Answer :
To solve the problem and find the values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 15\)[/tex] in the function [tex]\(f(x) = 4|x - 5| + 3\)[/tex], follow these steps:
1. Start with the equation:
[tex]\[
f(x) = 4|x - 5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to isolate the absolute value expression:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide both sides of the equation by 4:
[tex]\[
|x - 5| = 3
\][/tex]
4. At this point, recognize that the expression [tex]\(|x - 5| = 3\)[/tex] can split into two separate equations because absolute value equates to a positive and a negative scenario:
[tex]\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\][/tex]
5. Solve each equation separately:
- For [tex]\(x - 5 = 3\)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For [tex]\(x - 5 = -3\)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
6. Thus, the values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 15\)[/tex] are:
[tex]\[
x = 8 \quad \text{and} \quad x = 2
\][/tex]
Therefore, the correct choice is [tex]\(x = 2, x = 8\)[/tex].
1. Start with the equation:
[tex]\[
f(x) = 4|x - 5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to isolate the absolute value expression:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide both sides of the equation by 4:
[tex]\[
|x - 5| = 3
\][/tex]
4. At this point, recognize that the expression [tex]\(|x - 5| = 3\)[/tex] can split into two separate equations because absolute value equates to a positive and a negative scenario:
[tex]\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\][/tex]
5. Solve each equation separately:
- For [tex]\(x - 5 = 3\)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For [tex]\(x - 5 = -3\)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
6. Thus, the values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 15\)[/tex] are:
[tex]\[
x = 8 \quad \text{and} \quad x = 2
\][/tex]
Therefore, the correct choice is [tex]\(x = 2, x = 8\)[/tex].
