Math  /  Algebra

QuestionSimplify the expression. 222x+44\frac{2}{22 x+44}
Select the correct choice below and, if necessary, fill in the answer box to complete your choio A. 222x+44=\frac{2}{22 x+44}= \square (Simplify your answer.) B. The expression cannot be simplified.

Studdy Solution

STEP 1

What is this asking? We need to simplify a fraction with *x* in it, and if we can't simplify it, we need to say so! Watch out! Don't forget to factor completely!
Sometimes there are hidden common factors lurking in expressions.

STEP 2

1. Factor the denominator.
2. Simplify the fraction.

STEP 3

Let's **factor the denominator**, 22x+4422x + 44.
We're doing this because factoring helps us see if there are any common factors we can simplify in the fraction.
Notice that both 22x22x and 4444 are divisible by **2**, **11**, and **22**.
The *greatest common factor* is **22**.

STEP 4

So, we can rewrite 22x+4422x + 44 as 22(x+2)22(x + 2).
We're dividing each term by **22** and writing it outside the parentheses.
Remember, 22x22 \cdot x is 22x22x, and 22222 \cdot 2 is 4444.
So, 22(x+2)22(x+2) is the same as 22x+4422x + 44.

STEP 5

Now, our fraction looks like this: 222(x+2)\frac{2}{22(x+2)}.
We replaced the denominator with its factored form.
This is helpful because now we can clearly see if there are any common factors between the numerator and the denominator.

STEP 6

We see that both the numerator (22) and the denominator (22(x+2)22(x+2)) are divisible by **2**.
Let's divide both by **2**!
In the numerator, 22 divided by 22 is 11.
In the denominator, 2222 divided by 22 is 1111, so we get 11(x+2)11(x+2).
We're essentially multiplying the fraction by 1/21/2\frac{1/2}{1/2}, which is the same as multiplying by 11, so we're not changing the value of the fraction, just its appearance!

STEP 7

Our simplified fraction is 111(x+2)\frac{1}{11(x+2)}.
Woohoo!

STEP 8

The simplified expression is 111(x+2)\frac{1}{11(x+2)}.
So the answer is A. 222x+44=111(x+2)\frac{2}{22x + 44} = \frac{1}{11(x+2)}.

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