A rational equation is an equation that contains one or more rational expressions, which are fractions with variables in the denominator. These equations appear in rates, proportions, work problems, mixture problems, and formulas involving inverse relationships. The main challenge is that some variable values are not allowed because they make a denominator equal to zero.
Solving them carefully helps you avoid answers that look correct but do not actually work.
Key Facts
- A rational equation contains at least one rational expression, such as 3/(x + 2) = 5/x.
- Excluded values come from denominator = 0, so x + 2 = 0 gives x = -2 as not allowed.
- To clear denominators, multiply every term on both sides by the least common denominator.
- If a/b = c/d and b ≠ 0 and d ≠ 0, then ad = bc.
- After clearing denominators, solve the resulting linear, quadratic, or polynomial equation.
- Always check solutions in the original equation because clearing denominators can create extraneous solutions.
Vocabulary
- Rational expression
- A rational expression is a fraction whose numerator and denominator are polynomials.
- Rational equation
- A rational equation is an equation that contains one or more rational expressions.
- Least common denominator
- The least common denominator is the simplest expression that all denominators in an equation divide into evenly.
- Excluded value
- An excluded value is a value of the variable that makes at least one denominator equal to zero.
- Extraneous solution
- An extraneous solution is a value found during solving that does not satisfy the original equation.
Common Mistakes to Avoid
- Forgetting to list excluded values first is wrong because a final answer might make a denominator zero and must be rejected.
- Multiplying only some terms by the least common denominator is wrong because every term on both sides must be multiplied to keep the equation equivalent.
- Canceling across addition or subtraction is wrong because factors can cancel only when they multiply the entire numerator and denominator.
- Checking in the cleared equation instead of the original equation is wrong because extraneous solutions often still satisfy the cleared equation.
Practice Questions
- 1 Solve 2/x + 3/4 = 5/2. State any excluded values and check your answer.
- 2 Solve 1/(x - 3) + 2/x = 5/(x(x - 3)). State any excluded values and identify any extraneous solutions.
- 3 A student solves (x + 1)/(x - 2) = 3/(x - 2) by multiplying by x - 2 and gets x + 1 = 3, so x = 2. Explain why this answer must be rejected.