A calculus of the absurd
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4.12 Inequalities
In a similar way to how equations express that two mathematical objects must be the same, an inequality expresses that two mathematical objects must be different in a specific way. There are a number of different types of inequalities, for example
\(\seteqnumber{0}{4.}{151}\)
\begin{align}
a > b & \quad \text {Read \say {$a$ is greater than $b$}} \\ a < b & \quad \text {Read \say {$a$ is less than $b$}} \\ a \geqq b & \quad \text {Read \say {$a$ is greater than or equal to $b$}} \\ a \geqq b & \quad
\text {Read \say {$a$ is less than or equal to $b$}} \\
\end{align}
The less than symbol can be written as \(\le \) or \(\leqq \) - both symbols mean the same thing.
As with equations, inequalities can be manipulated according to a set of rules while maintaining the properties of the inequality.
4.12.1 Problems in applying functions to inequalities
One has to apply functions to inequalities. For example, where it is fine to add the same numbers to both sides of an inequality, multiplication can often cause problems! For example, while
Read: “\(2\) is less than \(3\)”
\(\seteqnumber{0}{4.}{156}\)
\begin{equation}
2 < 3
\end{equation}
It is not the case that the inequality will be true if we multiply both sides by a negative number.
\(\seteqnumber{0}{4.}{157}\)
\begin{equation}
-2 \nless -3
\end{equation}
This same problem occurs when we take the reciprocal3737 The reciprocal function is defined as \(f(x) = \frac {1}{x}\). of both sides of an inequality. For example, it is true that
\(\seteqnumber{0}{4.}{158}\)
\begin{equation}
8 > 4
\end{equation}
However, if we take the reciprocal of both sides, this inequality ceases to hold:
\(\seteqnumber{0}{4.}{159}\)
\begin{equation}
\frac {1}{8} \ngtr \frac {1}{4}
\end{equation}
Why is this the case? It helps to look at the graph of the reciprocal function here
On the diagram, I have marked two different inputs3838 Yes, these happen to be two specific points on the curve, but the principle extends to the entire domain of the reciprocal function - including the negative
numbers to the reciprocal function (these are the vertical lines running from the \(x\)-axis to the curve) and the corresponding outputs (these are the horizontal inputs running from the curve to the \(y\)-axis). The main thing to notice is that although the orange input was smaller
when it was inputted, the actual output is larger than that of the larger red input (but smaller output).
These kinds of informal arguments are very useful when solving problems, but it is also possible to prove this in a more formal manner (see Section 10.6.1 for specifics), but this is not
the place.