Answer :
To solve the problem, we need to find the values of [tex]\( x \)[/tex] such that the function [tex]\( f(x)=4|x-5|+3 \)[/tex] equals 15. Here's how you can do it step by step:
1. Set the function equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value:
Subtract 3 from both sides:
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value expression:
Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Set up two separate equations to solve for [tex]\( x \)[/tex]:
When dealing with absolute values, remember that [tex]\( |x-5|=3 \)[/tex] leads to two possibilities:
- [tex]\( x - 5 = 3 \)[/tex]
- [tex]\( x - 5 = -3 \)[/tex]
5. Solve each equation:
- For [tex]\( x - 5 = 3 \)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For [tex]\( x - 5 = -3 \)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
6. Conclusion:
The values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the answer is [tex]\( x=2, x=8 \)[/tex].
1. Set the function equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value:
Subtract 3 from both sides:
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value expression:
Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Set up two separate equations to solve for [tex]\( x \)[/tex]:
When dealing with absolute values, remember that [tex]\( |x-5|=3 \)[/tex] leads to two possibilities:
- [tex]\( x - 5 = 3 \)[/tex]
- [tex]\( x - 5 = -3 \)[/tex]
5. Solve each equation:
- For [tex]\( x - 5 = 3 \)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For [tex]\( x - 5 = -3 \)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
6. Conclusion:
The values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the answer is [tex]\( x=2, x=8 \)[/tex].
