Solve the equation. \[ |5 b+7|=|b+3| \]

To solve the equation \( |5b + 7| = |b + 3| \), we'll analyze the expression by breaking it down into different cases based on the properties of absolute values. ### Case 1: \( 5b + 7 = b + 3 \) Setting the insides of the absolute values equal gives us: \[ 5b + 7 = b + 3 \] Subtract \( b \) from both sides: \[ 4b + 7 = 3 \] Subtract 7 from both sides: \[ 4b = -4 \] Divide by 4: \[ b = -1 \] ### Case 2: \( 5b + 7 = -(b + 3) \) For this case, we set one expression equal to the negative of the other: \[ 5b + 7 = -b - 3 \] Add \( b \) to both sides: \[ 6b + 7 = -3 \] Subtract 7 from both sides: \[ 6b = -10 \] Divide by 6: \[ b = -\frac{5}{3} \] ### Case 3: \( -(5b + 7) = b + 3 \) Now, we consider the case where we reverse the sign of the first absolute value: \[ -(5b + 7) = b + 3 \] This can be simplified to: \[ -5b - 7 = b + 3 \] Add \( 5b \) to both sides: \[ -7 = 6b + 3 \] Subtract 3 from both sides: \[ -10 = 6b \] Divide by 6: \[ b = -\frac{5}{3} \] ### Case 4: \( -(5b + 7) = -(b + 3) \) Finally, let’s consider the case where both absolute values are negative: \[ -(5b + 7) = -(b + 3) \] This simplifies to: \[ -5b - 7 = -b - 3 \] Rearranging gives: \[ -5b + b = -3 + 7 \] This results in: \[ -4b = 4 \] So: \[ b = -1 \] ### Summary of Solutions The solutions from our cases are: - \( b = -1 \) - \( b = -\frac{5}{3} \) Thus, the complete solution set is: \[ b = -1 \quad \text{and} \quad b = -\frac{5}{3} \]