To determine which phrase correctly matches the given algebraic expression [tex]\( 2(x-7)+10 \)[/tex], let's break it down.
1. Expression Analysis:
The algebraic expression is [tex]\( 2(x-7) + 10 \)[/tex].
- Parentheses: First, observe the term inside the parentheses [tex]\( (x-7) \)[/tex]. This indicates a difference between [tex]\( x \)[/tex] and 7.
- Multiplication: The entire term inside the parentheses is being multiplied by 2.
- Addition: Finally, 10 is being added to the result of the multiplication.
2. Phrase Matching:
Let's analyze each phrase step-by-step:
- Phrase 1: "Two times the sum of [tex]\( x \)[/tex] and seven plus ten"
This phrase implies the expression should look like [tex]\( 2(x+7) + 10 \)[/tex], where [tex]\( x \)[/tex] and 7 are summed first before being multiplied by 2 and then adding 10. Clearly, this does not match our expression.
- Phrase 2: "Two times the difference of [tex]\( x \)[/tex] and seven plus ten"
This indicates we take the difference of [tex]\( x \)[/tex] and 7, multiply it by 2, and then add 10. This precisely matches our expression [tex]\( 2(x-7) + 10 \)[/tex].
- Phrase 3: "Two times [tex]\( x \)[/tex] minus seven plus ten"
This phrase translates to [tex]\( 2x - 7 + 10 \)[/tex], where [tex]\( x \)[/tex] is multiplied by 2 first, then 7 is subtracted, and 10 is added. This does not match our expression.
- Phrase 4: "Two times [tex]\( x \)[/tex] minus the sum of seven and ten"
This means [tex]\( 2x - (7 + 10) \)[/tex], where we first take the sum of 7 and 10, and then subtract it from [tex]\( 2x \)[/tex]. This also does not match our expression.
Therefore, the phrase that matches the algebraic expression [tex]\( 2(x-7)+10 \)[/tex] is:
"Two times the difference of [tex]\( x \)[/tex] and seven plus ten"
So, the correct answer is:
Two times the difference of [tex]\( x \)[/tex] and seven plus ten.
