Answer :
To find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] given [tex]\( f(x) = 4|x-5| + 3 \)[/tex], we need to set up and solve the equation:
[tex]\[ 4|x-5| + 3 = 15 \][/tex]
Step 1: Subtract 3 from both sides of the equation.
[tex]\[ 4|x-5| = 12 \][/tex]
Step 2: Divide both sides by 4 to isolate the absolute value.
[tex]\[ |x-5| = 3 \][/tex]
Step 3: Solve the absolute value equation. The equation [tex]\( |x-5| = 3 \)[/tex] means:
1. [tex]\( x-5 = 3 \)[/tex]
2. [tex]\( x-5 = -3 \)[/tex]
Step 4: Solve each equation separately.
1. For [tex]\( x-5 = 3 \)[/tex]:
[tex]\[
x - 5 = 3 \\
x = 3 + 5 \\
x = 8
\][/tex]
2. For [tex]\( x-5 = -3 \)[/tex]:
[tex]\[
x - 5 = -3 \\
x = -3 + 5 \\
x = 2
\][/tex]
So, the values of [tex]\( x \)[/tex] for which [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].
[tex]\[ 4|x-5| + 3 = 15 \][/tex]
Step 1: Subtract 3 from both sides of the equation.
[tex]\[ 4|x-5| = 12 \][/tex]
Step 2: Divide both sides by 4 to isolate the absolute value.
[tex]\[ |x-5| = 3 \][/tex]
Step 3: Solve the absolute value equation. The equation [tex]\( |x-5| = 3 \)[/tex] means:
1. [tex]\( x-5 = 3 \)[/tex]
2. [tex]\( x-5 = -3 \)[/tex]
Step 4: Solve each equation separately.
1. For [tex]\( x-5 = 3 \)[/tex]:
[tex]\[
x - 5 = 3 \\
x = 3 + 5 \\
x = 8
\][/tex]
2. For [tex]\( x-5 = -3 \)[/tex]:
[tex]\[
x - 5 = -3 \\
x = -3 + 5 \\
x = 2
\][/tex]
So, the values of [tex]\( x \)[/tex] for which [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].
