -3\)[/tex]:
Add 5 to both sides:
[tex]\[
x = -3 + 5 = 2
\][/tex]

6. Final solutions:
The values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(f(x) = 15\)[/tex] are [tex]\(x = 2\)[/tex] and [tex]\(x = 8\)[/tex].

Therefore, the correct values from the options given are [tex]\(x = 2, x = 8\)[/tex].

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