Solve for \( y \). \[ |4 y+5|=|4 y+9| \]

To solve the equation \( |4y + 5| = |4y + 9| \), we first consider the two cases where the expressions inside the absolute values can be equal or opposites of each other. ### Case 1: \[ 4y + 5 = 4y + 9 \] Subtracting \( 4y \) from both sides yields: \[ 5 = 9 \] This case is not possible, so we discard it. ### Case 2: \[ 4y + 5 = -(4y + 9) \] Expanding the right side: \[ 4y + 5 = -4y - 9 \] Adding \( 4y \) to both sides: \[ 8y + 5 = -9 \] Now, subtracting 5 from both sides: \[ 8y = -14 \] Dividing by 8 gives: \[ y = -\frac{14}{8} = -\frac{7}{4} \] ### Let's double-check: If \( y = -\frac{7}{4} \): - Calculate \( 4y + 5 \): \[ 4(-\frac{7}{4}) + 5 = -7 + 5 = -2 \] - Calculate \( 4y + 9 \): \[ 4(-\frac{7}{4}) + 9 = -7 + 9 = 2 \] - Check the absolute values: \[ |-2| = 2 \quad \text{and} \quad |2| = 2 \] Both sides are equal, confirming the solution is valid. Thus, the solution is: \[ y = -\frac{7}{4} \]