Inequality Proofs (Leaving Cert Mathematics): Flashcards

Prove that the expression is greater than or equal to zero.

It is always greater than or equal to zero, i.e., $x^2 \geq 0$ for any real number $x$.

$a^2 + b^2 \geq 2ab$.

Rewrite the inequality to isolate variables if possible.

$(a-b)^2 \geq 0$.

Show $x^2 - x + 1 > 0$.

It's expressed as $(x-1/2)^2 + 3/4 > 0$.

The squared term is positive and $3/4$ is also positive.

Using the method of AM-GM inequality.

It simplifies the expression and reveals positivity.