Rational Inequalities

Rational Inequalities

A rational inequality contains only rational functions, such as P(x)/Q(x) < 0. The interval method is the universal approach.

The Interval Method Theorem

Suppose the function f(x) is continuous on the entire number line and becomes zero at points x₁, x₂, ..., xₙ. Then on each interval between these points, the function preserves its sign.

Steps of the Interval Method

1. Move all terms to the left side. Reduce to a common denominator. The right side should become 0.
2. Find all values of the variable for which the numerator and denominator become zero.
3. Mark these points on the number line. They divide the line into intervals where the rational function keeps a constant sign.
4. Determine the sign on one interval, preferably an outer interval.
5. Determine signs on remaining intervals: when passing through a root of odd multiplicity, the sign changes; when passing through a root of even multiplicity, the sign stays the same.
6. The solution set is the union of intervals with the required sign.

Example

Solve: ((x² + 2x − 3) / (x² + 4x + 3)) × (1 − 3/(x + 1)) ≤ 0.

Factor: x² + 2x − 3 = (x + 3)(x − 1), x² + 4x + 3 = (x + 1)(x + 3).

Combined: ((x + 3)(x − 1)(x − 2)) / ((x + 1)²(x + 3)) ≤ 0.

Numerator zeros: x = −3, 1, 2. Denominator zeros: x = −1, −3. Mark on number line, determine signs.

Important: Do not cancel factors before analyzing — this changes the domain! The value x = −3 was excluded from the original because it makes the denominator zero. After cancellation, it would incorrectly appear to belong.

Answer: x ∈ [1; 2].