Explain how the distributive property and the commutative property of addition are used to show 2(x+1)+2x is equivalent to 4x+2
2(x+1)+2x
2x+2+2x
2x+2x+2
(2+2)x+2
4x+2
To show that 2(x + 1) + 2x is equivalent to 4x + 2, we will need to use the distributive property and the commutative property of addition. Let's break it down step by step.
Step 1: Apply the distributive property by multiplying the 2 to both terms inside the parentheses:
2(x + 1) + 2x = 2 * x + 2 * 1 + 2x
Step 2: Simplify the multiplication:
2x + 2 + 2x = 4x + 2
In this step, we can use the commutative property of addition to rearrange the terms:
2x + 2 + 2x = 2x + 2x + 2
Step 3: Combine like terms:
2x + 2x + 2 = 4x + 2
Therefore, we have successfully shown that 2(x + 1) + 2x is equivalent to 4x + 2 by applying the distributive property and using the commutative property of addition.
To understand how the distributive and commutative properties are used to show the equivalence between 2(x+1)+2x and 4x+2, let's break down the steps.
Step 1: Start with the given expression, 2(x+1)+2x.
Step 2: Apply the distributive property. The distributive property states that multiplying a number by a sum is the same as multiplying the number by each term of the sum and then adding the results.
For the term 2(x+1), we multiply each term inside the parentheses by 2:
2 * x + 2 * 1 = 2x + 2.
Now the expression becomes 2x + 2 + 2x.
Step 3: Use the commutative property of addition. The commutative property says that the order of terms in an addition can be rearranged without changing the sum.
Since addition is commutative, we can rearrange the terms in any order and still get the same value. So, we can rewrite the expression as: 2x + 2x + 2.
Step 4: Combine like terms. Combine the two terms with the variable x:
2x + 2x = 4x.
Now the expression becomes 4x + 2.
Therefore, by applying the distributive property and using the commutative property of addition, we have shown that 2(x+1)+2x is equivalent to 4x+2.