Note- Anytime adding or subtracting fractions, first find the least common denominator.

 

 

Look at a couple of expressions and find the least common denominator of them.

 

 

 

The first thing to do is to ignore whatever is on top. The only time to look at the top is if it will simplify, if it will then simplify it first. This problem will not simplify so look at the denominators (x +1) and (x – 2). Since these are in parentheses, do not look at what is in the inside of them, but rather look at the entire parentheses. These are not exactly alike so use both (x+1)(x-2) as the common denominator. This may sound vague and confusing but will be explained by looking at a few examples.

 

Note- If the terms were exactly alike, include one of them in the answer, an example of this will be shown later.

 

Find the least common denominator

 

 

Remember the rule that stated that whatever was inside of the parentheses had to be exactly alike? These parentheses are not alike because one is positive and one is negative, so use both of them. Therefore, making the common denominator (x - 3)(x +3).

 

Find the least common denominator

 

 

 

 

 

The least common denominator in this case is 2x(x + 1). These two denominators had nothing in common so use both of them. Now one might think that they had an x in common but the x is part of a term (x+1). Therefore look at the entire term (x+1) and see if it matches perfectly.

 

Note- In the previous problems there have been parentheses in the denominator in many of the fractions. Many of the problems may not have parentheses. It is fine at the beginning of the problem to put parentheses around an entire denominator or numerator if it helps to understand the problem better.

 

Find the least common denominator of the following two terms.

 

 

 

 

 

The first things to look at are the numbers and variables (letters) on the outside of the parentheses. This includes all those that are not being added or subtracted to anything but are being multiplied. This would include the 8y on the left and the 5 on the right side. Next, find the least common multiple of the two numbers (5 and 8). The least common multiple is the smallest number that both of those two numbers go into. In this case it is 40. Be careful! The least common multiple is not always the product of the two numbers. Now look at what else the numbers have with them. On the left there is a y and on the right there is a (y – 1). These do not have anything in common so use everything for the least common denominator. That would make the least common denominator for this problem

40 y (y-1).

 

Find the least common denominator of

 

and .

 

 

 

 

 

Like the previous example, first find the least common multiple of the two numbers on bottom that are being multiplied by everything else. That would be the 4 and the 5. The least common multiple of those two numbers is 20. Look at the next items after these two numbers on the left, it is a², and on the right it is a. This is where it gets a little tricky. Unlike factoring where one tries to find what the have in common, try to do just the opposite here. Ask the question, “How many a’s are needed to account for both sides (the right and the left).” It is clear to see that there are two a’s on the left (a²) and one (a) on the right. To account for every a on the left, 2 a’s are needed or (a²), since there is only one a on the right, do not worry about it because it is accounted for with the two a’s required for the left.

 

That means at least 20a² will be in the common denominator. But wait, unaccounted for are the (a - 1) on the left and the (a + 2) on the right. Since these have addition and subtraction in them, work them just like the previous examples to this one. They are not exactly alike so include both of them in the common denominator.

 

By combining everything needed for the common denominator, one gets the result of 20 a² (a-1)(a+2).

 

 

Find the least common denominator of

 

and .

 

 

 

 

Look at the bottom on the left hand side, it is easy to realize that (x –3) and an (x + 1) will be needed. Looking at the denominator on the right, an (x + 1) will be needed. Keep in mind that there is an (x+1) from the left side so there is no reason to include another one. That would make the least common denominator (x-3)(x+1).

 

Find the least common denominator of

 

and .

 

 

 

Look at the two numbers in the denominators. On the left is an assumed 1, and on the right there is a 3. The least common multiple of 1 and 3 is 3. Next, look at the y’s, on the left there are three y’s, and on the right there is only one. Remember enough will be needed to account for both sides. By using three y’s it would be enough for the y’s on the left and more than enough for the y on the right. So that means, use three y’s or y³. Now look at the z’s, there are 2 on the left and 4 on the right. Needed are 4 z’s or z⁴. All that is left now is the parentheses, on the left a (z-1), and on the right a(z-1) and (z+1). In order to account for all the parentheses, (z-1) and (z+1) will be needed. By combining all of these things that are needed one gets 3 y³ z⁴(z-1)(z+1) for the least common denominator.

 

Find the least common denominator of

 

and .

 

 

 

 

Find the least common multiple of 4 and 6 it is 12(remember that it is the smallest number that both 4 and 6 go into). Since there are five x’s on the right and just one on the left 5 x’s are needed. With 6 y’s on the right it will take 6 y’s. In parentheses to cover the left side it will take (x+2) (x+1) on the right side, one needs to add (x-1) to that, since (x+2) is already accounted for on the left side. This would make the least common denominator 12x⁵ y⁶ (x+2)(x+1)(x-1).

 

Note- It is very important at this point to recognize the difference between an expression and an equation. An equation is anything with an equals sign in it. An expression does not have an equals, less than, or greater than sign.

 

 

 

 

The simplification problems that will be done will almost always be expressions. That is, they do not have an equals sign in them. The tricky thing about trying to simplify an expression, unlike an equation, is that the only thing that it can be multiplied by is 1. If one multiplies it by anything other than 1, the value of the expression has changed and that is not allowed.

 

 

 

 

The steps to follow when simplifying rational expressions with addition and subtraction in them are as follows:

1)      Find the least common denominator.

2)      Multiply everything in the expression by the least common denominator (over itself).

3)      Simplify the results and put everything over the least common denominator.

 

 

 

 

Example

Simplify

 

 

 

 

Find the least common denominator, it is (x-1)(x+1). Multiply everything by the least common denominator. To simplify things begin by looking at the first term. when it is multiplied by (x-1)(x+1).

 

 

In multiplying this type of problem it is much easier to recognize a few things and take a few shortcuts. Use some simplification techniques. Since there is an (x-1) on top and one on bottom, the first thing to do is to cancel them.

 

 

To further simplify, distribute the 3 and get 3x + 3.

 

 

The first term is simplified. Move onto simplifying the second term.

 

Earlier it was stated that one can only multiply an expression by 1, obviously here it was not 1 being multiplied by, but rather (x-1)(x+1) at the very end of this problem this will be fixed and explained.

In order to simply the second term, multiply it by the least common denominator.

 

 

Here the (x+1)’s cancel out.

 

Simplify.

 

 

The second term is fully simplified so put it together with the first term.

 

Simplify like terms.

 

 

Answer!(almost) It is simplified up to here.

 

 

 

Earlier it was covered that one needs to multiply by 1. This expression was multiplied by the least common denominator, which was (x-1)(x+1). This is not equal to 1. What was actually being done earlier was that it was being multiplied by , the only difference is that the bottom part was not shown. Now the reason that one has to include a bottom part is because it has to multiply by 1 and remember that anything divided by itself is 1. Now for the first and only time in this problem, account for the bottom. Take the result (5x +1) and put it over the least common denominator.

 

 

 

 

Answer!

 

 

 

 

Here is a summary of the steps to simply a problem like this.

 

Step 1) Find the least common denominator and multiply everything

by it.

Step 2) Simplify before multiplying, then multiply

Step 3) Simplify the top and put it all over the least common

denominator.

 

 

 

Example

Simplify

 

The least common denominator of this one would be 6x(x-1). Multiply everything by 6x(x-1). Here is the first term.

 

 

 

Notice how everything on the top, except the 5, will cancel out because it is the exact same as everything that is on bottom. Therefore all that is left is the 5. Here is the second term multiplied by the least common denominator

 

 

 

On multiplying these, notice that the (x-1)’s cancel out. That means that all is left is the 2(6x) and the result of that is 12x.

 

Although this is the order that they came in, they need to be put in the correct order.

That would be, the term with the variable goes first and also include the last step of putting it over the least common denominator.

 

Answer!

 

 

 

 

 

 

Example

Simplify

 

The least common denominator of these two would be 20x². Multiply the first term by the least common denominator.

 

 

 

 

 

Before multiplying be sure and simplify all possible. The first thing to simplify is the 20 and the 5. Think of the largest number that can go into 20 and 5, and that would be 5. Factoring 5 out of each term (which leaves a 1 on bottom and a 4 on top) and canceling out the x², since there is one on top and one on bottom.

 

 

 

Multiply this out.

 

 

 

This is the result from the first term being multiplied by the least common denominator.

 

 

 

In the simplification process, notice that the two numbers, 4 and 20, can be simplified because 4 goes into both numbers. The 4 on bottom becomes a 1 and the 20 factors into a 5 (Since 4 goes into 20, 5 times). Also, when looking at the x’s there is one on top and one on bottom that can cancel. Remember that only one x from the top can cancel, since there is only one x on bottom.

 

 

 

Go ahead and multiply what is left. Notice the parentheses around the x + 1. Multiply by the entire term (x + 1), and not just the x or 1.

 

 

 

This is the result of multiplying the second term by the least common denominator so put it with the result from the first term.

 

 

 

 

Simplify by combining the like terms (12x and 5x).

 

 

 

 

The top is simplified. The final step is to put it over the least common denominator.

 

 

 

Answer!

 

 

 

 

 

 

Example

Simplify

 

 

 

 

The least common denominator is (x+2)(x-3). Multiply everything by (x+2)(x-3). Here is the first term multiplied by that.

 

 

 

 

The (x+2)’s cancel out since there is one on top and one on bottom.

 

 

 

Multiply.

 

 

 

 

 

Combine like terms.

 

 

 

 

 

The first term is simplified move on to the multiplication of the second term.

 

 

 

 

The (x-3)’s cancel out.

 

 

 

 

Multiply.

 

 

 

 

 

Simplify.

 

 

 

 

Combine the results of the first and the second term.

 

 

 

 

Simplify.

 

 

 

Put the result on top of the least common denominator.

 

 

 

 

Answer!

 

 

 

 

 

 

 

 

Example

Simplify

 

 

 

The least common denominator is (x-4)(x+1). Multiply the first term by the least common denominator.

 

 

 

 

The (x-4)’s cancel out.

 

 

 

Multiply.

 

 

 

Multiply the second term by the least common denominator. When doing this recognize that the 3 is negative.

 

 

 

 

The (x+1)’s cancel out.

 

 

 

Multiply.

 

 

 

 

Put this together with the first term.

 

 

 

 

 

Simplify by combining like terms.

 

 

 

 

 

Put this over the least common denominator.

 

 

 

 

 

Answer!

 

 

 

 

 

 

Example

Simplify

 

 

 

 

 

The least common denominator is 2x² (x-1). Multiply the first term by the least common denominator.

 

 

 

 

Everything cancels out on top and bottom except the 5.

 

 

 

 

This is all that is left after multiplying the first term because everything cancelled out. The next step is to multiply the second term by the least common denominator.

 

 

 

The (x-1)’s cancel out.

 

 

 

 

Multiply.

 

 

 

It would be natural to think that this is the result of the second term being multiplied by the least common denominator. But remember that there is one thing that has been ignored so far, it is the fact that the second term in the original problem was being subtracted.

 

 

 

This is the subtracting of the second term. Distribute out the negative.

 

 

 

Now the second term is multiplied so combine it with the first term.

 

 

 

Rewrite with the exponents first and put over the least common denominator.

 

 

 

Answer!

 

 

 

 

 

 

 

Example

Simplify

 

 

 

 

The least common denominator of this problem is 10x² (x+3)(x-5). Multiply the first term by the least common denominator.

 

 

 

 

The 10 and 2 will factor down because a 2 goes into both of them. That would make the 2 a 1 and the 10 a 5. Do not include the 1 in the rest of the problem because it is being multiplied by other numbers and 1 multiplied by anything else does not change it. One x on top and one x on bottom cancel out. The (x+3)’s cancel as well as the (x-5)’s.

 

 

Put the 3x-2 in parentheses to make sure and multiply the whole thing (or every term) by 5x.

 

 

 

 

The first term is now multiplied, multiply the second term by the least common denominator.

The first thing to realize is that the 5 and 10 will simplify into 1 and 2, the 2x’s cancel along with the (x+3)’s.

 

 

 

Now multiply the (4x-5) and the 2. Make sure and keep the result in parentheses because the entire results must be multiplied by (x-5).

 

 

 

Multiply.

 

 

 

Remember that the second term is being subtracted.

 

 

 

Distribute out the negative.

 

 

 

Combine this with the result from the first term and put it over the least common denominator.

 

 

 

Answer!