5(2x+6)=

Answer 1

To simplify the expression \(5(2x + 6)\), you can distribute the 5 to both terms inside the parentheses:

\[ 5(2x) + 5(6) \]

Calculating each term gives:

\[ 10x + 30 \]

So, the simplified expression is:

\[ 10x + 30 \]

Answer 2

Olivia is trying to simplify the expression 2+(x+5)
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 3

Olivia's reasoning contains an error in her use of the Distributive Property in Step 1. Let's analyze each step of her reasoning:

1. Step 1: She 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 is used to distribute a coefficient across terms inside parentheses (e.g., \(a(b + c) = ab + ac\)). In this case, there's no coefficient to distribute.

2. Step 2: She then applies the Commutative Property, which states that the order of addition doesn't matter. However, since her prior step was incorrect, this step does not logically follow.

3. Step 3: The Associative Property allows for the regrouping of terms, but again, the prior steps have led to a misunderstanding, so this step is also flawed.

4. Step 4: Finally, she claims the expression simplifies to \(9 + x\), which is incorrect based on her previous steps.

The mistake lies primarily in Step 1. The expression \(2 + (x + 5)\) can be directly simplified to \(x + 7\) without needing to incorrectly apply the properties of algebra. The correct simplification should directly combine \(2\) and \(5\) as \(2 + 5 = 7\), leading to:

\[ 2 + (x + 5) = x + 7 \]

So the error is in her initial assumption that she could distribute terms in that way.

Answer 4

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

Olivia's mistake occurs in Step 1. Let's break down each step to identify the error:

1. Step 1: Olivia claims to use the Distributive Property to rewrite \(2 + (x + 5)\) as \((2 + x) + (2 + 5)\). This is incorrect because the Distributive Property applies to expressions like \(a(b + c)\) but not in the way she is using it here. There is no distribution happening in the original expression; she should simply combine the constants. The correct way to simplify \(2 + (x + 5)\) is to add the constants directly, resulting in \(x + 7\).

2. Step 2: Olivia applies the Commutative Property, stating she can rearrange the expression. Although this property is valid, the previous step was incorrect, making the rearrangement not relevant.

3. Step 3: Olivia uses the Associative Property to group terms, but again this follows an incorrect foundation from Step 1.

4. Step 4: Finally, she combines the terms and concludes the expression is equivalent to \(9 + x\). This result is incorrect based on her initial mistake.

In summary, Olivia's reasoning is flawed starting from Step 1, where she misapplied the Distributive Property. The correct approach would simply combine \(2 + (x + 5)\) to yield \(x + 7\).