Solve for all values of \( c \) in simplest form. \[ 1=|2 c+1| \]

To solve the equation \( 1 = |2c + 1| \), we will consider the definition of absolute value. The expression \( |x| = a \) leads to two cases: \( x = a \) or \( x = -a \). 1. **Case 1:** \( 2c + 1 = 1 \) Subtracting 1 from both sides, we have: \[ 2c = 0 \] Dividing by 2 gives: \[ c = 0 \] 2. **Case 2:** \( 2c + 1 = -1 \) Subtracting 1 from both sides gives: \[ 2c = -2 \] Dividing by 2 results in: \[ c = -1 \] Thus, the solutions for \( c \) are \( c = 0 \) and \( c = -1 \). The final values of \( c \) in simplest form are: \[ \boxed{0 \text{ and } -1} \]