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], 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:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide both sides by 4 to solve for the absolute value expression:
[tex]\[
|x - 5| = 3
\][/tex]
4. The absolute value equation [tex]\( |x - 5| = 3 \)[/tex] gives two possible equations:
a) [tex]\(x - 5 = 3\)[/tex]
b) [tex]\(x - 5 = -3\)[/tex]
5. Solve each equation for [tex]\(x\)[/tex]:
- For [tex]\(x - 5 = 3\)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For [tex]\(x - 5 = -3\)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
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 correct answer 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:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide both sides by 4 to solve for the absolute value expression:
[tex]\[
|x - 5| = 3
\][/tex]
4. The absolute value equation [tex]\( |x - 5| = 3 \)[/tex] gives two possible equations:
a) [tex]\(x - 5 = 3\)[/tex]
b) [tex]\(x - 5 = -3\)[/tex]
5. Solve each equation for [tex]\(x\)[/tex]:
- For [tex]\(x - 5 = 3\)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For [tex]\(x - 5 = -3\)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
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 correct answer is [tex]\((x = 2, x = 8)\)[/tex].