Rational expressions & equations
The idea
A rational expression is a ratio of polynomials, such as 3/(x − 2), and it inherits its rules from ordinary fractions: common denominators to add, multiply straight across, divide by flipping. What is new is the domain — any value that makes a denominator zero is simply not allowed, and those excluded values silently shape every problem. You already handle fractions and polynomial algebra; here the two merge, and bookkeeping the forbidden values becomes part of the work.
To solve a rational equation, factor every denominator, record the excluded values, then multiply both sides by the least common denominator to clear the fractions. That clearing step can manufacture solutions that were never legal, because multiplying by an expression that equals zero at some x erases information. So the final move is non-negotiable: test each candidate against the excluded list and discard any that land on it. The misconception to avoid is canceling terms instead of factors — you may cancel only whole factors shared by an entire numerator and denominator, never individual terms inside a sum.
Worked example
Solve 3/(x − 2) + 1/x = 4/(x² − 2x).
- Factor the right-hand denominator: x² − 2x = x(x − 2). Every denominator is now built from x and x − 2, so the excluded values are x = 0 and x = 2.
- Multiply both sides by the least common denominator x(x − 2): the left side becomes 3x + (x − 2) and the right side becomes 4.
- Solve the cleared equation: 3x + x − 2 = 4 gives 4x = 6, so x = 3/2.
- Check against the exclusions: 3/2 is neither 0 nor 2, so it survives.
- Verify numerically: at x = 3/2 the left side is 3/(−1/2) + 1/(3/2) = −6 + 2/3 = −16/3, and the right side is 4/((9/4) − 3) = 4/(−3/4) = −16/3. Both sides match.
Answer. The only solution is x = 3/2.
Check your understanding
- Why can multiplying both sides of an equation by a variable expression create extraneous solutions, and at exactly which step does information get lost?
- How do the excluded values of a rational expression show up on its graph?
- Why is it illegal to cancel the x in (x + 3)/x while canceling the x in 3x/x is perfectly fine?
- How would you explain to a friend the difference between simplifying a rational expression and solving a rational equation?
Build the foundations first
Rational expressions & equations builds on these concepts. If any feel shaky, start there.