Answer :
To solve the problem, we need to find the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex] equals 15. Let's walk through the solution step by step:
1. Set the function equal to 15:
We have the equation:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression:
Subtract 3 from both sides to get:
[tex]\[
4|x-5| = 12
\][/tex]
3. Simplify the equation:
Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Solve the absolute value equation:
The absolute value equation [tex]\( |x-5| = 3 \)[/tex] can be broken into two separate cases because the absolute value of a number represents its distance from zero on the number line.
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
5. Conclusion:
The values of [tex]\( x \)[/tex] for which [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:
We have the equation:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression:
Subtract 3 from both sides to get:
[tex]\[
4|x-5| = 12
\][/tex]
3. Simplify the equation:
Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Solve the absolute value equation:
The absolute value equation [tex]\( |x-5| = 3 \)[/tex] can be broken into two separate cases because the absolute value of a number represents its distance from zero on the number line.
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
5. Conclusion:
The values of [tex]\( x \)[/tex] for which [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].
