To solve the equation
[tex]\[ |4y - 2| + 7 = 37, \][/tex]
let's proceed with the following steps:
1. Isolate the absolute value expression.
[tex]\[ |4y - 2| = 37 - 7 \][/tex]
[tex]\[ |4y - 2| = 30 \][/tex]
2. Remove the absolute value by considering both possible cases for \(4y - 2\):
[tex]\[ 4y - 2 = 30 \quad \text{or} \quad 4y - 2 = -30 \][/tex]
Let's solve each case separately:
### Case 1: \(4y - 2 = 30\)
Solve for \( y \):
[tex]\[ 4y - 2 = 30 \][/tex]
Add 2 to both sides:
[tex]\[ 4y = 32 \][/tex]
Divide both sides by 4:
[tex]\[ y = 8 \][/tex]
### Case 2: \(4y - 2 = -30\)
Solve for \( y \):
[tex]\[ 4y - 2 = -30 \][/tex]
Add 2 to both sides:
[tex]\[ 4y = -28 \][/tex]
Divide both sides by 4:
[tex]\[ y = -7 \][/tex]
So the solutions for the equation \(|4y - 2| + 7 = 37\) are:
[tex]\[ y = 8 \quad \text{or} \quad y = -7 \][/tex]
The correct answer is:
[tex]\[ y = 8 \quad \text{or} \quad y = -7 \][/tex]
