Answer :
To solve the problem where the function [tex]\( f(x) = 4|x - 5| + 3 \)[/tex] equals 15, follow these steps:
1. Set up the equation:
Start with the equation [tex]\( f(x) = 15 \)[/tex].
So, [tex]\( 4|x - 5| + 3 = 15 \)[/tex].
2. Isolate the absolute value:
Subtract 3 from both sides to isolate the term with the absolute value:
[tex]\[ 4|x - 5| = 15 - 3 \][/tex]
[tex]\[ 4|x - 5| = 12 \][/tex]
3. Solve for the absolute value:
Divide both sides by 4 to further isolate the absolute value:
[tex]\[ |x - 5| = \frac{12}{4} \][/tex]
[tex]\[ |x - 5| = 3 \][/tex]
4. Consider both scenarios for the absolute value equation:
Since [tex]\( |x - 5| = 3 \)[/tex], this equation can be split into two separate linear equations:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 3 + 5 \][/tex]
[tex]\[ x = 8 \][/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = -3 + 5 \][/tex]
[tex]\[ x = 2 \][/tex]
5. Conclusion:
The values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Hence, the correct answer from the choices provided is [tex]\( x = 2, x = 8 \)[/tex].
1. Set up the equation:
Start with the equation [tex]\( f(x) = 15 \)[/tex].
So, [tex]\( 4|x - 5| + 3 = 15 \)[/tex].
2. Isolate the absolute value:
Subtract 3 from both sides to isolate the term with the absolute value:
[tex]\[ 4|x - 5| = 15 - 3 \][/tex]
[tex]\[ 4|x - 5| = 12 \][/tex]
3. Solve for the absolute value:
Divide both sides by 4 to further isolate the absolute value:
[tex]\[ |x - 5| = \frac{12}{4} \][/tex]
[tex]\[ |x - 5| = 3 \][/tex]
4. Consider both scenarios for the absolute value equation:
Since [tex]\( |x - 5| = 3 \)[/tex], this equation can be split into two separate linear equations:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 3 + 5 \][/tex]
[tex]\[ x = 8 \][/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = -3 + 5 \][/tex]
[tex]\[ x = 2 \][/tex]
5. Conclusion:
The values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Hence, the correct answer from the choices provided is [tex]\( x = 2, x = 8 \)[/tex].
