Inequality Proofs Revision Notes for Leaving Cert Mathematics

Inequality Proofs

In some cases you will be asked to prove that a certain quantity is greater than another. The key here is to deduce the inequality to an expression will a squared, because a squared is always greater than zero. This is better demonstrated with an example.

Example

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Prove that for all real numbers $a$ and $b$ that $a^2+b^2 \ge2ab$

Bring all terms to one side.

a^2+b^2-2ab \ge 0

Factorise

(a-b)^2 \ge 0

The squared term is guaranteed to be positive, so its always greater or equal to $0$ .

Example

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Prove that for all real numbers $x$ that $x^2+1>x$ .

Bring all terms to one side.

x^2-x+1>0

Write in completed square form :

\left(x-\frac{1}{2} \right)^2+\frac{3}{4}>0

The squared term is guaranteed to be positive and also we adding $\tfrac{3}{4}$ which is also positive, which makes the whole expression on the left positive.

Inequality Proofs (Leaving Cert Mathematics): Revision Notes