Solving Fractional-Rational Inequalities
What is a Fractional-Rational Inequality?
A fractional-rational inequality is an inequality that contains a variable in the denominator of a fraction. It has the general form:
$$ \frac{P(x)}{Q(x)} \lessgtr 0 $$
where $P(x)$ and $Q(x)$ are polynomials, and $Q(x) \neq 0$.
⚠️ Critical Rule: Never Multiply Both Sides by a Variable Expression!
Unlike regular inequalities, you cannot simply multiply both sides by the denominator because:
- The denominator might be positive or negative depending on the value of $x$.
- Multiplying by a negative expression reverses the inequality sign.
- The denominator might equal zero, which is undefined.
Instead, we use the Method of Intervals (also called the Sign Chart Method).