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 need to solve the equation [tex]\( 4|x-5| + 3 = 15 \)[/tex].

Here's a step-by-step solution:

1. Start with the given equation:
[tex]\[
4|x-5| + 3 = 15
\][/tex]

2. Isolate the absolute value term:
[tex]\[
4|x-5| + 3 - 3 = 15 - 3
\][/tex]
[tex]\[
4|x-5| = 12
\][/tex]

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

4. Remove the absolute value by considering both cases:
[tex]\( |x-5| = 3 \)[/tex] means [tex]\( x-5 = 3 \)[/tex] or [tex]\( x-5 = -3 \)[/tex].

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

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

Thus, the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].

Conclusion:

The correct values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are:
[tex]\[
x = 2 \, \text{and} \, x = 8
\][/tex]

So, the answer is:
[tex]\[
x = 2, \, x = 8
\][/tex]

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