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Solving Rational Equations

A rational expression is an expression which has at least one fractional term in it. An equation containing one or more rational expression is known as rational equation. Rational equation can have one or more than one variables. But here, we are going to discuss rational equation in one variable. Few examples of rational equations are:

  • $\frac{x}{5}$ = $2x+5$
  • $\frac{t+2}{t-1}$ = $\frac{2t}{t-1}$$-4$
  • $\frac{2(x+2)}{3} - \frac{4x}{9}$ = $\frac{1}{x-1}$

How to Solve Rational Equations?

Given below are few cases in solving rational equations.
Case 1: When the equation is in the form $\frac{a}{b}=\frac{c}{d}$, cross multiply equation and find the answer by manipulating the variables.
Case 2: When the equation is in the form $\frac{a}{b}+\frac{c}{d}=\frac{e}{f}$ (or other similar forms), find the LCM (lowest common multiple) of denominators and then simplify.
Case 3: Cross check by substituting the result into the given equation.

We can illustrate above cases with an example:

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

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

LCM (x + 1, x - 1) = x$^2$ - 1

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

$\frac{x^{2}+2x+1-2x^{2}+2x}{x^{2}-1}$ = $-1$
$-x^{2}+4x+1=-x^{2}+1$
4x = 0
x = 0

Cross check: Put x = 0 in the given equation.
$\frac{0+1}{0-1}-\frac{0}{0+1}$ = $-1$
-1 = -1
Therefore, x = 0 is a solution.

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