How to Add and Subtract Rational Expressions

By knowing a few simple rules you can easily add and subtract Rational Expressions. In this blog post, we will introduce you step by step guide on how to add and subtract rational expressions.

Related Topics

• How to Multiply Rational Expressions
• How to Divide Rational Expressions
• How to Solve Rational Equations
• How to Simplify Complex Fractions
• How to Graph Rational Expressions

A step-by-step guide to Adding and Subtracting Rational Expressions

For adding and subtracting rational expressions:

• Find the least common denominator (LCD).
• Write each expression using the LCD.
• Add or subtract the numerators.
• Simplify as needed

Examples

Adding and Subtracting Rational Expressions – Example 1:

Solve: \(\frac{4}{2x+3}+\frac{x-2 }{2x+3}=\)

Solution:

The denominators are equal. Then, use fractions addition rule: \(\frac{a}{c}±\frac{b}{c}=\frac{a ± b}{c}→\frac{4}{2x+3}+\frac{x-2}{2x+3}=\frac{4+(x-2) }{2x+3}=\frac{x+2}{2x+3}\)

Adding and Subtracting Rational Expressions – Example 2:

Solve \(\frac{x + 4}{x – 5}+\frac{x – 4}{x + 6}\)=

Solution:

Find the least common denominator of\( (x-5)\) and \((x+6): (x-5)(x+6) \)
Then: \(\frac{x + 4}{x – 5}+\frac{x – 4}{x + 6}=\frac{(x+4)(x+6)}{(x-5)(x+6)}+\frac{(x – 4)(x-5)}{(x + 6)(x-5)}=\frac{(x+4)(x+6)+(x – 4)(x-5)}{(x + 6)(x-5)}\)
Expand: \((x+4)(x+6)+(x-4)(x-5)=2x^2+x+44\)
Then: \(\frac{(x+4)(x+6)+(x – 4)(x-5)}{(x + 6)(x-5)}=\frac{2x^2+x+44}{(x +6)(x-5)}=\frac{2x^2+x+44}{x^2+x-30}\)

Adding and Subtracting Rational Expressions – Example 3:

Solve: \(\frac{3}{x+4}+\frac{x-2 }{x+4}\)=

Solution:

Use fraction addition rule: \(\frac{a}{c}±\frac{b}{c}=\frac{a ± b}{c}→\frac{3}{x+4}+\frac{x-2}{x+4}=\frac{3+(x-2) }{x+4}=\frac{x+1}{x+4}\)

Adding and Subtracting Rational Expressions – Example 4:

Solve: \(\frac{x + 4}{x – 8}+ \frac{x }{x + 6}\)=

Solution:

Least common denominator of \((x-8)\) and \((x+6): (x-8)(x+6)\)
Then: \(\frac{(x+4)(x+6)}{(x-8)(x+6)}+\frac{x(x-8)}{(x + 6)(x-8)}=\frac{(x+4)(x+6)+x(x-8)}{(x + 6)(x-6)}\)
Expand: \((x+4)(x+6)+x(x-8)=2x^2+2x+24\)
Then: \(\frac{x + 4}{x – 8}+ \frac{x }{x + 6}=\frac{2x^2+2x+24}{(x +6)(x-8)}\)

Exercises for Simplifying Fractions

Add and Subtract Rational Expressions.

1. \(\color{blue}{\frac{3}{x+1}-\frac{4x}{x+1}=}\)
2. \(\color{blue}{\frac{x+8}{x+1}+\frac{x-9}{x+2}=}\)
3. \(\color{blue}{\frac{6x}{x+5}+\frac{x+2}{x+7}=}\)
4. \(\color{blue}{\frac{15}{x+6}-\frac{x+1}{x^{2}-36}=}\)
5. \(\color{blue}{\frac{x+4}{x+3}-\frac{5x}{x-3}=}\)
6. \(\color{blue}{\frac{x+8}{x-4}+\frac{x-5}{x^{2}-16}=}\)

1. \(\color{blue}{\frac{3-4x}{x+1}}\)
2. \(\color{blue}{\frac{2x^2+2x+7}{(x+1)(x+2)}}\)
3. \(\color{blue}{\frac{7x^2+49x+10}{(x+5)(x+7)}}\)
4. \(\color{blue}{\frac{14x-91}{(x+6)(x-6)}}\)
5. \(\color{blue}{\frac{-4x^2-14x-12}{(x+3)(x-3)}}\)
6. \(\color{blue}{\frac{x^2+13x+27}{(x+4)(x-4)}}\)