Answer :
Sure! Let's solve the equation [tex]\( f(x) = 15 \)[/tex] for the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex].
1. Start with the given equation:
[tex]\[
f(x) = 15
\][/tex]
Substitute [tex]\( f(x) = 4|x-5| + 3 \)[/tex]:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to isolate the absolute value expression:
[tex]\[
4|x-5| = 12
\][/tex]
3. Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. The absolute value equation [tex]\( |x-5| = 3 \)[/tex] means that [tex]\( x - 5 \)[/tex] can be either 3 or -3:
[tex]\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\][/tex]
5. Solve each equation separately:
- For [tex]\( x - 5 = 3 \)[/tex]:
[tex]\[
x = 3 + 5
\][/tex]
[tex]\[
x = 8
\][/tex]
- For [tex]\( x - 5 = -3 \)[/tex]:
[tex]\[
x = -3 + 5
\][/tex]
[tex]\[
x = 2
\][/tex]
6. Therefore, the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
The correct answer is:
[tex]\[
\boxed{x = 2, x = 8}
\][/tex]
1. Start with the given equation:
[tex]\[
f(x) = 15
\][/tex]
Substitute [tex]\( f(x) = 4|x-5| + 3 \)[/tex]:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to isolate the absolute value expression:
[tex]\[
4|x-5| = 12
\][/tex]
3. Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. The absolute value equation [tex]\( |x-5| = 3 \)[/tex] means that [tex]\( x - 5 \)[/tex] can be either 3 or -3:
[tex]\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\][/tex]
5. Solve each equation separately:
- For [tex]\( x - 5 = 3 \)[/tex]:
[tex]\[
x = 3 + 5
\][/tex]
[tex]\[
x = 8
\][/tex]
- For [tex]\( x - 5 = -3 \)[/tex]:
[tex]\[
x = -3 + 5
\][/tex]
[tex]\[
x = 2
\][/tex]
6. Therefore, the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
The correct answer is:
[tex]\[
\boxed{x = 2, x = 8}
\][/tex]
