To simplify the expression \(3(x + 4) - 2(2x - 5)\), we will follow a step-by-step approach. Let's break it down: ### 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, \(-2(2x - 5)\): - Multiply \(-2\) by \(2x\): \(-2 \cdot 2x = -4x\) - Multiply \(-2\) by \(-5\): \(-2 \cdot -5 = 10\) (Note that multiplying two negatives gives a positive) - So, \(-2(2x - 5) = -4x + 10\) ### Step 2: Combine the results Now we can combine the results from both distributions: \[ 3(x + 4) - 2(2x - 5) = (3x + 12) + (-4x + 10) \] ### Step 3: Combine like terms Next, we will combine the like terms (the terms that contain \(x\) and the constant terms): 1. Combine the \(x\) terms: - \(3x - 4x = -1x\) 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: \[ -x + 22 \] ### Answer The correct option is **B. \(-x + 22\)**. ### Explanation of Other Options - **Option A: \(x + 22\)**: This option is incorrect because it suggests that the coefficient of \(x\) is positive, which contradicts our calculation where we found it to be \(-x\). - **Option C: \(-x + 10\)**: This option is incorrect because it has the wrong constant term. We calculated the constant term to be \(22\), not \(10\). - **Option D: \(x + 10\)**: This option is incorrect for the same reasons as option A; it incorrectly states the coefficient of \(x\) as positive and also has the wrong constant term. ### Common Pitfalls - **Distributing Negatives**: A common mistake is forgetting to distribute the negative sign correctly, especially when dealing with expressions like \(-2(2x - 5)\). - **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). - **Sign Errors**: Be careful with signs when adding or subtracting terms, as they can change the outcome significantly. ### Revision Summary - Distribute coefficients carefully to each term inside parentheses. - Combine like terms systematically to simplify expressions. - Pay attention to signs, especially when distributing negatives. - Double-check your final expression to ensure all terms are accounted for correctly.