example of problem solving involving rational equations

Solving Rational Equations · Examples

Listen up, fraction fans! In today’s lesson, you will learn and practice solving rational equations. As you will see, these are any equation involving a fraction, also known as a rational number in math talk!

By the end, you will know the difference between rational and irrational numbers and have two tricks for solving rational equations.

You could even tackle one of the tricky challenges to form a rational equation using the Pythagorean theorem , or to simplify an expression involving some radicals!

What is a Rational Equation? How to Solve Rational Equations Step 1: Eliminate the Denominators Step 2: Simplify the Equation Step 3: Solve the Equation Step 4: Check Solutions Practice & Challenges Question 1 Question 2 Challenge 1 Challenge 2 Worksheet To Sum Up (Pun Intended!)

What is a Rational Equation?

A rational equation is simply an equation involving a rational number.

A ratio -nal number can be written as a ratio of two integers – an irratio -nal number cannot.

Most of the numbers you know and love such as \(\Large\frac{2}{7}\), \(\Large\frac{1}{2}\) and \(-\Large\frac{20817}{43}\) are rational. Some common irrational numbers are π, \(\sqrt{2}\) and Euler’s number, e. These cannot be written as a fraction of integers.

Numberphile has an interesting video about All the Numbers , which categorizes number types, including rational and irrational numbers .

Technically speaking, basic equations like x+2=5 are rational because each term is a rational number. However, the rational equations you will solve today won’t be so easy!

An example of what you will more likely see in an exam is something like this:

Each term is shown as a fraction.

Rational equations can also include radicals:

Or other operations such as division:

Luckily, the technique you learn now will work for every type of rational equation!

How to Solve Rational Equations

The method to solve these equations is pretty much the same for every type of rational equation. You’ll see questions of varying difficulty in this lesson; don’t be afraid to tackle the challenges later on!

Step 1: The Denominator Elimination Round!

First, you need to deal with the elephant in the room: what should you do with the denominators!?

Solving rational equations is just like solving any other equation once you complete this step.

If it’s a simple case, where you have one fraction being equal to one other fraction, you can cross multiply .

Multiply both sides by the values of both denominators. In this example, both sides are multiplied by 3, then 5.

The 3 cancels with the left denominator and the 5 cancels with the right denominator, leaving you with 5(x+4)=3×2.

See why it’s called cross multiplying?

The product of the left denominator and right numerator equals the product of the right denominator and left numerator !

The more general way to deal with the denominators is to find their lowest common multiple (LCM) . This is the smallest number which all denominators divide neatly, leaving no remainder.

If you cannot find the LCM by inspection – if you cannot “just see it” – you need to factor every denominator like you would with a polynomial.

If you have more than one constant term, you may need to find their prime factors.

The LCM is the smallest combination of each denominator’s factors.

You’ll now see a worked example to illustrate!

Remember, you can only cross multiply when each side has only one fraction, so in this case, your first step is to find the LCM.

The only factors of 3x you know for certain are 3 and x. The only factor you know of x is just x, and 4 is a constant so you can use it as it is.

Write down each denominator’s polynomial factors into rows, with similar terms lined up in the same column.

You need to include both 3 and 4 because neither is a factor of the other. You don’t need both copies of x because x is a factor of itself! So the LCM is 12x.

You might find another example of finding the LCM with the same technique helpful.

You’re now ready to eliminate the denominators by multiplying both sides by the LCM.

Step 2: Simplify the Equation

Multiply each term by the LCM. Continuing from the last example, you have:

You now have a regular equation with no fractions, which should be familiar ground!

Step 3: Solve the Equation

Solving rational equations usually produces a simple polynomial equation. Hopefully, you’ve solved lots of these before!

You could complete the square, factor the terms by inspection, or use the quadratic formula.

This example can be solved by factoring the polynomial, having found that x+2 and x+4 are factors.

You could also solve the equation by completing the square:

Or by using the quadratic formula with a=1, b=6 and c=8:

Each way of solving the simplified rational equation is valid, but you will find that some are quicker than others!

Step 4: Check Every Solution

It is important to check that your solutions are complete, meaning you’ve found all of them and that they don’t give any weird numbers when substituted into the original equation.

In the worked example, you were left with a quadratic equation and found two distinct roots.

Quadratic equations either have two distinct solutions, one repeated solution, or no real solution so the solution x=-2 or x=-4 is complete.

You must be careful that none of the rational terms in the original equation have a zero in the denominator.

Do this by going back to the beginning and substituting your answers into the denominators!

The denominators in the worked example are 3x, x, and 4. Replacing x with -2 or -4 doesn’t give you zero in any of them, so you’re safe here!

A solution that gives a zero-denominator is not allowed. That’s because dividing by zero is “illegal” in math!

Any number divided by zero gives an error on a calculator. Ever wondered why that is?

This is your time to shine – try solving rational equations for yourself and, if you’re feeling confident, tackle the challenges too.

As they say, practice makes perfect! Use the worked example for guidance if you get stuck.

Find x in the following rational equation:

The equation is two equal fractions so you can cross-multiply. You could also simplify \(\Large\frac{15}{3}\normalsize\) to 5, but this does not change the final answer.

Each term is divisible by 9. Simplify the equation by dividing both sides by 9:

This form is called the difference of two squares because it can be factored like this:

So the solution is x=±3.

These must be all the solutions because quadratic equations have a maximum of two distinct real roots.

Neither denominator in the original rational equation has an x term, so substituting any value for x makes no difference to their values – there is no chance of them being zero!

This means the solutions x=3 and x=-3 are valid.

Solve the following rational equation:

There are three fractions so you cannot cross-multiply.

See that the second denominator is the difference of two squares?

Multiply each term by the LCM and simplify.

So its solution is -5, right?… STOP RIGHT THERE! Don’t forget, we can’t divide by zero!

If you put x=±5 into the original equation, at least one of the denominators is always zero, so the original equation has no solutions.

Challenge 1

Can you spot the mistake in the following example? Hint: there has been some cheating with radicals!

If you need a refresher on radicals , check out our lesson on multiplying them. That will get you on the right track!

The mistake is that radicals cannot be subtracted like normal terms.

Instead, you must square both sides of the equation to remove the radical. Similar terms can then be combined as usual.

Still confused? You can find lots of interactive questions on Lumen Learning . Radicals often pop up in rational equations, so getting comfortable with radicals is super helpful for exam success!

Challenge 2

Find the value of x, by using the Pythagorean theorem on the following right-angled triangle:

If you need a refresher on the Pythagorean theorem or are interested in the man himself, check out our lesson. Do the worksheets and you’ll be acing triangle questions in no time !

The Pythagorean theorem states that:

Where c is the length of the hypotenuse, and a and b are the other side lengths.

This gives the rational equation:

Simplifying, you find:

The LCM is 36 so the denominators are removed by dividing each term by this:

It’s always fun when different areas of math link together!

To Sum Up (Pun Intended!)

In today’s lesson on solving rational equations, you first saw the difference between rational and irrational numbers.

Rational numbers are “nice” because they can be written as a fraction of integers. Remember that all integers are rational because they can be written with a denominator of 1!

Irrational numbers are a little more abstract. They include weird but incredibly beautiful numbers like π and e, which cannot be written as a fraction of integers.

Rational equations are solved by eliminating the denominator in every term, then simplifying and solving as normal.

Denominators can be removed by cross-multiplication if there is only one fraction on either side or by finding the LCM if the equation is more complicated.

Don’t be shy, leave a comment below if you have any questions or need help!

Still curious about rational numbers, or eager for an extra challenge? Check out our lesson on the rational root theorem , which combines algebra and equation solving.

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Unit 13: Rational expressions, equations, & functions

About this unit.

This topic covers:

Simplifying rational expressions

• Multiplying, dividing, adding, & subtracting rational expressions
• Rational equations
• Graphing rational functions (including horizontal & vertical asymptotes)

Modeling with rational functions

Rational inequalities, partial fraction expansion, intro to rational expressions.

• Intro to rational expressions (Opens a modal)
• Reducing rational expressions to lowest terms (Opens a modal)
• Simplifying rational expressions: common monomial factors (Opens a modal)
• Simplifying rational expressions: common binomial factors (Opens a modal)
• Simplifying rational expressions: opposite common binomial factors (Opens a modal)
• Simplifying rational expressions (advanced) (Opens a modal)
• Simplifying rational expressions: grouping (Opens a modal)
• Simplifying rational expressions: higher degree terms (Opens a modal)
• Simplifying rational expressions: two variables (Opens a modal)
• Simplifying rational expressions (old video) (Opens a modal)
• Reduce rational expressions to lowest terms: Error analysis 4 questions Practice
• Reduce rational expressions to lowest terms 4 questions Practice
• Simplify rational expressions (advanced) 4 questions Practice

Multiplying & dividing rational expressions

• Multiplying & dividing rational expressions: monomials (Opens a modal)
• Multiplying rational expressions (Opens a modal)
• Dividing rational expressions (Opens a modal)
• Multiplying rational expressions: multiple variables (Opens a modal)
• Dividing rational expressions: unknown expression (Opens a modal)
• Multiply & divide rational expressions: Error analysis 4 questions Practice
• Multiply & divide rational expressions 4 questions Practice
• Multiply & divide rational expressions (advanced) 4 questions Practice

Adding & subtracting rational expressions

• Adding & subtracting rational expressions: like denominators (Opens a modal)
• Intro to adding & subtracting rational expressions (Opens a modal)
• Intro to adding rational expressions with unlike denominators (Opens a modal)
• Adding rational expression: unlike denominators (Opens a modal)
• Subtracting rational expressions: unlike denominators (Opens a modal)
• Least common multiple (Opens a modal)
• Least common multiple: repeating factors (Opens a modal)
• Subtracting rational expressions: factored denominators (Opens a modal)
• Least common multiple of polynomials (Opens a modal)
• Adding & subtracting rational expressions (Opens a modal)
• Subtracting rational expressions (Opens a modal)
• Add & subtract rational expressions: like denominators 4 questions Practice
• Add & subtract rational expressions (basic) 4 questions Practice
• Least common multiple 4 questions Practice
• Add & subtract rational expressions: factored denominators 4 questions Practice
• Add & subtract rational expressions 4 questions Practice

Nested fractions

• Nested fractions (Opens a modal)
• Nested fractions 4 questions Practice

Solving rational equations

• Rational equations intro (Opens a modal)
• Equations with one rational expression (advanced) (Opens a modal)
• Equations with rational expressions (Opens a modal)
• Equations with rational expressions (example 2) (Opens a modal)
• Equation with two rational expressions (old example) (Opens a modal)
• Equation with two rational expressions (old example 2) (Opens a modal)
• Equation with two rational expressions (old example 3) (Opens a modal)
• Rational equations intro 4 questions Practice
• Rational equations (advanced) 4 questions Practice
• Rational equations 4 questions Practice

Direct and inverse variation

• Intro to direct & inverse variation (Opens a modal)
• Recognizing direct & inverse variation (Opens a modal)
• Recognizing direct & inverse variation: table (Opens a modal)
• Direct variation word problem: filling gas (Opens a modal)
• Direct variation word problem: space travel (Opens a modal)
• Inverse variation word problem: string vibration (Opens a modal)
• Proportionality constant for direct variation (Opens a modal)
• Recognize direct & inverse variation 4 questions Practice

End behavior of rational functions

• End behavior of rational functions (Opens a modal)
• End behavior of rational functions 4 questions Practice

Discontinuities of rational functions

• Discontinuities of rational functions (Opens a modal)
• Analyzing vertical asymptotes of rational functions (Opens a modal)
• Rational functions: zeros, asymptotes, and undefined points 4 questions Practice
• Analyze vertical asymptotes of rational functions 4 questions Practice

Graphs of rational functions

• Graphing rational functions according to asymptotes (Opens a modal)
• Graphs of rational functions: y-intercept (Opens a modal)
• Graphs of rational functions: horizontal asymptote (Opens a modal)
• Graphs of rational functions: vertical asymptotes (Opens a modal)
• Graphs of rational functions: zeros (Opens a modal)
• Graphs of rational functions (old example) (Opens a modal)
• Graphing rational functions 1 (Opens a modal)
• Graphing rational functions 2 (Opens a modal)
• Graphing rational functions 3 (Opens a modal)
• Graphing rational functions 4 (Opens a modal)
• Graphs of rational functions 4 questions Practice
• Analyzing structure word problem: pet store (1 of 2) (Opens a modal)
• Analyzing structure word problem: pet store (2 of 2) (Opens a modal)
• Combining mixtures example (Opens a modal)
• Rational equations word problem: combined rates (Opens a modal)
• Rational equations word problem: combined rates (example 2) (Opens a modal)
• Rational equations word problem: eliminating solutions (Opens a modal)
• Reasoning about unknown variables (Opens a modal)
• Reasoning about unknown variables: divisibility (Opens a modal)
• Structure in rational expression (Opens a modal)
• Mixtures and combined rates word problems 4 questions Practice
• Rational inequalities: one side is zero (Opens a modal)
• Rational inequalities: both sides are not zero (Opens a modal)
• Intro to partial fraction expansion (Opens a modal)
• Partial fraction expansion (Opens a modal)
• Partial fraction expansion: repeated factors (Opens a modal)
• Partial fraction expansion 4 questions Practice

Rational Equations

Solving rational equations.

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

Examples of How to Solve Rational Equations

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

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

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

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.

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

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

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

• The LCD here is [latex]\left( {x – 3} \right)[/latex]. Use it as a multiplier to both sides of the rational equation.
• I hope you get this linear equation after performing some cancellations.

Distribute the constant [latex]9[/latex] into [latex]\left( {x – 3} \right)[/latex].

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

It should work so yes, [latex]x = 2[/latex] is the final answer.

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

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.

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

Quick note : 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.

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

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

Focusing on the denominators, the LCD should be [latex]6x[/latex]. Why?

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

• The LCD is [latex]6x[/latex]. Distribute to both sides of the given rational equation.
• It should look like after careful cancellation of similar terms.
• Distribute the constant into the parenthesis.
• The variable [latex]x[/latex] can be combined on the left side of the equation.
• Since there’s only one constant on the left, I will keep the variable [latex]x[/latex] to the opposite side.
• So I subtract both sides by [latex]5x[/latex].
• Divide both sides by [latex]-2[/latex] to isolate [latex]x[/latex].
• Yep! We got the final answer.

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

Whenever you see a trinomial in the denominator, always factor it out to identify the unique terms. By simple factorization, I found that [latex]{x^2} + 4x – 5 = \left( {x + 5} \right)\left( {x – 1} \right)[/latex]. 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 a 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 below and make sure you check your answers for extraneous values.

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

• The LCD is [latex]\left( {x + 5} \right)\left( {x – 5} \right)[/latex]. Distribute this into the rational equation.
• 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?

• I expanded both sides of the equation using FOIL. You should have a similar setup up to this point. Now combine like terms (the [latex]x[/latex]) in both sides of the equation.
• 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.
• Subtract both sides by [latex]{x^2}[/latex].
• The problem is reduced to a regular linear equation from a quadratic.
• To isolate the variable [latex]x[/latex] on the left side implies adding both sides by [latex]6x[/latex].
• Move all constant to the right.
• Add both sides by [latex]30[/latex].
• Finally, divide both sides by [latex]5[/latex] and we are done.

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

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

• Factor out the denominators.
• Multiply both sides by the LCD obtained above.

Be careful now with your cancellations.

• You should end up with something like this when done right.
• 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.

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

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

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

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

Multiply the constants into the parenthesis.

• Combine similar terms
• Keep the variable to the left side by subtracting [latex]x[/latex] on both sides.
• Keep constants to the right.
• Add both sides by [latex]8[/latex] to solve for [latex]x[/latex]. Done!

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

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.
• 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 [latex]3x[/latex].
• 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 [latex]x^2+bx+c=0[/latex] since the trinomial is easily factorable by inspection .

• The factors of [latex]{x^2} – 5x + 4 = \left( {x – 1} \right)\left( {x – 4} \right)[/latex]. You can check it by the FOIL method .
• Use the Zero Product Property to solve for [latex]x[/latex].

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.

You might also like these tutorials:

• Adding and Subtracting Rational Expressions
• Multiplying Rational Expressions
• Solving Rational Inequalities