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 find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] given the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex], we'll solve the equation step by step.

### Step 1: Set up the equation

We start by setting [tex]\( f(x) = 15 \)[/tex]:

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

### Step 2: Isolate the absolute value

Subtract 3 from both sides to isolate the term with the absolute value:

[tex]\[ 4|x - 5| = 12 \][/tex]

### Step 3: Solve for the absolute value expression

Divide both sides by 4:

[tex]\[ |x - 5| = 3 \][/tex]

The equation [tex]\( |x - 5| = 3 \)[/tex] means that the expression inside the absolute value, [tex]\( x - 5 \)[/tex], can be either 3 or -3.

### Step 4: Solve the two equations

1. Equation 1: [tex]\( x - 5 = 3 \)[/tex]

Add 5 to both sides:

[tex]\[ x = 8 \][/tex]

2. Equation 2: [tex]\( x - 5 = -3 \)[/tex]

Add 5 to both sides:

[tex]\[ x = 2 \][/tex]

### Step 5: Verify the solutions

To double-check, substitute both solutions back into the original function to ensure they satisfy [tex]\( f(x) = 15 \)[/tex].

- For [tex]\( x = 8 \)[/tex]:
[tex]\[
f(8) = 4|8 - 5| + 3 = 4 \times 3 + 3 = 12 + 3 = 15
\][/tex]

- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 4|2 - 5| + 3 = 4 \times 3 + 3 = 12 + 3 = 15
\][/tex]

Both solutions satisfy the original equation.

### Conclusion

The values of [tex]\( x \)[/tex] that make [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].

The correct answer is [tex]\( x=2, x=8 \)[/tex].