How to Solve a Rational Equation

A Rational Equation is an Equation that has Rational-Terms, in which the Numerators and the Denominators of those Terms are either Constants/Variables. An Example of a Rational Equation is,...(2/x)+(3/4)=1. Another Example is...(x-3)/2 = 3/(x+2). In this Article we will show a Moderately Easy method to solve these examples, and this Method can also be applied to other similar Rational Equations.

Things You'll Need

• Paper and
• Pencil.

Instructions

1. To solve for x, in the Equation (2/x)+(3/4)=1, we should first assume that x ≠ 0, since x is a Denominator, and division by zero ( 0 ), is not defined.

2. There are several different ways in which this problem can be solved. One way (a very good way ), is to clear fractions, that is, try to rewrite the Rational Equation so that there are no Denominators, or the Denominators are all equal to one ( 1 ).

3. In order to clear fractions, we find the Least Common Denominator (LCD).
   ( Please see the Article 'How to Find the LCD' by this same Author, Z-MATH ), and Multiply each term of the Rational Equation by the LCD. The LCD for this Equation, (2/x)+(3/4)=1, is 4x.

4. We now Multiply Each term of the Rational Equation (2/x)+(3/4)=1, by 4x.
   That is,....(2/x)4x + (3/4)4x = (1)4x. Which is equal to the following:
   (8x)/x + (12x)/4 = 4x. since x ≠ 0, then the first term can be reduced to lowest terms by dividing x by x giving us the term 8, and similarly the second term would be 3x, and the third term in the Equation is 4x.

5. So the Equation (2/x)+(3/4)=1, can be expressed as,...8 + 3x = 4x. We now Subtract 3x from both sides of the Equation and the result is ...
   8 + 3x - 3x = 4x - 3x which is equal to 8 = x. The solution to the Rational Equation (2/x)+(3/4)=1, is x = 8.

6. We check to see if the solution x = 8, is the correct one by substituting, x = 8 into (2/x)+(3/4)=1. Here we see that (2/8)+(3/4) is equal to (1/4)+(3/4)= (4/4)=1. So x = 8, is the correct Answer.

7. We are going to use the same Method of Clearing Fractions that we used in solving the above problem, to solve the second example. The problem is,... (x-3)/2 = 3/(x+2). We should assume that x ≠ -2, since the Denominator (x+2) will be Zero (0), if x = -2, and again division by Zero (0) is not defined.

8. The LCD for the Rational Equation: (x-3)/2 = 3/(x+2), is 2(x+2). We will now multiply each term of the Rational Equation by this LCD. That is,.......... 2(x+2)(x-3)/2 = 2(x+2)3/(x+2), which is equal to
   (x+2)(x-3)2/2 = (2)(3)(x+2)/(x+2) which is (x+2)(x-3) = (2)(3), by reducing each term to lowest term.

9. SO The Rational Equation (x-3)/2 = 3/(x+2) can be expressed as
   (x+2)(x-3) = 6, which is x^2-x-6=6, which equals x^2-x-12=0. This is a Quadractic Equation which can be solved by factoring, so we have
   (x+3)(x-4)=0. That is x = -3 or x = 4. (Please see the Article 'How to solve a Quadratic Equation by factoring' by this same author, Z-MATH).

10. WE now check both solutions, x=-3, then x=4, to see if one, or both, or neither, solve the original Rational Equation. First we check x=-3.
    by substituting x=-3, into (x-3)/2 = 3/(x+2), we get (-3-3)/2 = -6/2=-3.
    and 3/(-3+2) = 3/(-1) = -3. So x=-3 is a solution of the Equation.
    Similarly by substituting x=4, we get (4-3)/2 = 1/2, and 3/(4+2) = 3/6 = 1/2. So x=4 is also a solution to the Rational Equation.