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Answer :
To solve the problem, we need to find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex], given the function [tex]\( f(x) = 4|x - 5| + 3 \)[/tex].
Here are the steps to find the solution:
1. Set the function equal to 15:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression:
Subtract 3 from both sides:
[tex]\[
4|x - 5| + 3 - 3 = 15 - 3
\][/tex]
Simplifies to:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Solve for the absolute value:
Divide by 4 on both sides to isolate [tex]\( |x - 5| \)[/tex]:
[tex]\[
\frac{4|x - 5|}{4} = \frac{12}{4}
\][/tex]
Simplifies to:
[tex]\[
|x - 5| = 3
\][/tex]
4. Remove the absolute value:
This equation [tex]\( |x - 5| = 3 \)[/tex] means [tex]\( x - 5 \)[/tex] can be either 3 or -3. Therefore, we have two cases to solve:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
5. Solve each case separately:
- For Case 1: [tex]\( x - 5 = 3 \)[/tex]
[tex]\[
x = 3 + 5
\][/tex]
[tex]\[
x = 8
\][/tex]
- For Case 2: [tex]\( x - 5 = -3 \)[/tex]
[tex]\[
x = -3 + 5
\][/tex]
[tex]\[
x = 2
\][/tex]
6. Verification:
We now have two values for [tex]\( x \)[/tex], which are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex].
Thus, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are:
[tex]\[
x = 2 \quad \text{and} \quad x = 8
\][/tex]
Therefore, the correct option is:
[tex]\[
x = 2, x = 8
\][/tex]
Here are the steps to find the solution:
1. Set the function equal to 15:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression:
Subtract 3 from both sides:
[tex]\[
4|x - 5| + 3 - 3 = 15 - 3
\][/tex]
Simplifies to:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Solve for the absolute value:
Divide by 4 on both sides to isolate [tex]\( |x - 5| \)[/tex]:
[tex]\[
\frac{4|x - 5|}{4} = \frac{12}{4}
\][/tex]
Simplifies to:
[tex]\[
|x - 5| = 3
\][/tex]
4. Remove the absolute value:
This equation [tex]\( |x - 5| = 3 \)[/tex] means [tex]\( x - 5 \)[/tex] can be either 3 or -3. Therefore, we have two cases to solve:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
5. Solve each case separately:
- For Case 1: [tex]\( x - 5 = 3 \)[/tex]
[tex]\[
x = 3 + 5
\][/tex]
[tex]\[
x = 8
\][/tex]
- For Case 2: [tex]\( x - 5 = -3 \)[/tex]
[tex]\[
x = -3 + 5
\][/tex]
[tex]\[
x = 2
\][/tex]
6. Verification:
We now have two values for [tex]\( x \)[/tex], which are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex].
Thus, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are:
[tex]\[
x = 2 \quad \text{and} \quad x = 8
\][/tex]
Therefore, the correct option is:
[tex]\[
x = 2, x = 8
\][/tex]
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