Solving rational equations

An equation that contains at least on rational expression is called a rational equation. You solve a rational equation as you solve any other equation.

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Example

$$\frac{x-3}{2x}+\frac{3}{x}=5\Rightarrow \frac{x-3}{2x}+\frac{{\color{blue} 2}\cdot 3}{{\color{blue} 2}\cdot x}=5\Rightarrow$$

$$\frac{x-3+6}{2x}=5\Rightarrow \frac{x+3}{2x}=5\Rightarrow$$

$$\frac{{\color{red}{ \not}}{2x}\left (x+3 \right )}{{\color{red} {\not}}{2x}}=5\cdot 2x=10x\Rightarrow$$

$$x+3=10x\Rightarrow {\color{red} {\not}}{x}+3-{\color{red} {\not}}{x}=10x-x\Rightarrow$$

$$3=9x\Rightarrow \frac{3}{9}=\frac{9x}{9}\Rightarrow \frac{1}{3}=x$$

Don't forget to check your solution and make sure that your answer is not an excluded value. When multiplying two rational expressions there is always a risk of getting false solutions or extraneous solutions.

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Video lesson

Solve the rational equation

$$\frac{x+3}{x}-\frac{x-1}{x+3}=23$$