Answer :
To find the values of [tex]\(x\)[/tex] for which the function [tex]\(f(x) = 4|x-5| + 3\)[/tex] equals 15, let's 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. Solve for the absolute value:
Subtract 3 from both sides to isolate the term with the absolute value:
[tex]\[ 4|x-5| = 12 \][/tex]
3. Divide by 4:
Divide each side by 4 to simplify the equation:
[tex]\[ |x-5| = 3 \][/tex]
4. Consider the scenarios for the absolute value:
The equation [tex]\(|x-5| = 3\)[/tex] means that the expression inside the absolute value has two possible cases:
[tex]\(x - 5 = 3\)[/tex]
[tex]\(x - 5 = -3\)[/tex]
5. Solve each scenario:
- For the first scenario, [tex]\(x - 5 = 3\)[/tex]:
[tex]\[ x = 5 + 3 = 8 \][/tex]
- For the second scenario, [tex]\(x - 5 = -3\)[/tex]:
[tex]\[ x = 5 - 3 = 2 \][/tex]
Therefore, the values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 15\)[/tex] are [tex]\(x = 8\)[/tex] and [tex]\(x = 2\)[/tex]. Thus, the correct answer 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. Solve for the absolute value:
Subtract 3 from both sides to isolate the term with the absolute value:
[tex]\[ 4|x-5| = 12 \][/tex]
3. Divide by 4:
Divide each side by 4 to simplify the equation:
[tex]\[ |x-5| = 3 \][/tex]
4. Consider the scenarios for the absolute value:
The equation [tex]\(|x-5| = 3\)[/tex] means that the expression inside the absolute value has two possible cases:
[tex]\(x - 5 = 3\)[/tex]
[tex]\(x - 5 = -3\)[/tex]
5. Solve each scenario:
- For the first scenario, [tex]\(x - 5 = 3\)[/tex]:
[tex]\[ x = 5 + 3 = 8 \][/tex]
- For the second scenario, [tex]\(x - 5 = -3\)[/tex]:
[tex]\[ x = 5 - 3 = 2 \][/tex]
Therefore, the values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 15\)[/tex] are [tex]\(x = 8\)[/tex] and [tex]\(x = 2\)[/tex]. Thus, the correct answer is:
[tex]\(x = 2, x = 8\)[/tex]
