Addition and Subtraction of Rational Expressions

A rational expression is a fraction in which the numerator and denominator are polynomials. If P(x) and Q(x) are polynomials, then a rational expression is defined as:

, where Q(x) ≠ 0.

We can perform various operations on rational expressions just like fractions. So, to add and subtract rational expressions, we use the same rules for adding and subtracting fractions as for adding and subtracting just numbers.

Conditions for Adding and Subtracting

For adding and subtracting rational expressions, go through the following steps:

Step 1: Factorize the denominator.

Step 2: Find the least common multiple of the denominator and rewrite each fraction with the common denominator.

Step 3: Add or subtract the numerators of the two rational expressions.

Step 4: If possible, then factorize again.

Step 5: Write the final answer in its simplified form.

Example: Solve the following expression-

Step 1: Fatorise the denominator
The denominators are already factored. The first denominator is 4x, and the second denominator is 2x.

Step 2: Find the least common denominator (LCD)
The denominators are 4x and 2x. The least common denominator between 4x and 2x is 4x because 4 is the least common multiple of 4 and 2.

Step 3: Rewrite each fraction with the common denominator
We will rewrite both fractions with the denominator 4x.

For the first fraction , it already has the denominator 4x, so we don't need to change it.

For the second fraction ​, we need to multiply both the numerator and denominator by 2 to make the denominator 4x.

Now, the expression looks like:

Step 4: Subtract the numerators

Since both fractions have the same denominator, we can subtract the numerators:

Now, simplify the numerator by distributing the negative sign to the second term:

(x + 3) - (4x - 2) = x + 3 - 4x + 2 = -3x + 5

Thus, the expression becomes:

of

Solved Examples

Example 1: Solve 5/(3y + 4) + (y - 6)/(3y + 4). 

Given expression:
5/(3y + 4) + (y - 6)/(3y + 4)

We can observe that both fractions have a common denominator.

5/(3y + 4) +(y - 6)/(3y + 4) = {5 + (y - 6)}/(3y + 4)
= (y - 1)/(3y + 4)

Thus, 5/(3y + 4) + (y - 6)/(3y + 4) = (y - 1)/(3y + 4)

Example 2: Solve (2x + 7)/(x + 5) - (6x - 1)/(x - 2).

Given expression:
(2x + 7)/(x + 5) - (6x - 1)/(x - 2)

We can observe that both fractions don't have a common denominator

So, find the least common multiple of the denominators, i.e., (x + 5) and (x - 2): (x + 5)(x - 2)

Now, (2x + 7)/(x + 5) - (6x - 1)/(x - 2)
[(2x + 7)(x - 2)/(x + 5)(x - 2)] - [(6x - 1)(x + 5)/(x + 5)(x - 2)]
[(2x² - 4x + 7x - 14)/(x + 5)(x - 2)] - [(6x² + 30x - x - 5)/(x + 5)(x - 2)]
[(2x² + 3x - 14)/(x + 5)(x - 2)] - [(6x² + 29x - 5)/(x + 5)(x - 2)] 
[(2x² + 3x - 14) - (6x² + 29x - 5)]/(x + 5)(x - 2)
(-4x² - 26x - 9)/(x + 5)(x - 2)
-(4x² + 26x + 9)/(x + 5)(x - 2)

Hence, (2x + 7)/(x + 5) - (6x - 1)/(x - 2) = -(4x² + 26x + 9)/(x + 5)(x - 2)

Example 3: Solve 4x/(7x - 2) + (19 - 2x)/(7x - 2).

Given expression:
4x/(7x - 2) + (19 - 2x)/(7x - 2)

We can observe that both fractions have a common denominator

4x/(7x-2) + (19 - 2x)/(7x-2) = (4x + 19 - 2x)/(7x - 2)
= (2x + 19)/(7x-2)

Hence, 4x/(7x - 2) + (19 - 2x)/(7x - 2) = (2x + 19)/(7x - 2)

Example 4: Solve 8/(x-3) - 4x/(x²-9).

Given expression:
8/(x - 3) - 4x/(x² - 9)

We can observe that both fractions don't have a common denominator

(x² - 9) = (x - 3)(x + 3)   {Since, a² - b² = (a - b)(a + b)}

So, find the least common multiple of the denominators, i.e., (x - 3) and (x - 3)(x + 3): (x - 3)(x + 3)

8/(x - 3) - 4x/(x² - 9) = [8(x + 3)/(x - 3)(x + 3)] - [4x/(x - 3)(x + 3)]
[8x + 24/(x² - 9)] - [4x/(x² - 9)]
(8x + 24 - 4x)/(x² - 9)
(4x + 24)/(x² - 9)
4(x + 6)/(x² - 9)

Hence, (8/x-3)-(4x/x²-9) = 4(x + 6)/(x²-9)

Example 5: Solve (6u - v/7u) + (3u+5v/8v).

Given expression:

(6u - v/7u) + (3u + 5v/8v)

We can observe that both fractions don't have a common denominator

So, find the least common multiple of the denominators, i.e., 7u and 8v: 56uv

(6u - v/7u) + (3u + 5v/8v)
= [(6u - v)(8v)/(7u)(8v)] - [(3u + 5v)(7u)/(8v)(7u)]
= [(48uv - 8v²)/56uv] - [(21u² + 35uv)/56uv]
= [(48uv - 8v²) - (21u² + 35uv)/56uv]
= (-8v2 + 13uv - 21u²)/56uv
= -(21u² - 13uv + 8v²)/56uv

Hence, (6u - v/7u) + (3u+5v/8v) = -(21u²-13uv+8v²)/56uv

Practice Problem

1. Solve

2. Solve:

3. Solve:

4. Solve:

5. Solve:

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