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 = 7.5[/tex]
B. [tex]x = 0.5, x = 7.5[/tex]
C. [tex]x = 2, x = 8[/tex]
D. [tex]x = 1.5, x = 8[/tex]

Answer :

To solve the problem of finding the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] when [tex]\( f(x) = 4|x-5| + 3 \)[/tex], let's go through the steps:

1. Set the Function Equal to 15:

Start by setting the function equal to 15:

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

2. Isolate the Absolute Value:

Subtract 3 from both sides to isolate the absolute value expression:

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

3. Divide to Solve for the Absolute Value:

Divide both sides by 4:

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

4. Create the Two Possible Equations:

Because the absolute value of a number [tex]\( |a| = b \)[/tex] implies two possible scenarios—[tex]\( a = b \)[/tex] or [tex]\( a = -b \)[/tex]—we can set up two separate equations:

- Equation 1: [tex]\( x - 5 = 3 \)[/tex]
- Equation 2: [tex]\( x - 5 = -3 \)[/tex]

5. Solve Each Equation:

For the first equation [tex]\( x - 5 = 3 \)[/tex], add 5 to both sides:

[tex]\[
x = 3 + 5 = 8
\][/tex]

For the second equation [tex]\( x - 5 = -3 \)[/tex], add 5 to both sides:

[tex]\[
x = -3 + 5 = 2
\][/tex]

6. Conclusion:

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

Thus, the solution corresponds to the option: [tex]\( x = 2, x = 8 \)[/tex].

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