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 :

Let's solve the problem step-by-step:

We are given the function [tex]\( f(x) = 4|x - 5| + 3 \)[/tex] and need to find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex].

1. Set the Equation:

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 term with the absolute value:
[tex]\[
4|x - 5| = 15 - 3
\][/tex]
[tex]\[
4|x - 5| = 12
\][/tex]

3. Solve for the Absolute Value:

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

4. Solve the Absolute Value Equation:

The equation [tex]\( |x - 5| = 3 \)[/tex] implies two scenarios:

- [tex]\( x - 5 = 3 \)[/tex]
- [tex]\( x - 5 = -3 \)[/tex]

Solve each equation for [tex]\( x \)[/tex]:

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

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

5. Conclusion:

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

Therefore, the answer is [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex].

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