To determine which phrase matches the algebraic expression [tex]\(2(x + 7) + 10\)[/tex], let's break down the expression step by step.
### Step-by-Step Simplification
1. Start with: [tex]\(2(x + 7) + 10\)[/tex]
2. Distribute the 2 inside the parentheses:
[tex]\[
2 \cdot x + 2 \cdot 7 + 10
\][/tex]
[tex]\[
= 2x + 14 + 10
\][/tex]
3. Combine like terms:
[tex]\[
2x + 24
\][/tex]
So, the simplified form of the expression [tex]\(2(x + 7) + 10\)[/tex] is [tex]\(2x + 24\)[/tex].
### Interpreting the Original Expression
Now, let's interpret the original algebraic expression [tex]\(2(x+7) + 10\)[/tex]:
- Inside the parentheses: [tex]\(x + 7\)[/tex] represent the sum of [tex]\(x\)[/tex] and 7.
- The 2 outside the parentheses: indicates that this entire sum is multiplied by 2.
- After this multiplication: 10 is added to the result.
So, the phrase that matches this interpretation is:
Two times the sum of [tex]\(x\)[/tex] and seven plus ten
### Verifying Other Choices
- Two times the difference of [tex]\(x\)[/tex] and seven plus ten: This would correspond to [tex]\(2(x - 7) + 10\)[/tex].
- Two times [tex]\(x\)[/tex] minus seven plus ten: This would correspond to [tex]\(2x - 7 + 10\)[/tex].
- Two times [tex]\(x\)[/tex] plus the sum of seven and ten: This would correspond to [tex]\(2x + (7 + 10)\)[/tex].
### Correct Answer
None of the other choices match the expression [tex]\(2(x + 7) + 10\)[/tex].
Therefore, the correct phrase is:
Two times the sum of [tex]\(x\)[/tex] and seven plus ten.
