Inequalities

Basic model

We now introduce the concept of inequality. In the most basic form, an inequality is the definition of an order between the numbers (or elements of our set), that gives some structure to the elements in relation to each other.

Let's be a bit less abstract and give the most basic example:

$$3 < 5$$

This means that the number 3 is "less than" the number 5. In our unary notation with "xs" or "s(s(s(...." it would simply indicate the number with the most amount of "s" or "x". As the definition feels a bit circular, since it requires the sense of the word "most", we can give a more difficult one and less dependent on previous definitions.

$$a < b$$

It means that if we were to count up starting from the number "a", we would reach "b" only after counting up $b-a$ times. If "b" and "a" are not known, we only know that we'll reach someday "b" starting from "a" and counting up (therefore $b-a$ is also unknown).

In the previous example this means that we take our "3" and count up, sure that one day we'll reach "5". This happens exactly after counting up "5-3" times.

Adding and removing a value from both sides of the inequality does not change its value. Instead, multiplying can change the truth value of the inequality when the value we multiply by is negative. As an example:

$$3 < 5$$ $$3*(-1) < 5*(-1)$$ $$-3 < -5$$

But if we were to count up from "-3", we would reach instead "-2","-1","0",... without ever reaching "-5". Therefore the order inverts when multiplying (or dividing!) by a negative number.

$$-3 > -5$$

The $>$ symbol is the same as the $<$ symbol, except with its arguments inverted. There are also the "or equal variants", where $\leq$ means "less or equal", while $\geq$ means "greater or equal". These allows inequalities to set "less strict" boundaries, rather than being strictly superior. For example:

$$3 \leq 3$$ $$5 \geq 3$$

With variables

Introducing "x" into the mix, we can now have expressions as

$$3 < x < 5$$

where the true "logical form" would be expressed as

$$3 < x \land x < 5$$

So we separate the two $<$ symbols, into two inequalities, and can treat each separately. By the way, this also was the case for equality, as

$$3=x=2+1$$

is simply a conjunction (logical and) with the following written form

$$3=x \land x=2+1$$

How to solve each equality kind of depends on the form of the equation. Generally we would like a form

$$x = \ldots$$

where the expression at the right of the equal (the dots) is any expression that doesn't involve "x". If it did then, we would not gain any insight as what the true value of "x" would be, as it would be self referential.

For the moment, the only subset of equations we really know how to solve is "linear equations". That means equations such that a variable "x" is not being multiplied by another instance of "x".

Let's now consider "a" and "b" as constant numbers (meaning their value is already set in stone, and we do not need to find it ourselves, and it can be anything):

$$a*x+b = 0$$

Any equation of the linear variety can be then "solved" by the following:

• Subtract "b":

$$a*x+b - b = 0 - b$$ $$a*x = - b$$

• Divide by "a":

$$\frac{a*x}{a} = \frac{-b}{a}$$ $$\frac{a}{a}*x = - \frac{b}{a}$$ $$x=-\frac{b}{a}$$

Considering instead an equation like:

$$x*x = 4$$

We cannot simply divide by "x", as otherwise we would end up with

$$x = \frac{4}{x}$$

which is arguably worse, as we're nowhere close to understanding what "x" could be. When considering an equation with a "x*x" term (that isn't being simplified of course, so no form like $(x*x)/x$) we call that one "quadratic".

Why quadratic?

The exact reason for the term is found in the next article about exponentials, but we can link the common etymology of the term "quadro" to "square", and in a bit of a spoiler for geometry, finding the area of a square is simply multiplying the side by itself, "x*x". Since I'm introducing concepts I've not yet explained here, this paragraph is purely optional.

The actual solutions to the above equation are both "-2" and "2". We can "fact check" by substituting the "x" with our number:

$$\begin{aligned} x * x = 4 \implies & 2 * 2 = 4 & \implies 4 = 4 \\ x * x = 4 \implies & (-2) * (-2) = 4 & \implies 4 = 4 \end{aligned}$$

This means that "non-linear" (anything that is not linear) equations can have multiple solutions. There are not always straightforward solutions, so finding these is going to require creativity and a lot of work.

Back to inequalities

Unlike equalities, even linear inequalities can have multiple values. Take a form such as:

$$x < 5$$

This means that to make that statement true, "x" can have any value like "3","0", "4/5", but not "5", "7" ... Since there are endless numbers that we can make that are lower than 5 (but not in the natural set!), then the statement has a number of answers that we cannot really write down, therefore we are satisfied by just reaching the form $x<5$.

Let's try something more abstract, with "a" and "b" as our "placeholder/constant" numbers.

$$a*x + b < 0$$

Same steps as before..

$$a*x + b - b < - b$$ $$a*x < - b$$ $$x < - \frac{b}{a}$$

Okay, let's try with $a=1$ and $b=2$ ...

$$x + 2 < 0$$ $$x < -\frac{2}{1}$$ $$x < - 2$$

And if I were to substitute $x=-3$...

$$-3 + 2 < 0$$ $$-1 < 0$$

Everything works out!

What if "a" was negative? In our steps above, when dealing with equality, we have a step where we divide "a". In the equality this did not change the truth value, but here we need to swap things around (specifically the direction of the inequality)!

So, with $a = -2$ and $b = 6$:

$$-2*x + 6 < 0$$ $$x > - \frac{6}{-2}$$ $$x > - - \frac{6}{2}$$ $$x > - - 3$$ $$x > 3$$

And if we pick $x=4$:

$$-2*4 + 6 < 0$$ $$-8 + 6 < 0$$ $$-2 < 0$$

If we picked $x<3$ our inequality would have been false. When we get to graphing, remember this property, whether our "a" is positive or negative.