To solve the expression \( 3(x + 4) - 2(x - 1) \) and determine which of the provided options is equivalent, we will follow a step-by-step approach. ### Step 1: Distribute the Terms First, we need to distribute the constants outside the parentheses to the terms inside the parentheses. 1. **Distributing \(3\) to \( (x + 4) \)**: \[ 3(x + 4) = 3x + 12 \] 2. **Distributing \(-2\) to \( (x - 1) \)**: \[ -2(x - 1) = -2x + 2 \] ### Step 2: Combine the Results Now, we combine the results from the distributions: \[ 3(x + 4) - 2(x - 1) = (3x + 12) + (-2x + 2) \] ### Step 3: Simplify the Expression Next, we combine like terms: 1. Combine the \(x\) terms: \[ 3x - 2x = 1x \quad \text{(or simply } x\text{)} \] 2. Combine the constant terms: \[ 12 + 2 = 14 \] Putting it all together, we have: \[ 3(x + 4) - 2(x - 1) = x + 14 \] ### Final Answer Thus, the expression simplifies to: \[ x + 14 \] ### Correct Option The correct option is **D. \( x + 14 \)**. ### Explanation of Other Options - **Option A: \( x + 10 \)**: This is incorrect because the constant term after simplification is \(14\), not \(10\). - **Option B: \( x + 6 \)**: This is also incorrect for the same reason; the constant term does not match. - **Option C: \( 5x + 10 \)**: This option is incorrect because it suggests a different coefficient for \(x\) and a different constant term, which does not match our simplified expression. ### Common Pitfalls - **Forgetting to distribute correctly**: Always ensure that you distribute the constants to all terms inside the parentheses. - **Combining like terms incorrectly**: Be careful to only combine terms that are similar (i.e., \(x\) terms with \(x\) terms and constant terms with constant terms). - **Neglecting the negative sign**: When distributing a negative, ensure that it affects the sign of the terms inside the parentheses. ### Revision Summary - Distribute constants to terms inside parentheses carefully. - Combine like terms accurately to simplify expressions. - Always double-check your final expression against the options provided. - Be mindful of signs when distributing negative constants. By following these steps, you can confidently simplify expressions and identify equivalent forms.