Decision map — what type of inequality am I facing?

Recognising the structure of an irrational inequality, before calculating, saves a great deal of work. Here is the sequence of questions to ask yourself.

In brief — Decision map

When facing an irrational inequality, ask yourself, in this order:

1. Is it an always $\ge 0$ sum? If so, the answer is immediate: the inequality $\ge 0$ is always true within the CE, the one $<0$ has no solutions.
2. Are both sides always $\ge 0$? If so — that is, if they are two radicals, or a root and an absolute value, or two absolute values — square directly without splitting.
3. Otherwise, in which direction is the inequality?
   • $\sqrt{A}\le B$: Type I, a system with three conditions.
   • $\sqrt{A}\ge B$: Type II, a union of two cases.
4. If there is a fraction, combine everything in a sign table.
5. At the end, intersect with the CE. Always.

The first two questions serve to hunt out the shortcuts; only if they fail do we fall back on the general method of the two types. The last step, the intersection with the CE, must never be forgotten.

Links

Topics: Irrational inequalities
Concepts: Existence conditions · Irrational inequality
Methods: Irrational inequality by cases · Non-negative sides shortcut
Skills: Reasoning by cases · Solving inequalities