Answer :
To find the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 4|x - 5| + 3 \)[/tex] equals 15, we need to solve the equation:
[tex]\[ 4|x - 5| + 3 = 15 \][/tex]
Step 1: Subtract 3 from both sides.
Start by isolating the absolute value term:
[tex]\[ 4|x - 5| + 3 - 3 = 15 - 3 \][/tex]
Which simplifies to:
[tex]\[ 4|x - 5| = 12 \][/tex]
Step 2: Divide by 4.
Next, divide both sides of the equation by 4 to further isolate the absolute value:
[tex]\[ |x - 5| = \frac{12}{4} \][/tex]
This simplifies to:
[tex]\[ |x - 5| = 3 \][/tex]
Step 3: Solve the absolute value equation.
The equation [tex]\( |x - 5| = 3 \)[/tex] means that the expression inside the absolute value, [tex]\( x - 5 \)[/tex], can be either 3 or -3. So we have two cases to consider:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
Case 1: Solve [tex]\( x - 5 = 3 \)[/tex]
Add 5 to both sides:
[tex]\[ x = 3 + 5 \][/tex]
[tex]\[ x = 8 \][/tex]
Case 2: Solve [tex]\( x - 5 = -3 \)[/tex]
Add 5 to both sides:
[tex]\[ x = -3 + 5 \][/tex]
[tex]\[ x = 2 \][/tex]
We have found two values for [tex]\( x \)[/tex] that satisfy the original equation: [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/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.
Start by isolating the absolute value term:
[tex]\[ 4|x - 5| + 3 - 3 = 15 - 3 \][/tex]
Which simplifies to:
[tex]\[ 4|x - 5| = 12 \][/tex]
Step 2: Divide by 4.
Next, divide both sides of the equation by 4 to further isolate the absolute value:
[tex]\[ |x - 5| = \frac{12}{4} \][/tex]
This simplifies to:
[tex]\[ |x - 5| = 3 \][/tex]
Step 3: Solve the absolute value equation.
The equation [tex]\( |x - 5| = 3 \)[/tex] means that the expression inside the absolute value, [tex]\( x - 5 \)[/tex], can be either 3 or -3. So we have two cases to consider:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
Case 1: Solve [tex]\( x - 5 = 3 \)[/tex]
Add 5 to both sides:
[tex]\[ x = 3 + 5 \][/tex]
[tex]\[ x = 8 \][/tex]
Case 2: Solve [tex]\( x - 5 = -3 \)[/tex]
Add 5 to both sides:
[tex]\[ x = -3 + 5 \][/tex]
[tex]\[ x = 2 \][/tex]
We have found two values for [tex]\( x \)[/tex] that satisfy the original equation: [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex].
The correct answer is [tex]\( x = 2, x = 8 \)[/tex].
