Activity 1-2: Inequalities

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What inequalities do you know?

What do you think the most basic inequality of all might be?

Maybe …

the Triangle Inequality.

The length of any one side of a triangle is less than the sum of the other two.

Notice that a triangle has another basic inequality;

a < b < c A < B < C

a < b + c, b < a + c, c < a + b.

Travelling from A to B direct is shorter than travelling from A to B via C;

we are saying ‘the shortest distance between any two points is a straight line’.

Standard inequalities like theseare of great use to the mathematician.

More arise from this question: How do we find the average

of two non-negative numbers a and b?

How are these ordered? Does the order of size depend on a and b?

Task: try to come up with a proof that AM ≥ GM for all non-negative a and b.

When does equality hold?

Now try to show that GM ≥ HM for all non-negative a and b. When does equality hold?

We can see that equality only

holds in each casewhen a = b.

We can often come up with a diagram that demonstrates an inequality.

What inequality does the following diagram illustrate?

How about this?

Hint:calculate

OA,AB,AC.

So AM GM HM.

First reflect on this diagram.

So we have that ab + bc + ac a2 + b2 + c2.

Can we prove the AM-GM inequality for three numbers?That is, if a, b, c > 0, does 3abc ≤ a3 + b3 + c3 hold?

Now reflect on this diagram...

Carom is written by Jonny Griffiths, hello@jonny-griffiths.net

With thanks to Claudi Alsina and Roger B. Nelsen,authors of When Less is More; Visualising Basic Inequalities.