Solving Rational Equations  Step by Step 
A rational equation is a type of equation where it involves at least one rational expression, a fancy name for fraction. The best approach to address this type of equation is to eliminate all the denominators using the idea of LCD (least common denominator). By doing so, the leftover equation to deal with is usually either linear or quadratic.
In this lesson, I want to go over ten (10) worked examples with various levels of difficulty. I believe that most of us learn math by looking at many examples. Here we go!
Direction: Solve each rational equation, and make sure you check your answers for extraneous values.
Example 1: Solve the rational equation .
Would it be nice if the denominators are not there? Well, we can't simply vanish them without any valid algebraic step. The approach is to find the Least Common Denominator (also known Least Common Multiple) and use that to multiply both sides of the rational equation. It results to the removal of the denominators,leaving us with regular equations that we already know how to solve such as linear and quadratic. That is the essence of solving rational equations.
Example 2: Solve the rational equation .
The first step in solving rational equation is always to find the "silver bullet" known as LCD. So for this problem, finding the LCD is simple.
Here we go...
 Try to express each denominator as unique powers of prime numbers, variables and/or terms.
 Multiply together the ones with the highest exponents for each unique prime number, variable and/or terms to get the required LCD.

The LCD is 9x. Distribute it to both sides of the equation to eliminate the denominators. 

Simplify. 

To keep the variables on the left side, subtract both sides by 63. 

The resulting equation is just a onestep equation. Divide both sides by the coefficient of x. 

That is it! Check the value x = −39 back into the main rational equation and it should convince you that it works. 
Example 3: Solve the rational equation.
It looks like the LCD is already given. We have a unique and common term (x − 3) for both of the denominators. The number 9 has the trivial denominator of 1 so I will disregard it. Therefore the LCD must be (x − 3).
Example 4: Solve the rational equation .
I hope that you can tell now what's the LCD for this problem by inspection. If not, you'll be fine. Just keep going over few examples and it will make more sense as you go along.
 Try to express each denominator as unique powers of prime numbers, variables and/or terms.
 Multiply together the ones with the highest exponents for each unique prime number, variable and/or terms to get the required LCD.
Example 5: Solve the rational equation.
Focusing on the denominators, the LCD should be 6x. Why?
Remember, multiply together "each copy" of the prime numbers or variables with the highest powers.
Example 6: Solve the rational equation .
Whenever you see a trinomial in the denominator, always factor it out to identify the unique terms. By simple factorization, I found that x^{2} + 4x − 5 = (x + 5) (x −1). Not too bad?
Finding the LCD just like in previous problems...
 Try to express each denominator as unique powers of prime numbers, variables and/or terms. In this case, we have terms in the form of binomials.
 Multiply together the ones with the highest exponents for each unique copy of prime number, variable and/or terms to get the required LCD.

Before I distribute the LCD into the rational equations, factor out the denominators completely.
This aids in the cancellations of the commons terms later. 

Multiply each side by the LCD. 

Wow! It's amazing how quickly the "clutter" of the original problem has been cleaned up. 

Get rid of the parenthesis by the distributive property.
You should end up with a very simple equation to solve. 
Example 7: Solve the rational equation .
Since the denominators are two unique binomials, it makes sense that the LCD is just their product.
Example 8: Solve the rational equation .
This one looks a bit intimidating. But if we stick to the basics, like finding the LCD correctly, and multiplying it across the equation carefully, we should realize that we can control this "beast" quite easily.
 Expressing each denominator as unique powers of terms
 Multiply each unique terms with the highest power to obtain the LCD
Example 9: Solve the rational equation .
Let's find the LCD for this problem, and use it to get rid of all the denominators.
 Express each denominator as unique powers of terms.
 Multiply the each unique term with the highest power to determine the LCD.
Example 10: Solve the rational equation .
Start by determining the LCD. Express each denominator as powers of unique terms. Then multiply together the expressions with the highest exponents for each unique term to get the required LCD.
So then we have...


Factor out the denominators completely. 


Distribute the LCD found above into the rational equation to eliminate all the denominators. 


Distribute the constant into the parenthesis. 


Critical step: We are dealing with a quadratic equation here. Therefore keep everything (both variables and constants) on one side forcing the opposite side to equal zero. 


I can make the left side equal to zero by subtracting both sides by 3x. 


At this point, it is clear that we have a quadratic equation to solve.
Always start with the simplest method before trying anything else. I will utilize factoring method since the trinomial is easily factorable by inspection. 


Yep, we are right! The factors of x^{2}−5x+4 = (x−1) (x−4). You can check it by FOIL method. 


Use the zeroproduct property to solve for x.
Set each factor equal to zero, then solve each simple onestep equation.
Again, always check the solved answers back into the original equations to make sure they are valid. 

