Irrational inequalities

In the second year we learnt to solve irrational equations and the basic versions of the inequalities in the two canonical types $\sqrt{A(x)} > B(x)$ and $\sqrt{A(x)} < B(x)$. In the third year the next step is to tackle the more complex inequalities: mixed cases, several radicals at once, shortcuts when both sides are $\ge 0$, fractional inequalities with radicals.

The whole difficulty is concentrated in one fact: squaring is not an innocent operation when working with inequalities. Squaring preserves the inequality only if both sides are non-negative; otherwise it can reverse its direction or introduce false solutions. This is why the classical method splits into two cases, and depending on the direction of the inequality these combine in one way or the other. The chapter isolates two fundamental types, shows their shortcuts, and closes with a decision map and a complete example of a fractional inequality with radicals.

Sezioni

• Type I: the root smaller than an expression
• Type II: the root greater than an expression
• Shortcuts when both sides are always non-negative
• Sums of always non-negative quantities
• Fractional inequalities with radicals: sign study
• Application: lines at an assigned distance

Approfondimenti

• A reminder about the existence conditions
• Decision map: what type of inequality am I facing?

Esercizi

• All the chapter’s worked exercises