The method for the smaller root — Type I

Consider an inequality in which the root is on the left-hand side and the right-hand side is an expression without a radical: $\sqrt{A(x)}\le B(x).$

Let us reason: the left-hand side $\sqrt{A(x)}$, when it exists, is always $\ge 0$. If $B(x)$ is negative, then a non-negative quantity can never be less than or equal to a negative one: the inequality has no solutions in that region. We must therefore require $B(x)\ge 0$. Once we have ensured that both sides are non-negative, we can square while preserving the direction.

In brief — Method for $\sqrt{A(x)}\le B(x)$

The inequality is equivalent to the system $\begin{cases} A(x)\ge 0 & \text{(CE: esiste la radice)} \\ B(x)\ge 0 & \text{(il secondo membro è non negativo)} \\ A(x)\le \bigl(B(x)\bigr)^2 & \text{(elevamento al quadrato)} \end{cases}$ The three conditions must be intersected. If the inequality is strict ($<$), strict inequalities hold throughout; if it is $\le$, everything stays with the equality.

The three conditions tell a precise story: the first guarantees that the root exists, the second that the comparison makes sense, the third actually compares the values. Omitting any one of them leads to errors.

Links

Topics: Irrational inequalities
Concepts: Existence conditions · Irrational inequality
Methods: Irrational inequality by cases · Squaring
Skills: Reasoning by cases · Solving inequalities