Proofs Involving Inequalities

When working with inequalities, there are a number of logical relationships we can use to create new inquealities from an initial premise. These relationships are based on the inherent ordering that inequalities place on numbers on the number line.

For example, if we are told that $a < b$ and $b < c$, then what can we say about the relationship between $a$ and $c$:

• $a < b$ tells us that $b$ lies to the right of $a$ on the number line
• $b < c$ tells us that $c$ lies to the right of $b$ on the number line

If we consider the location of $b$ on the number line, we see that $a$ must lie to its left, and $c$ must lie on its right. Therefore, we can be certain that $c$ lies to the right of $a$ on the number line, and so we can conclude that

$a < c$

On the other hand, if we are told that $a < b$ and $b > c$, then we do not have enough information to assert any propositions about the relationship between $a$ and $c$

• $a < b$ tells us that $b$ lies to the right of $a$ on the number line
• $b > c$ tells us that $c$ lies to the left of $b$ on the number line
  • $c$ could be further left of $b$ than $a$ is, which would mean that $c < a$
  • $c$ could be closer to $b$ than $a$ is, which would mean that $a < c$
  • $c$ could even be equal to $a$

With this in mind, we do need to be careful when combining information from multiple inequalities. We also need to be careful when we consider weak inequalities like $\leq$ and $\geq$. For example, suppose we are told:

• $a \leq b$, which tells us that $b$ lies to the right of $a$ or is in fact equal to a
• $b < c$, which tells us that $c$ lies to the left of $b$ but they are not equal

When we compare $a$ and $c$, we can still be certain that $a < c$ and that they are not equal. This is because, even though $b$ can be equal to $a$, $c$ cannot be equal to $b$ and therefore to $a$.

But, if we were instead told that $a \leq b$ and $b \leq c$, then it is possible that $c$ can be equal to $b$ and therefore can also be equal to $a$, so we conclude that $a \leq c$. We refer to these as the transitive properties of inqualities. These are summarised in the table below

Inequality 1	Inequality 2	Conclusion
$a < b$	$b < c$	$a < c$
$a < b$	$b \leq c$	$a < c$
$a \leq b$	$b < c$	$a < c$
$a \leq b$	$b \leq c$	$a \leq c$
$a < b$	$b > c$	Unknown
$a < b$	$b \geq c$	Unknown
$a \leq b$	$b > c$	Unknown
$a \leq b$	$b \geq c$	Unknown

...

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