Fractional Quadratic Inequalities #
A fractional inequality has the form \(\frac{N(x)}{D(x)} \gtrless 0\). Solve by: (1) studying the numerator zeros, (2) studying the denominator zeros (excluded!), (3) building a sign table, (4) selecting intervals with the required sign.
Worked Example #
$$\frac{x^2 - 4}{x - 1} > 0$$Numerator: \(x = \pm 2\). Denominator: \(x = 1\) (excluded). Sign table → Solution: \(-2 < x < 1 \cup x > 2\)
Key rules: never divide by the denominator; denominator zeros are always excluded; if \(\geq\) or \(\leq\), numerator zeros are included but denominator zeros are not.
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