Solve the equation \( |2 x+1|=10 \)

To solve the equation \( |2x + 1| = 10 \), we'll consider the definition of absolute value. The absolute value of an expression equals a positive number, which means the expression inside the absolute value can be either positive or negative. Therefore, we can set up two separate equations: 1. **Case 1:** \( 2x + 1 = 10 \) 2. **Case 2:** \( 2x + 1 = -10 \) Let's solve each case step by step. --- ### **Case 1:** \( 2x + 1 = 10 \) 1. **Subtract 1 from both sides:** \[ 2x = 10 - 1 \] \[ 2x = 9 \] 2. **Divide both sides by 2:** \[ x = \frac{9}{2} \] \[ x = 4.5 \] --- ### **Case 2:** \( 2x + 1 = -10 \) 1. **Subtract 1 from both sides:** \[ 2x = -10 - 1 \] \[ 2x = -11 \] 2. **Divide both sides by 2:** \[ x = \frac{-11}{2} \] \[ x = -5.5 \] --- ### **Conclusion** The solutions to the equation \( |2x + 1| = 10 \) are: \[ x = \frac{9}{2} \quad \text{and} \quad x = -\frac{11}{2} \] Or in decimal form: \[ x = 4.5 \quad \text{and} \quad x = -5.5 \] You can verify these solutions by plugging them back into the original equation: - **For \( x = 4.5 \):** \[ |2(4.5) + 1| = |9 + 1| = |10| = 10 \] - **For \( x = -5.5 \):** \[ |2(-5.5) + 1| = |-11 + 1| = |-10| = 10 \] Both solutions satisfy the original equation.