If you're seeing this message, it means we're having trouble loading external resources on our website.

# 有理表达式的加减

## 热身： $\frac{3}{x-2}-\frac{2}{x+1}$‍

$\begin{array}{rl}& \phantom{=}\frac{3}{x-2}-\frac{2}{x+1}\\ \\ & =\frac{3}{x-2}\left(\frac{x+1}{x+1}\right)-\frac{2}{x+1}\left(\frac{x-2}{x-2}\right)\\ \\ & =\frac{3\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}-\frac{2\left(x-2\right)}{\left(x+1\right)\left(x-2\right)}\\ \\ & =\frac{3\left(x+1\right)-2\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}\\ \\ & =\frac{3x+3-2x+4}{\left(x-2\right)\left(x+1\right)}\\ \\ & =\frac{x+7}{\left(x-2\right)\left(x+1\right)}\end{array}$

### 看看你的知识掌握地如何

$\frac{5x}{x+3}+\frac{4}{x+2}=$

## 最小公分母

### 数字分数

$\begin{array}{rl}& \phantom{=}\frac{3}{4}+\frac{1}{6}\\ \\ & =\frac{3}{2\cdot 2}+\frac{1}{2\cdot 3}\\ \\ & =\frac{3}{2\cdot 2}\left(\frac{3}{3}\right)+\frac{1}{2\cdot 3}\left(\frac{2}{2}\right)\\ \\ & =\frac{9}{12}+\frac{2}{12}\\ \\ & =\frac{11}{12}\end{array}$

### 变量表达式

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

$\underset{\begin{array}{rl}& \text{需要一个}\\ & \text{因式}\\ & \left(x+3\right)\end{array}}{\underset{⏟}{\frac{2}{\left(x-2\right)\left(x+1\right)}}}+\underset{\begin{array}{rl}& \text{需要一个}\\ & \text{因式}\\ & \left(x-2\right)\end{array}}{\underset{⏟}{\frac{3}{\left(x+1\right)\left(x+3\right)}}}$

$\begin{array}{rl}& \phantom{=}\frac{2}{\left(x-2\right)\left(x+1\right)}+\frac{3}{\left(x+1\right)\left(x+3\right)}\\ \\ & =\frac{2}{\left(x-2\right)\left(x+1\right)}\left(\frac{x+3}{x+3}\right)+\frac{3}{\left(x+1\right)\left(x+3\right)}\left(\frac{x-2}{x-2}\right)\\ \\ & =\frac{2\left(x+3\right)}{\left(x-2\right)\left(x+1\right)\left(x+3\right)}+\frac{3\left(x-2\right)}{\left(x+1\right)\left(x+3\right)\left(x-2\right)}\\ \\ & =\frac{2\left(x+3\right)+3\left(x-2\right)}{\left(x-2\right)\left(x+1\right)\left(x+3\right)}\\ \\ & =\frac{2x+6+3x-6}{\left(x-2\right)\left(x+1\right)\left(x+3\right)}\\ \\ & =\frac{5x}{\left(x-2\right)\left(x+1\right)\left(x+3\right)}\end{array}$

### 看看你对知识掌握得如何

$\frac{1}{x\left(x-6\right)}+\frac{3}{\left(x+1\right)\left(x-6\right)}=$

$\frac{3x}{2\left(x-1\right)}-\frac{4}{\left(x-1\right)\left(x+2\right)}=$

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

## 为什么要使用最小公分母？

$\begin{array}{rl}\frac{3}{4}+\frac{1}{6}& =\frac{3}{4}\left(\frac{6}{6}\right)+\frac{1}{6}\left(\frac{4}{4}\right)\\ \\ & =\frac{18}{24}+\frac{4}{24}\\ \\ & =\frac{22}{24}\\ \\ & =\frac{11}{12}\end{array}$