The sign-table strategy

The richest case is the one in which the inequality involves a ratio of expressions with radicals, of the form $\frac{N(x)}{D(x)}\le 0$. As with the fractional inequalities studied in Year 2, the strategy is always the same.

Theorem — Strategy for fractional inequalities with radicals

1. Write the global existence conditions (both on the radicals and on the fact that the denominator does not vanish).
2. Study separately the sign of the numerator and that of the denominator.
3. Combine them in a sign table.
4. Intersect with the existence conditions to produce the final solution.

The sign study of a single factor (in the numerator or the denominator) is almost always a Type I or Type II inequality, or a “shortcut” case from the previous sections: the techniques seen so far become the building blocks with which the overall sign study is constructed.

Links

Topics: Irrational inequalities
Concepts: Existence conditions · Fractional inequality · Irrational inequality · Sign table
Methods: Sign study
Skills: Solving inequalities