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 = 15, x = 8[/tex]
C. [tex]x = 2, x = 7.5[/tex]
D. [tex]x = 0.5, x = 7.5[/tex]

Answer :

We start with the equation:

[tex]$$
4|x-5| + 3 = 15.
$$[/tex]

Step 1: Isolate the absolute value

Subtract 3 from both sides:

[tex]$$
4|x-5| = 15 - 3 = 12.
$$[/tex]

Step 2: Divide by 4

Divide both sides by 4 to solve for the absolute value term:

[tex]$$
|x-5| = \frac{12}{4} = 3.
$$[/tex]

Step 3: Solve the absolute value equation

The equation [tex]$|x-5| = 3$[/tex] can be split into two cases:

1. [tex]$x - 5 = 3$[/tex], which gives:
[tex]$$
x = 3 + 5 = 8.
$$[/tex]

2. [tex]$x - 5 = -3$[/tex], which gives:
[tex]$$
x = -3 + 5 = 2.
$$[/tex]

Thus, the solutions are [tex]$x=2$[/tex] and [tex]$x=8$[/tex].

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