Inequality Refresher

Method 1 : Find a counterexample.

To prove that a universal statement (a statement in which it claims that it's true for all $x$ ) is false, we just need to find a value of $x$ such that the inequality $x\sqrt x \geq x$ can't be fulfilled.

Choose $x=\dfrac{1}{4}$ , we have $x\sqrt{x}=\dfrac{1}{8}<x$ which contradicts the claim.

Therefore, the inequality is FALSE .

Method 2 : Convert the inequality into a polynomial inequality.

Because $x>0$ , then we can let $y = \sqrt{x}$ , and upon substitution, the inequality is equivalent of $y^2 \cdot y \geq y^2$ or $y^2(y-1) \geq 0$ .

However, this inequality is only true when $y - 1 \geq 0$ . So a possible value of $y$ that doesn't satisfy this inequality is $y = \dfrac12$ , or $x = \left(\dfrac12\right)^2 = \dfrac14$ , as shown in Method 1.

Therefore, the inequality is FALSE .

$x\sqrt{x} \geq x$ holds true only for $x \in [1, +\infty)$

The expression in false for $0<x<1$ .