Answer :
Let's solve the problem step by step to find the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 4|x - 5| + 3 \)[/tex] equals 15.
1. Set the equation:
We start by setting [tex]\( f(x) \)[/tex] equal to 15:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the absolute value term:
Subtract 3 from both sides:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Solve for the absolute value:
Divide both sides by 4:
[tex]\[
|x - 5| = 3
\][/tex]
4. Create two equations:
The equation [tex]\( |x - 5| = 3 \)[/tex] leads to two separate equations:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
5. Solve each equation:
- For [tex]\( x - 5 = 3 \)[/tex]:
Add 5 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For [tex]\( x - 5 = -3 \)[/tex]:
Add 5 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
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 equation:
We start by setting [tex]\( f(x) \)[/tex] equal to 15:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the absolute value term:
Subtract 3 from both sides:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Solve for the absolute value:
Divide both sides by 4:
[tex]\[
|x - 5| = 3
\][/tex]
4. Create two equations:
The equation [tex]\( |x - 5| = 3 \)[/tex] leads to two separate equations:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
5. Solve each equation:
- For [tex]\( x - 5 = 3 \)[/tex]:
Add 5 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For [tex]\( x - 5 = -3 \)[/tex]:
Add 5 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
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].
