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Answer :
To solve for the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 15 \)[/tex] given the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex], follow these steps:
1. Set the function equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to isolate the absolute value term:
[tex]\[
4|x-5| = 15 - 3 = 12
\][/tex]
3. Divide both sides by 4 to further solve for the absolute value:
[tex]\[
|x-5| = \frac{12}{4} = 3
\][/tex]
4. Set up the two equations for the absolute value expression:
- [tex]\( x - 5 = 3 \)[/tex]
- [tex]\( x - 5 = -3 \)[/tex]
5. 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]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex]. So, the correct option is [tex]\( x = 2, x = 8 \)[/tex].
1. Set the function equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to isolate the absolute value term:
[tex]\[
4|x-5| = 15 - 3 = 12
\][/tex]
3. Divide both sides by 4 to further solve for the absolute value:
[tex]\[
|x-5| = \frac{12}{4} = 3
\][/tex]
4. Set up the two equations for the absolute value expression:
- [tex]\( x - 5 = 3 \)[/tex]
- [tex]\( x - 5 = -3 \)[/tex]
5. 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]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex]. So, the correct option is [tex]\( x = 2, x = 8 \)[/tex].
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