# 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

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