Answered

Angela used multiplication to simplify the expression by distributing the $\frac{1}{2}$ through the parentheses.



Angela's work:

Angela's expression: $\frac{1}{2}(6 x+2)$

Simplified expression: $3x+2$



Which statement is true of Angela's work?


AShe correctly distributed the $\frac{1}{2}$ through the parentheses.
BDivision, rather than multiplication, is the operation to use when distributing.
CAngela needed to multiply $\frac{1}{2}$ by each of the terms inside the parentheses.

Verified Answer :

  • The distributive property \(a(b + c) = ab + ac\) requires multiplying the factor outside the parentheses by each term inside.
  • Applying this correctly: \(\frac{1}{2}(6x + 2) = \frac{1}{2} \cdot 6x + \frac{1}{2} \cdot 2 = 3x + 1\).
  • Angela failed to multiply the constant \(2\) by \(\frac{1}{2}\), resulting in \(3x + 2\) instead of \(3x + 1\).
  • The correct statement is: \(\boxed{\text{Angela needed to multiply } \frac{1}{2} \text{ by each of the terms inside the parentheses.}}\)

Explanation

  1. Analyzing the Expression and Distributive Property

The expression given is \(\frac{1}{2}(6x + 2)\). To simplify this using the distributive property, we must multiply the term outside the parentheses, \(\frac{1}{2}\), by each term inside the parentheses. The distributive property is formally stated as \(a(b + c) = ab + ac\). Applying this to our expression, we have:
\[\frac{1}{2}(6x + 2) = \left(\frac{1}{2} \cdot 6x\right) + \left(\frac{1}{2} \cdot 2\right)\]

  1. Evaluating Angela's Work

Now we perform the individual multiplications:

  1. First term: \(\frac{1}{2} \cdot 6x = 3x\).
  2. Second term: \(\frac{1}{2} \cdot 2 = 1\).

Combining these, the correctly simplified expression is \(3x + 1\). Comparing this to Angela's result of \(3x + 2\), we can see that she correctly calculated the first term (\(3x\)) but failed to apply the distributive property to the second term (the constant \(2\)).

  1. Conclusion

Based on our analysis, Angela did not multiply the second term inside the parentheses by \(\frac{1}{2}\). She left the constant \(2\) unchanged. Therefore, the statement that accurately describes the error in her work is that Angela needed to multiply \(\frac{1}{2}\) by each of the terms inside the parentheses. The other options are incorrect because she did not distribute correctly and multiplication is indeed the correct operation for distribution.

Examples

The distributive property is a fundamental tool in algebra, much like how you might distribute ingredients in a recipe. If you have a recipe that requires 3 batches of a mix containing 2 cups of flour and 1 cup of sugar, you must multiply the quantity of each ingredient by 3 to get the total amount needed: \(3(2 \text{ cups flour} + 1 \text{ cup sugar}) = 6 \text{ cups flour} + 3 \text{ cups sugar}\). Failing to multiply every term—like forgetting to multiply the sugar—would lead to an incorrect batch! This same logic applies when solving algebraic expressions.

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