Solving Irrational Inequalities
A comprehensive guide to inequalities involving radicals and fractional exponents
1. Definition
Irrational Inequality: An inequality that contains a variable under a radical sign ($ \sqrt[n]{\cdot} $) or raised to a fractional exponent ($ x^{p/q} $).
The central strategy is to reduce the irrational inequality to an equivalent rational inequality or a system of rational inequalities by raising both sides to an appropriate power.
However, we must be extremely careful: raising both sides of an inequality to a power is not always a reversible operation. We need to track domains, sign conditions, and monotonicity to ensure equivalence.
Examples of Irrational Inequalities
- $ \sqrt{x - 2} < 3 $
- $ \sqrt[3]{7 - x^2} < 1 - x $
- $ \sqrt{2x - 5} < 4 - x $
- $ \sqrt[4]{6x^2 + 25x + 8} > 2 + x $
Key Differences from Equations
- Solutions are intervals, not isolated points
- Domain restrictions are critical
- Even vs. odd index changes the method
- Direction of inequality matters when raising to a power