A brief Introduction to Inequalities Anthony Erb Lugo June 13, 2011
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A Trivial Inequality
IDEA 1: Take any real number, say x for example, and square it. No matter what x you chose, the result, x2 , is always non-negative, i.e. x2 ≥ 0. This is known as the Trivial Inequality, and is the base for many, if not all, inequality problems. USAGE: When attempting to use this inequality, try to rearrange the problem so that there is a zero on the right hand side, and then factor the expression on the left hand side in a way that it’s made up of “squares”. Example 1.1: Let a and b be real numbers. Prove that, a2 + b2 ≥ 2ab Proof. Note that by subtracting 2ab on both sides we get, a2 − 2ab + b2 ≥ 0 Or, (a − b)2 ≥ 0 Which is true due to the Trivial Inequality. Example 1.2: Let a, b and c be real numbers. Prove that, a2 + b2 + c2 ≥ ab + bc + ac Proof. We start by moving all of the terms to the left, a2 + b2 + c2 − ab − bc − ac ≥ 0 By multiplying by 2 we can see that, 2(a2 + b2 + c2 − ab − bc − ac) = (a − b)2 + (a − c)2 + (b − c)2 ≥ 0 Thus our original inequality is true. Alternatively, you could notice, from Example 1.1, that the following inequalities are true, a2 + b2 ≥ 2ab b2 + c2 ≥ 2bc a2 + c2 ≥ 2ac Hence their sum, 2(a2 + b2 + c2 ) ≥ 2(ab + bc + ac) is also true, so all that is left is to do is divide by 2 and we’re done.