Answer :
To find the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex] equals 15, we will go through the following steps:
1. Set the function equal to 15:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to simplify the equation:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide both sides by 4 to isolate the absolute value term:
[tex]\[
|x - 5| = 3
\][/tex]
4. Solve the absolute value equation [tex]\( |x - 5| = 3 \)[/tex]. This equation means that the expression inside the absolute value, [tex]\( x - 5 \)[/tex], can be either 3 or -3. Therefore, we get two cases to consider:
[tex]\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\][/tex]
5. Solve each case 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]
Thus, the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
The correct answer is:
[tex]\[ x = 2, x = 8 \][/tex]
1. Set the function equal to 15:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to simplify the equation:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide both sides by 4 to isolate the absolute value term:
[tex]\[
|x - 5| = 3
\][/tex]
4. Solve the absolute value equation [tex]\( |x - 5| = 3 \)[/tex]. This equation means that the expression inside the absolute value, [tex]\( x - 5 \)[/tex], can be either 3 or -3. Therefore, we get two cases to consider:
[tex]\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\][/tex]
5. Solve each case 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]
Thus, the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
The correct answer is:
[tex]\[ x = 2, x = 8 \][/tex]