Mathwords logoMathwords

Rational Equation

Rational Equation
Fractional Equation

An equation which has a rational expression on one or both sides of the equal sign. Sometimes rational equations have extraneous solution.

 

 

See also

Rational function

Key Formula

P(x)Q(x)=R(x)S(x)\frac{P(x)}{Q(x)} = \frac{R(x)}{S(x)}
Where:
  • P(x),R(x)P(x), R(x) = Polynomial expressions in the numerators
  • Q(x),S(x)Q(x), S(x) = Polynomial expressions in the denominators, where Q(x) ≠ 0 and S(x) ≠ 0
  • xx = The variable you solve for; any value that makes a denominator zero is excluded from the domain

Worked Example

Problem: Solve the rational equation: 3/x + 1/4 = 5/x
Step 1: Identify the denominators and state the restriction. The denominators are x and 4, so x ≠ 0.
Denominators: x and 4⇒x≠0\text{Denominators: } x \text{ and } 4 \quad \Rightarrow \quad x \neq 0
Step 2: Find the least common denominator (LCD). The LCD of x and 4 is 4x.
LCD=4x\text{LCD} = 4x
Step 3: Multiply every term on both sides by the LCD to clear the fractions.
4x⋅3x+4x⋅14=4x⋅5x⇒12+x=204x \cdot \frac{3}{x} + 4x \cdot \frac{1}{4} = 4x \cdot \frac{5}{x} \quad \Rightarrow \quad 12 + x = 20
Step 4: Solve the resulting equation for x.
x=20−12=8x = 20 - 12 = 8
Step 5: Check: x = 8 does not violate the restriction x ≠ 0, and substituting back gives 3/8 + 1/4 = 3/8 + 2/8 = 5/8, which equals 5/8. The solution checks out.
38+14=58✓\frac{3}{8} + \frac{1}{4} = \frac{5}{8} \quad \checkmark
Answer: x = 8

Another Example

This example shows that a rational equation can have no solution at all — not because of an extraneous solution, but because the algebra produces a contradiction. This contrasts with the first example, which had a straightforward valid solution.

Problem: Solve the rational equation: x/(x − 2) = 4/(x − 2) + 1
Step 1: Identify the restriction. The denominator is (x − 2), so x ≠ 2.
x≠2x \neq 2
Step 2: The LCD is (x − 2). Multiply every term by (x − 2).
(x−2)⋅xx−2=(x−2)⋅4x−2+(x−2)⋅1(x - 2) \cdot \frac{x}{x-2} = (x-2) \cdot \frac{4}{x-2} + (x-2) \cdot 1
Step 3: Simplify both sides after the fractions are cleared.
x=4+(x−2)⇒x=x+2x = 4 + (x - 2) \quad \Rightarrow \quad x = x + 2
Step 4: Subtract x from both sides. This gives 0 = 2, which is a contradiction. The equation has no solution.
0=2(contradiction)0 = 2 \quad \text{(contradiction)}
Answer: No solution. The equation is a contradiction, so no value of x satisfies it.

Frequently Asked Questions

What is an extraneous solution in a rational equation?
An extraneous solution is a value you obtain algebraically that does not actually satisfy the original equation, usually because it makes a denominator equal to zero. When you multiply both sides by the LCD, you sometimes introduce solutions that weren't valid in the original equation. That is why you must always check your answers by substituting them back into the original equation.
How do you solve a rational equation step by step?
First, identify all denominators and note the values of the variable that make any denominator zero (these are your restrictions). Next, find the least common denominator of all fractions. Multiply every term on both sides of the equation by the LCD to eliminate all fractions. Then solve the resulting polynomial equation and check each solution against your restrictions.
What is the difference between a rational equation and a rational expression?
A rational expression is a single fraction where the numerator and denominator are polynomials, such as (x + 1)/(x − 3). A rational equation, on the other hand, sets two expressions equal to each other and asks you to find the values of x that make the equation true. You simplify rational expressions, but you solve rational equations.

Rational Equation vs. Rational Expression

Rational EquationRational Expression
DefinitionAn equation containing at least one fraction with a variable in the denominatorA fraction where the numerator and denominator are polynomials
Contains an equal sign?Yes — it is an equation to solveNo — it is an expression to simplify or evaluate
GoalFind the values of the variable that satisfy the equationSimplify, add, subtract, multiply, or divide the expression
Extraneous solutions?Possible — must check all solutions against domain restrictionsNot applicable (no equation to solve)
Example3/x + 1 = 5/x3/x + 5/x = 8/x

Why It Matters

Rational equations appear frequently in Algebra 2 and Precalculus courses, and they are tested on standardized exams like the SAT and ACT. They also model real-world situations involving rates, such as work-rate problems ("Two pipes fill a pool together") and problems involving distance, speed, and time. Mastering them builds essential skills for later topics like partial fraction decomposition and solving equations with logarithmic or trigonometric fractions.

Common Mistakes

Mistake: Forgetting to check for extraneous solutions after solving.
Correction: Always substitute each answer back into the original equation. If a solution makes any denominator zero, discard it — it is extraneous and not a valid solution.
Mistake: Multiplying only some terms by the LCD instead of every term on both sides.
Correction: When you multiply by the LCD, you must distribute it to every single term on both sides of the equation. Missing even one term will give you an incorrect equation to solve.

Related Terms