# Add and subtract rational expressions

If the two rational expressions that you want to add or subtract have the same denominator you just add/subtract the numerators which each other.

**Example**

$$\frac{x}{x-1}+\frac{2-x}{x-1}=\frac{x+2-x}{x-1}=\frac{2}{x-1}$$

When the denominators are not the same in all expressions that you want to add or subtract as in the example below you have to find a common denominator. The easiest way to do this is to multiply the denominators with each other, but that might not get the simplest computations and usually requires a lot of simplifying afterwards, but it's a method that always works if you're uncertain. A way to get the usually easiest computations is to find the least common denominator (LCD). The LCD is the least number that is a common multiple of the two or more numbers in the denominator.

**Example**

$$\frac{x-2}{x+1}+\frac{3}{x}$$

The LCD in this case is the same as the multiple of the two denominators i.e.

$$\frac{{\color{green} {x-2}}}{{\color{green} {x+1}}}+\frac{{\color{blue} {3}}}{{\color{blue} {x}}}=\frac{{\color{blue}{ x}}\left ( x-2 \right )}{{\color{blue} {x}}\left ( x+1 \right )}+\frac{3\left ( {\color{green} {x+1}} \right )}{x\left ( {\color{green} {x+1}} \right )}=$$

$$\frac{x\left ( x-2 \right )+3\left ( x+1 \right )}{x\left ( x+1 \right )}=\frac{x^{2}-2x+3x+3}{x^{2}+x}=$$

$$=\frac{x^{2}+x+3}{x^{2}+x} $$

**Video lesson**

Simplify the rational expression

$$\frac{x+3}{4x}-\frac{2}{x-1}$$