1. Activity 1-2: Inequalities
Activity 1-2: Inequalities

2. What inequalities do you know?
What do you think
the most basic inequality of all might be?
Maybe …
the Triangle Inequality.

3. a < b < c  A < B < C
Notice that a triangle
has another basic inequality;
a < b < c  A < B < C
The length of any one side of a triangle
is less than the sum of the other two.
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’.

4. Standard inequalities like these
are 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?

5. 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.

6. We can see that
equality only
holds in each case
when a = b.

7. We can often come up with a diagram that demonstrates an inequality.
What inequality does the following diagram illustrate?

9. How about this?
Hint:
calculate
OA,
AB,
AC.

10. So AM  GM  HM.

11. Can we prove the AM-GM inequality for three numbers?
That is, if a, b, c > 0, does 3abc ≤ a3 + b3 + c3 hold?
First reflect on this diagram.
So we have that
ab + bc + ac  a2 + b2 + c2.

12. Now reflect on this diagram...

13. With thanks to Claudi Alsina and Roger B. Nelsen,
authors of When Less is More; Visualising Basic Inequalities.
Carom is written by Jonny Griffiths,