To simplify the expression \( 3(x + 4) - 2(x - 5) \), we will follow a step-by-step approach. ### Step 1: Distribute the coefficients First, we need to distribute the numbers outside the parentheses to the terms inside the parentheses. 1. For the first part, \( 3(x + 4) \): - Multiply \( 3 \) by \( x \): \( 3 \cdot x = 3x \) - Multiply \( 3 \) by \( 4 \): \( 3 \cdot 4 = 12 \) - So, \( 3(x + 4) = 3x + 12 \) 2. For the second part, \( -2(x - 5) \): - Multiply \( -2 \) by \( x \): \( -2 \cdot x = -2x \) - Multiply \( -2 \) by \( -5 \): \( -2 \cdot -5 = 10 \) (Note: multiplying two negatives gives a positive) - So, \( -2(x - 5) = -2x + 10 \) ### Step 2: Combine the results Now we can combine the results from both distributions: \[ 3(x + 4) - 2(x - 5) = (3x + 12) + (-2x + 10) \] ### Step 3: Combine like terms Next, we will combine the like terms (the terms with \( x \) and the constant terms): 1. Combine the \( x \) terms: - \( 3x - 2x = 1x \) or simply \( x \) 2. Combine the constant terms: - \( 12 + 10 = 22 \) Putting it all together, we have: \[ x + 22 \] ### Final Result Thus, the simplified expression is: \[ x + 22 \] ### Correct Option The correct option is **A. \( x + 22 \)**. ### Explanation of Other Options - **B. \( x + 2 \)**: This option is incorrect because it does not account for the correct addition of the constant terms. The constants \( 12 \) and \( 10 \) add up to \( 22 \), not \( 2 \). - **C. \( x + 18 \)**: This option is also incorrect. It suggests that the constant terms add up to \( 18 \), which is not true since \( 12 + 10 = 22 \). - **D. \( x + 9 \)**: This option is incorrect as well. It implies that the constant terms add up to \( 9 \), which is incorrect for the same reason as above. ### Common Pitfalls - **Distributing Negatives**: Be careful when distributing negative coefficients. Remember that multiplying two negatives results in a positive. - **Combining Like Terms**: Ensure that you only combine terms that are similar (i.e., \( x \) terms with \( x \) terms and constant terms with constant terms). ### Revision Summary - Distribute coefficients carefully to each term inside the parentheses. - Combine like terms accurately to simplify expressions. - Always double-check your arithmetic when adding or subtracting constants. - Be cautious with negative signs during distribution. By following these steps, you can simplify expressions confidently and accurately!