Resolver ecuaciones racionales

Resolver ecuaciones racionales

A rational equation is a type of equation where it involves at least one rational expression, a fancy name for a fracción. 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!



Examples of How to Solve Rational Equations

Ejemplo 1: Solve the rational equation below and make sure you check your answers for extraneous values.

Resolver ecuaciones racionales

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 in 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.


  • The LCD is 6x. I will multiply both sides of the rational equation by 6x to eliminate the denominators. That’s our goal anyway – to make our life much easier.
Resolver ecuaciones racionales
  • You should have something like this after distributing the LCD.
Resolver ecuaciones racionales
  • I decided to keep the variable x on the right side. So remove the -5x on the left by adding both sides by 5x.
Resolver ecuaciones racionales
  • Simplify. It’s obvious now how to solve this one-step equation. Divide both sides by the coefficient of 5x.
Resolver ecuaciones racionales
  • Yep! The final answer is x = 2 after checking it back into the original rational equation. It yields a true statement.

Always check your “solved answers” back into the original equation to exclude extraneous solutions. This is a critical aspect of the overall approach when dealing with problems like Rational Equations and Radical Equations.



Ejemplo 2: Solve the rational equation below and make sure you check your answers for extraneous values.

Resolver ecuaciones racionales

The first step in solving a rational equation is always to find the “silver bullet” known as LCD. So for this problem, finding the LCD is simple.

Aquí vamos.

Try to express each denominator as unique powers of prime numbers, variables and/or terms.

Resolver ecuaciones racionales

Multiply together the ones with the highest exponents for each unique prime number, variable and/or terms to get the required LCD.


Resolver ecuaciones racionales
  • The LCD is 9x. Distribute it to both sides of the equation to eliminate the denominators.
Resolver ecuaciones racionales
  • Simplificar.
Resolver ecuaciones racionales
  • To keep the variables on the left side, subtract both sides by 63.
Resolver ecuaciones racionales
  • The resulting equation is just a one-step equation. Divide both sides by the coefficient of x.
Resolver ecuaciones racionales
  • That is it! Check the value x = - ,39 back into the main rational equation and it should convince you that it works.
Resolver ecuaciones racionales

Ejemplo 3: Solve the rational equation below and make sure you check your answers for extraneous values.


Resolver ecuaciones racionales

It looks like the LCD is already given. We have a unique and common term left( {x - 3} right) for both of the denominators. The number 9 has the trivial denominator of 1 so I will disregard it. Therefore the LCD must be left( {x - 3} right).

  • The LCD here is left( {x - 3} right). Use it as a multiplier to both sides of the rational equation.
Resolver ecuaciones racionales
  • I hope you get this linear equation after performing some cancellations.

Distribute the constant 9 into left( {x - 3} right).

Resolver ecuaciones racionales
  • Combine the constants on the left side of the equation.
Resolver ecuaciones racionales
  • Simplificar
Resolver ecuaciones racionales
  • Move all the numbers to the right side by adding 21 to both sides.
Resolver ecuaciones racionales
  • Simplificar
Resolver ecuaciones racionales
  • Not too bad. Again make it a habit to check the solved “answer” from the original equation.
Resolver ecuaciones racionales

It should work so yes, x = 2 is the final answer.


Ejemplo 4: Solve the rational equation below and make sure you check your answers for extraneous values.

Resolver ecuaciones racionales

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 a 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.

Resolver ecuaciones racionales

Multiply together the ones with the highest exponents for each unique prime number, variable and/or terms to get the required LCD.

Resolver ecuaciones racionales
  • The LCD is 4left( {x + 2} right). Multiply each side of the equations by it.
Resolver ecuaciones racionales
  • After careful distribution of the LCD into the rational equation, I hope you have this linear equation as well.

Nota rápida: If ever you’re faced with leftovers in the denominator after multiplication, that means you have an incorrect LCD.

Now, distribute the constants into the parenthesis on both sides.

Resolver ecuaciones racionales
  • Combine the constants on the left side to simplify it.
Resolver ecuaciones racionales
  • At this point, make the decision where to keep the variable.
Resolver ecuaciones racionales
  • Keeping the x to the left means we subtract both sides by 4.
Resolver ecuaciones racionales
  • Simplificar
Resolver ecuaciones racionales
  • Add both sides by 3x.
Resolver ecuaciones racionales
  • That’s it. Check your answer to verify its validity.
Resolver ecuaciones racionales

Ejemplo 5: Solve the rational equation below and make sure you check your answers for extraneous values.

Resolver ecuaciones racionales

Focusing on the denominators, the LCD should be 6x. Why?

Resolver ecuaciones racionales

Remember, multiply together “each copy” of the prime numbers or variables with the highest powers.

Resolver ecuaciones racionales
  • The LCD is 6x. Distribute to both sides of the given rational equation.
Resolver ecuaciones racionales
  • It should look like after careful cancellation of similar terms.

Distribute the constant into the parenthesis.

Resolver ecuaciones racionales
  • The variable x can be combined on the left side of the equation.
Resolver ecuaciones racionales
  • Since there’s only one constant on the left, I will keep the variable x to the opposite side.
Resolver ecuaciones racionales
  • So I subtract both sides by 5x.
Resolver ecuaciones racionales
  • Divide both sides by -2 to isolate x.
Resolver ecuaciones racionales
  • Yep! We got the final answer.
Resolver ecuaciones racionales

Ejemplo 6: Solve the rational equation below and make sure you check your answers for extraneous values.

Resolver ecuaciones racionales

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 = left( {x + 5} right)left( {x - 1} right). 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.

Resolver ecuaciones racionales

Multiply together the ones with the highest exponents for each unique copy of a prime number, variable and/or terms to get the required LCD.

Resolver ecuaciones racionales
  • Before I distribute the LCD into the rational equations, factor out the denominators completely.

This aids in the cancellations of the commons terms later.

Resolver ecuaciones racionales
  • Multiply each side by the LCD.
Resolver ecuaciones racionales
  • Wow! It’s amazing how quickly the “clutter” of the original problem has been cleaned up.
Resolver ecuaciones racionales
  • Get rid of the parenthesis by the distributive property.

You should end up with a very simple equation to solve.

Resolver ecuaciones racionales

Ejemplo 7: Solve the rational equation below and make sure you check your answers for extraneous values.

Resolver ecuaciones racionales

Since the denominators are two unique binomials, it makes sense that the LCD is just their product.

  • The LCD is left( {x + 5} right)left( {x - 5} right). Distribute this into the rational equation.
Resolver ecuaciones racionales
  • It results in a product of two binomials on both sides of the equation.

It makes a lot of sense to perform the FOIL method. Does that ring a bell?

Resolver ecuaciones racionales
  • I expanded both sides of the equation using FOIL. You should have a similar setup up to this point. Now combine like terms (the x) in both sides of the equation.
Resolver ecuaciones racionales
  • What’s wonderful about this is that the squared terms are exactly the same! They should cancel each other out. We could have bumped into a problem if their signs are opposite.
Resolver ecuaciones racionales
  • Subtract both sides by {x^2}.
Resolver ecuaciones racionales
  • The problem is reduced to a regular linear equation from a quadratic.
Resolver ecuaciones racionales
  • To isolate the variable x on the left side implies adding both sides by 6x.
Resolver ecuaciones racionales
  • Move all constant to the right.
Resolver ecuaciones racionales
  • Suma ambos lados por 30.
Resolver ecuaciones racionales
  • Finally, divide both sides by 5 and we are done.
Resolver ecuaciones racionales

Ejemplo 8: Solve the rational equation below and make sure you check your answers for extraneous values.

Resolver ecuaciones racionales

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

Resolver ecuaciones racionales

Multiply each unique terms with the highest power to obtain the LCD

Resolver ecuaciones racionales
  • Factoriza los denominadores.
Resolver ecuaciones racionales
  • Multiply both sides by the LCD obtained above.

Be careful now with your cancellations.

Resolver ecuaciones racionales
  • You should end up with something like this when done right.
Resolver ecuaciones racionales
  • Next step, distribute the constants into the parenthesis.

This is getting simpler in each step!

I would combine like terms on both sides also to simplify further.

Resolver ecuaciones racionales
  • This is just a multi-step equation with variables on both sides. Easy!
Resolver ecuaciones racionales
  • To keep x on the left side, subtract both sides by 10x.
Resolver ecuaciones racionales
  • Move all the pure numbers to the right side.
Resolver ecuaciones racionales
  • Resta ambos lados por 15.
Resolver ecuaciones racionales
  • A simple one-step equation.
Resolver ecuaciones racionales
  • Divide both sides by 5 to get the final answer. Again, don’t forget to check the value back into the original equation to verify.
Resolver ecuaciones racionales

Ejemplo 9: Solve the rational equation below and make sure you check your answers for extraneous values.

Resolver ecuaciones racionales

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.

Resolver ecuaciones racionales

Multiply each unique term with the highest power to determine the LCD.

Resolver ecuaciones racionales
  • Factor out the denominators completely
Resolver ecuaciones racionales
  • Distribute the LCD found above into the given rational equation to eliminate all the denominators.
Resolver ecuaciones racionales
  • We reduced the problem into a very easy linear equation. That’s the “magic” of using LCD.

Multiply the constants into the parenthesis.

Resolver ecuaciones racionales
  • Combine similar terms
Resolver ecuaciones racionales
  • Simplificar
Resolver ecuaciones racionales
  • Keep the variable to the left side by subtracting x on both sides.
Resolver ecuaciones racionales
  • Keep constants to the right.
Resolver ecuaciones racionales
  • Add both sides by 8 to solve for x. Done!
Resolver ecuaciones racionales

Ejemplo 10: Solve the rational equation below and make sure you check your answers for extraneous values.

Resolver ecuaciones racionales

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.

Resolver ecuaciones racionales

So then we have,

Resolver ecuaciones racionales
  • Factor out the denominators completely.
Resolver ecuaciones racionales
  • Distribute the LCD found above into the rational equation to eliminate all the denominators.
Resolver ecuaciones racionales
  • Distribute the constant into the parenthesis.
Resolver ecuaciones racionales
  • 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.
Resolver ecuaciones racionales
  • I can make the left side equal to zero by subtracting both sides by 3x.
Resolver ecuaciones racionales
  • 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 the factoring method of the form x^2+bx+c=0 since the trinomial is easily factorable by inspection.

Resolver ecuaciones racionales
  • The factors of {x^2} - 5x + 4 = left( {x - 1} right)left( {x - 4} right). You can check it by the FOIL method.
Resolver ecuaciones racionales
  • Use the Zero Product Property to solve for x.

Set each factor equal to zero, then solve each simple one-step equation.

Again, always check the solved answers back into the original equations to make sure they are valid.

Resolver ecuaciones racionales

Practica con hojas de trabajo

Usted también puede estar interesado en:

Sumar y restar expresiones racionales

Multiplicar expresiones racionales

Resolver desigualdades racionales



Añade un comentario de Resolver ecuaciones racionales
¡Comentario enviado con éxito! Lo revisaremos en las próximas horas.

End of content

No more pages to load