Olivia is trying to simplify the expression 2+(x+5)

Question

using properties of algebra. She has explained her reasoning in steps, but where did she go wrong?

I can use the Distributive Property to rewrite this expression as (2+x)+(2+5)

Then, I can use the Commutative Property to rewrite it again as (2+5)+(2+x)

I can also use the Associative Property to write the expression as (2+5+2)+x

Finally, I can simplify to show that the expression is equivalent to 9+x

(1 point)
Responses

Step 1
Step 1

Step 2
Step 2

Step 3
Step 3

Step 4

Answer 1

Olivia's error occurred in her first step. Let's break down where she went wrong:

1. Step 1 Error: Olivia states that she can use the Distributive Property to rewrite \( 2 + (x + 5) \) as \( (2 + x) + (2 + 5) \). This is incorrect because the Distributive Property applies to multiplying a term outside of parentheses by each term inside the parentheses, which is not the operation she is working with here.

The correct application of the Distributive Property would involve an expression like \( a(b + c) = ab + ac \), but that's not relevant in this context since there is no multiplication happening outside the parentheses.

2. Subsequent Steps: The errors made in Step 2 and Step 3 stem from the incorrect foundation established in Step 1. Since the expression was already misrepresented, the reasoning using the Commutative and Associative Properties becomes irrelevant and further incorrect.

To simplify \( 2 + (x + 5) \) correctly:

1. Apply the Associative Property to group the terms: \[ 2 + (x + 5) = (2 + 5) + x \]

2. Simplify: \[ (2 + 5) + x = 7 + x \]

So the correct simplified expression is \( 7 + x \), not \( 9 + x \).

In summary, Olivia went wrong in Step 1 by incorrectly applying the Distributive Property.