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GMAT: How to Add And Subtract Rational Expressions
Practice Adding And Subtracting Rational Expressions

Adding and subtracting rational expressions can feel daunting, especially when trying to find a common denominator. Let me show you the process I like to use. I think it will make adding and subtracting rational expressions more enjoyable!

Practice Adding And Subtracting Rational Expressions

Introduction

Remember back when we added and subtracted fractions? Well, a rational expression is simply a fraction with 'x's and numbers. We follow the same process for adding and subtracting rational expressions with a little twist. Now we may need to factor and FOIL to simplify the expression.

The process we will follow is:
  • Factor
  • Find the common denominator
  • Rewrite fractions using the common denominator
  • Put the entire numerator over the common denominator
  • Simplify the numerator
  • Factor and cancel, if possible
  • Write the final answer in simplified form

As we get started, let's also remember that to add or subtract fractions, we need a common denominator. Try this mnemonic to help you remember when you need a common denominator and when you don't:

Add Subtract Common Denominators, Multiply Divide None.

Auntie Sits Counting Diamonds, Mother Does Not.

Example #1

Let's look at our first example.

(x + 4)/(3x - 9) + (x- 5)/(6x- 18) First, we need to factor.

(3x - 9) = 3 (x- 3) and (6x - 18) = 6 (x - 3) After we replace the factored terms, our new expression looks like: (x + 4)/3 (x - 3) + (x - 5)/6 (x - 3)

To find our common denominator, we simply write down our denominators. From the first term we have 3 (x- 3) as our denominator. We write that down for our common denominator. When we look at the second expression's denominator, 6 (x - 3), we notice that 6 = 3 ×2. So the second expression has 2 ×3 (x- 3). We already have 3 (x - 3) written, so the only piece not used is 2. We write that down multiplied by 3 (x - 3). Our common denominator will be 2 ×3 (x - 3) or 6 (x - 3).

Our next step is to multiply each piece of the expression so we have 6 (x - 3) as our new denominator. In our first fraction, we need to multiply by 2 over 2. This will give me 2 (x + 4)/2 * 3(x - 3). Looking at the second fraction, I notice I already have 6 (x - 3) in the denominator, so I can leave this one alone.

Now let's write the entire numerator over our common denominator:

2(x + 4) + (x - 5)/6(x - 3) Let's simplify the numerator.

2(x + 4) = 2x + 8 2x + 8 + (x - 5)/6(x - 3) Collect like terms in the numerator.

3x + 3/6(x - 3) Factor the numerator if possible.

3x + 3 = 3 (x + 1)

The 3 over 6 reduces to 1 over 2. There isn't anything to slash or cancel, so we distribute in the numerator and denominator for our final answer: x + 1/2x - 6

Example #2

(x - 2)/(x + 5) + (x^2 + 5x + 6)/(x^2 + 8x + 15) First, we need to factor.

x^2 + 5x + 6 = (x + 3)(x + 2) x^2 + 8x + 15 = (x + 5)(x + 3)

After we replace the factored terms, our new expressions looks like:

(x - 2)/(x + 5) + (x + 3)(x + 2)/(x + 5)(x + 3)

To find our common denominator, we simply write down our denominators. From the first term, we have (x+ 5) as our denominator. In the second term, we have (x + 5) and (x + 3). Since we already have (x + 5) written as part of our common denominator, we will just write (x + 3). So, our common denominator is (x + 5)(x + 3).

Our next step is to multiply each piece of the expression, so we have (x + 5)(x + 3) as our new denominator. In the first fraction, we need to multiply by (x + 3) over (x + 3). This will give us (x - 2)(x + 3)/(x + 5)(x + 3) as our first fraction. Looking at the second fraction, I notice I already have (x + 5)(x + 3) in the denominator, so I can leave this one alone.

Now, let's write the entire numerator over our common denominator.

((x - 2)(x + 3) + (x + 3)(x + 2))/(x + 5)(x + 3) Let's simplify the numerator by writing the numerator over our common denominator and FOIL.

(x - 2)(x + 3) = (x^2 + x - 6) and (x + 3)(x + 2) = (x^2 + 5x + 6) Collect like terms in the numerator.

2x^2 + 6x Factor the numerator if possible.

2x(x + 3) Our expression now looks like:

2x(x + 3)/(x + 5)(x + 3) We can slash, or cancel, (x + 3) over (x + 3).

This gives us our final answer, 2x/(x + 5).

Example #3

(x^2 + 12x + 36)/(x^2 - x - 6) + (x + 1)/(3 - x) First, we need to factor. (x^2 + 12x + 36) = (x + 6)(x + 6) (x^2 - x + 6) = (x - 3)(x + 2)

After we replace the factored terms, our new expressions looks like: (x + 6)(x + 6)/(x - 3)(x + 2)) + (x + 1)/(3 - x)

To find our common denominator, we simply write down our denominators. From the first term, we have (x - 3)(x + 2) as our denominator. In the second term, we have (3 - x). I could write (3 - x) as part of the common denominator, but I know that -1 * (x - 3) = (3 - x). So, now it will match with the denominator (x - 3).

Now, our expression looks like:

(x + 6)(x + 6)/(x - 3)(x + 2)) + (x + 1)/-1(x - 3 And that -1? It can be put into the numerator. Remember, 1/-1 = -1/1 = -1. It doesn't matter where I put the -1 in the fraction as long as I have a +1 to match it.

So, our common denominator is (x - 3)(x + 2).

In the first fraction, I already have the common denominator (x - 3)(x + 2), so I leave that one alone. In the second fraction, I need to multiply by (x + 2) over (x + 2). This gives us the common denominator of (x - 3)(x + 2).

Our expression now looks like:

(x + 6)(x + 6)/(x - 3)(x + 2) + (-1)(x + 1)(x + 2)/(x - 3)(x + 2) Let's simplify the numerator by writing the numerator over our common denominator and using FOIL, which is First Outside Inside Last.

(x + 6)(x + 6) = x^2 + 12x + 36 and

(-1)(x + 1)(x + 2) = (-1)(x^2 + 3x + 2) = -x^2 - 3x- 2 Collect like terms in the numerator. Our expression now looks like: (9x + 34)/(x - 3)(x + 2)

The numerator doesn't factor, so our last step is to FOIL the denominator. Our final answer is (9x + 34)/(x^2 - x - 6).

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