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Subtracting Algebraic Fractions. A simple guide on how to take away fractions with algebra.

Updated on August 15, 2012


Before you attempt the subtraction of algebraic fractions make sure you can do the addition of algebraic fractions. To subtract algebraic fractions follow these 3 easy steps.

Step 1. Make the denominators of both algebraic fractions equal. Do this by multiplying the numerators and denominators of one or both fractions.

Step 2. Put the two fractions together over the same denominator.

Step 3. Expand the brackets on the numerator. Take extra care with the second bracket as you will have to multiply it by a negative number. This is the place where most pupils will go wrong. It is not usually required to expand the double brackets out on the denominator.

Step 4. Simplify the numerator.

Step 5. This last step might not be necessary. Sometimes you can factorise the numerator and cancel factors with the denominator.

Let’s take a look at a couple of examples of subtracting algebraic fractions:

Example 1

Simplify 7/(x-5) – 3/(x+2)

Step 1. Make the denominators of both algebraic fractions equal. Do this by multiplying the numerators and denominators of one or both fractions.

7/(x-5) – 3/(x+2)

= 7(x+2)/[(x-5)(x+2)] – 3(x-5)/[(x+2)(x-5)]

Step 2. Put the two fractions together over the same denominator.

[7(x+2) – 3(x-5)]/[(x+2)(x-5)]

Step 3. Expand the brackets on the numerator. Take extra care with the second bracket as you will have to multiply it by a negative number. This is the place where most pupils will go wrong. It is not usually required to expand the double brackets out on the denominator.

[7x + 14 -3x + 15]/[(x+2)(x-5)]

Step 4. Simplify the numerator

[4x + 29]/[(x+2)(x-5)]

Step 5. This last step might not be necessary. Sometimes you can factorise the numerator and cancel factors with the denominator.

The numerator cannot be factorised so this is your final answer:

[4x + 29]/[(x+2)(x-5)]

Example 2

Simplify 6/(x-3) – 1/(x-4)

Step 1. Make the denominators of both algebraic fractions equal. Do this by multiplying the numerators and denominators of one or both fractions.

6/(x-3) – 1/(x-4)

= 6(x-4)/[(x-3)(x-4)] – 1(x-3)/[(x-4)(x-3)]

Step 2. Put the two fractions together over the same denominator.

[6(x-4) – 1(x-3)]/[(x-4)(x-3)]

Step 3. Expand the brackets on the numerator. Take extra care with the second bracket as you will have to multiply it by a negative number. This is the place where most pupils will go wrong. It is not usually required to expand the double brackets out on the denominator.

[6x - 24 -x + 3]/[(x-4)(x-3)]

Step 4. Simplify the numerator

[5x - 21]/[(x-4)(x-3)]

Step 5. This last step might not be necessary. Sometimes you can factorise the numerator and cancel factors with the denominator.

The numerator cannot be factorised so this is your final answer:

[5x - 21]/[(x-4)(x-3)]

For some help on adding algebraic fractions click on the link below:

Addition Of Algebraic Fractions - how to add fractions with algebra

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