Solving Rational Equations

Learn how to solve equations containing rational expressions by finding the LCD and eliminating denominators.

Advanced30 minLesson
Printable worksheet

Definition

A rational equation is an equation that contains one or more rational expressions (fractions with variables in the denominator).
Examples of rational equations:
2x=13\frac{2}{x} = \frac{1}{3}
x+1x−2=5\frac{x+1}{x-2} = 5
3x+1+2x−1=5x2−1\frac{3}{x+1} + \frac{2}{x-1} = \frac{5}{x^2-1}
Strategy for solving:
  1. 1.Find the LCD (Least Common Denominator) of all fractions
  2. 2.Multiply every term by the LCD to clear denominators
  3. 3.Solve the resulting equation
  4. 4.Check for extraneous solutions (values that make any denominator zero)

Try it now

Which of the following is a rational equation?

Worked Examples

Solve: 5x=156\frac{5}{x} = \frac{15}{6}
1

Simplify the right side if possible

156=52\frac{15}{6} = \frac{5}{2} → 5x=52\frac{5}{x} = \frac{5}{2}
2

Cross multiply

5⋅2=5⋅x5 \cdot 2 = 5 \cdot x → 10=5x10 = 5x
3

Solve for x

x=105x = \frac{10}{5} → x=2x = 2
4

Check: Does x=2x = 2 make any denominator zero?

Denominators: x=2x = 2 and 66. Neither is zero when x=2x = 2. → Valid solution

Common Mistakes

Forgetting to check for extraneous solutions

Why it's wrong: When you multiply by an expression containing the variable, you might introduce solutions that don't work in the original equation.

Correct: Always substitute your answer back into the original equation to verify it doesn't make any denominator zero.

Not multiplying ALL terms by the LCD

Why it's wrong: Students often forget to multiply terms that are already integers or simple fractions.

Correct: Every term on both sides of the equation must be multiplied by the LCD, including constants like 33 or 12\frac{1}{2}.

Using the wrong LCD

Why it's wrong: Finding the LCD of algebraic expressions is harder than finding LCD of numbers.

Correct: Factor all denominators first, then include each factor the maximum number of times it appears in any one denominator.

Sign errors when distributing

Why it's wrong: After clearing denominators, students often make errors with negative signs during distribution.

Correct: Be extra careful with subtraction: −(x−3)=−x+3-(x-3) = -x + 3, not −x−3-x - 3.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Solving Rational Equations | Step-by-Step Examples

by Understand The Math

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Interactive Visual

Function graph

-6-4-202468-505xy(2, 2.5)x = 0y = 0

Tap or hover over the graph to read off values.

y = 5/xy = 15/6Asymptote
  • Intersection point: (2, 2.5)

Equation Solver

Solve for xx:

5x=156\frac{5}{x} = \frac{15}{6}
1
5x=156\frac{5}{x} = \frac{15}{6}
Find the excluded valuesA denominator can never be zero

Tips:

  • • Whatever you do to one side, do to the other
  • • Undo operations in reverse order (PEMDAS backwards)
  • • Check your answer by substituting back

Follow along as we solve the equation step by step.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
Which of the following is a rational equation?

Why It Matters

Rational equations appear throughout science, engineering, and everyday problem-solving:
  • Work problems: If one pipe fills a tank in 4 hours and another in 6 hours, how long together? This requires solving 14+16=1t\frac{1}{4} + \frac{1}{6} = \frac{1}{t}
  • Rate problems: Finding average speed for a round trip with different speeds
  • Optics: The lens equation 1f=1do+1di\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} relates focal length to object and image distances
  • Electronics: Parallel resistors combine as 1Rtotal=1R1+1R2\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}
Mastering rational equations unlocks your ability to solve these real-world applications!

Real World Applications

Work Rate Problems

When two workers or machines complete a task together, their combined rate is the sum of individual rates.

Example:

Pipe A fills a pool in 6 hours. Pipe B fills it in 4 hours. Together: 16+14=1t\frac{1}{6} + \frac{1}{4} = \frac{1}{t} gives t=2.4t = 2.4 hours.

1Try It Yourself

Maria can paint a room in 5 hours. Carlos can paint it in 3 hours. They work together.

How long does it take them to paint the room together?

Step 1: Write the mathematical expression

Set up the equation using work rates:

Average Speed Problems

Finding average speed for a round trip requires rational equations when speeds differ.

Example:

Drive 60 km at 30 km/h, return at 60 km/h. Average speed is NOT 45 km/h! Solve 120v=6030+6060\frac{120}{v} = \frac{60}{30} + \frac{60}{60} to get v=40v = 40 km/h.

2Try It Yourself

A cyclist rides 24 km uphill at 8 km/h, then returns downhill at 24 km/h.

What is the cyclist's average speed for the entire trip?

Step 1: Write the mathematical expression

Use: Average speed = Total distance / Total time

Electrical Circuits

Parallel resistors combine using the formula $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}$.

Example:

Two resistors of 6 ohms and 3 ohms in parallel: 1RT=16+13=12\frac{1}{R_T} = \frac{1}{6} + \frac{1}{3} = \frac{1}{2}, so RT=2R_T = 2 ohms.

Key Takeaways

  • 1A rational equation contains fractions with variables in the denominator
  • 2Solve by multiplying all terms by the LCD to clear denominators
  • 3Always identify restricted values (where denominators equal zero) before solving
  • 4Check all solutions - extraneous solutions must be rejected
  • 5If the result is quadratic, use factoring or the quadratic formula

Frequently Asked Questions

An extraneous solution is a value that satisfies the simplified equation but makes one of the original denominators equal to zero. It appears during the solving process but is not a valid solution to the original equation.
An extraneous solution is a value that satisfies the simplified equation but makes one of the original denominators equal to zero. It appears during the solving process but is not a valid solution to the original equation.
Multiplying by the LCD eliminates all denominators, converting the rational equation into a polynomial equation that's easier to solve. Think of it as clearing fractions.
Yes! If all solutions found are extraneous (they make a denominator zero), then the equation has no solution.
First factor all denominators completely. The LCD includes each unique factor raised to the highest power it appears in any denominator.

Glossary

Rational equation
An equation containing one or more rational expressions (fractions with variables in denominators)
LCD (Least Common Denominator)
The smallest expression that all denominators divide into evenly
Extraneous solution
A solution that emerges from solving but doesn't satisfy the original equation (makes a denominator zero)
Restricted value
A value of the variable that makes any denominator equal to zero; these are excluded from the domain
Cross multiplication
A shortcut for equations with one fraction on each side: if ab=cd\frac{a}{b} = \frac{c}{d}, then ad=bcad = bc

Formula Card

LCD Method

Multiply every term by LCD, then solve

Clear all denominators by multiplying every term by the least common denominator

Cross Multiplication

If ab=cd\frac{a}{b} = \frac{c}{d}, then ad=bcad = bc

A shortcut when there is one fraction on each side of the equation

Work Rate Formula

1t1+1t2=1ttogether\frac{1}{t_1} + \frac{1}{t_2} = \frac{1}{t_{together}}

Combined rate equals the sum of individual rates

Parallel Resistors

1RT=1R1+1R2\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}

Total resistance of parallel resistors in electrical circuits

Today's challengeOne problem, three tries

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