Answer :
Let's determine if the expressions [tex]\( x + 4 + x \)[/tex] and [tex]\( 6 + 2x - 2 \)[/tex] are equivalent by working through the problem step-by-step.
First, we evaluate the expressions when [tex]\( x = 5 \)[/tex].
### Evaluation of the First Expression
The first expression is:
[tex]\[ x + 4 + x \][/tex]
Substitute [tex]\( x = 5 \)[/tex] into the expression:
[tex]\[ 5 + 4 + 5 \][/tex]
Simplify this:
[tex]\[ 5 + 4 = 9 \][/tex]
[tex]\[ 9 + 5 = 14 \][/tex]
So, when [tex]\( x = 5 \)[/tex], the value of the first expression is 14. This matches the value Nancy found:
[tex]\[ x + 4 + x = 14 \][/tex]
### Evaluation of the Second Expression
The second expression is:
[tex]\[ 6 + 2x - 2 \][/tex]
Substitute [tex]\( x = 5 \)[/tex] into the expression:
[tex]\[ 6 + 2(5) - 2 \][/tex]
Simplify this:
[tex]\[ 2(5) = 10 \][/tex]
[tex]\[ 6 + 10 - 2 = 16 - 2 = 14 \][/tex]
So, when [tex]\( x = 5 \)[/tex], the value of the second expression is 14.
### Determining Equivalence
Since both expressions evaluate to 14 when [tex]\( x = 5 \)[/tex], we conclude that the expressions are equivalent for [tex]\( x = 5 \)[/tex].
Thus, the value of the second expression when [tex]\( x = 5 \)[/tex] is 14, and the two expressions are equivalent.
So the correct answer is:
"The value of the second expression is 14, so the expressions are equivalent."
First, we evaluate the expressions when [tex]\( x = 5 \)[/tex].
### Evaluation of the First Expression
The first expression is:
[tex]\[ x + 4 + x \][/tex]
Substitute [tex]\( x = 5 \)[/tex] into the expression:
[tex]\[ 5 + 4 + 5 \][/tex]
Simplify this:
[tex]\[ 5 + 4 = 9 \][/tex]
[tex]\[ 9 + 5 = 14 \][/tex]
So, when [tex]\( x = 5 \)[/tex], the value of the first expression is 14. This matches the value Nancy found:
[tex]\[ x + 4 + x = 14 \][/tex]
### Evaluation of the Second Expression
The second expression is:
[tex]\[ 6 + 2x - 2 \][/tex]
Substitute [tex]\( x = 5 \)[/tex] into the expression:
[tex]\[ 6 + 2(5) - 2 \][/tex]
Simplify this:
[tex]\[ 2(5) = 10 \][/tex]
[tex]\[ 6 + 10 - 2 = 16 - 2 = 14 \][/tex]
So, when [tex]\( x = 5 \)[/tex], the value of the second expression is 14.
### Determining Equivalence
Since both expressions evaluate to 14 when [tex]\( x = 5 \)[/tex], we conclude that the expressions are equivalent for [tex]\( x = 5 \)[/tex].
Thus, the value of the second expression when [tex]\( x = 5 \)[/tex] is 14, and the two expressions are equivalent.
So the correct answer is:
"The value of the second expression is 14, so the expressions are equivalent."
