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.



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.



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