Answer :
To solve the problem of finding the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex] equals 15, we can follow these steps:
1. Set up the equation:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value term:
Start by subtracting 3 from both sides of the equation:
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value:
Divide both sides by 4 to solve for the absolute value:
[tex]\[
|x-5| = 3
\][/tex]
4. Break down the absolute value equation:
The equation [tex]\( |x-5| = 3 \)[/tex] consists of two possible cases:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
5. Solve each case:
- For Case 1: [tex]\( x - 5 = 3 \)[/tex]
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For Case 2: [tex]\( x - 5 = -3 \)[/tex]
[tex]\[
x = -3 + 5 = 2
\][/tex]
6. Conclusion:
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 up the equation:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value term:
Start by subtracting 3 from both sides of the equation:
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value:
Divide both sides by 4 to solve for the absolute value:
[tex]\[
|x-5| = 3
\][/tex]
4. Break down the absolute value equation:
The equation [tex]\( |x-5| = 3 \)[/tex] consists of two possible cases:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
5. Solve each case:
- For Case 1: [tex]\( x - 5 = 3 \)[/tex]
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For Case 2: [tex]\( x - 5 = -3 \)[/tex]
[tex]\[
x = -3 + 5 = 2
\][/tex]
6. Conclusion:
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].
