# How To Solve Rational Equations

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Definition: - The equation which includes the rational expressions s is known as rational equation.

Examples of rational equations: -
(x – 1) (2 x + 2) / (x + 3) = x / 3
(5 x + 1) / (8 x – 3) – 1 = x / (8 x – 3) + 3 / (2 x + 5)
X / 5 = (2 + x) / 8

Question 1: - Solve the following rational equation for x:
(x + 1) / (x ^2 + 2 x + 1) = (2 x – 3) / (x + 1)

Solution: -
i)             Factorize the denominator of the L.H.S. rational expression
(x ^2 + 2 x + 1) = (x ^2 + 2 * x * 1 + 1^2) = (x + 1) ^2

ii)            Now you can cancel out the numerator to the numerator and denominator to the denominator.
(x + 1) / (x ^2 + 2 x + 1) = (2 x – 3) / (x + 1)
(x + 1) / (x + 1) ^ 2 = (2 x – 3) / (x +1)
Cancel out (x + 1) from the numerator and denominator of the L.H.S. rational expression.
1 / (x + 1) = (2 x – 3) / (x + 1).
Again cancel out (x + 1) from the denominator of the L.H.S. and from denominator of the R.H.S.
1 = 2 x – 3
1+3 = 2x-3+3
4=2x
X=4

Question 2: - If 1/x = x/4, find x.

Solution: -
1/x=x/4
X^2 = 4
X= ± 2