Well someone requested me to write a guide to solving different types of inequalities and I think this will be useful for all
Also before I start, if you guys want to request guides on any other mathematical topics, just let me know and I'll be happy to write one.
Firstly solving inequalities is very different from solving your garden variety equalities like
etc. Inequalities need to be treated with special consideration. In fact there is a whole branch of algebra in pure mathematics that dedicates itself to inequalities, there are some very famous inequalities, the CS inequality, AM-GM, Chebyshev inequality etc but we won't go into that detailed
Although if you want to know more about them, feel free to discuss them with me!
Here are some "skills" which you should confirm for yourself and remember! These are the fundamental arithmetic you need to remember when dealing with inequalities.
1. Addition: If
and
then
2. Multiplication: If
and
then
, if
then
.
3. Reciprocals: If
then
provided both
and
have the same sign. NOTE: change of sign!
Now I don't think we need to go through how to solve stuff like
and alike, those should be common knowledge. We will focus on 4 main types of inequalities which often causes alot of trouble:
1. Quadratic and higher degree inequalities
2. Rational inequalities
3. Square root inequalities
4. Absolute value (modulus) inequalities.
We will first focus on 1.
Quadratic and higher degree inequalitiesI will illustrate the main concept of how to solve these through examples, each example illustrates a tactic in which you need to remember!
Example 1:
Solve
for
Notice this question could also have had
rather than
, we will use the same tactic.
First always factorise the given quadratic (we will see what happens if you can't later on in the guide)
So we have
(x+4)>0)
.
Notice on the LHS we have 2 brackets which we can consider as 2 "numbers". If you can not see why, let
and
so now we have
.
Now if we just use some common sense, we can see that if
then
and 
but we could also have
and 
. Notice the word "and" this will be useful later.
So this means we have either
and
(just substituting back in what
and
represents)
or 
and
Let us solve the first possibility:
and
So we have
and
but common sense tells us anything greater than 2 MUST ALSO be greater than -4, so
is redundant and our answer is simply just
. In more formal terms, we used the conjunctive "and" between our 2 inequalities, "and" means we take the intersection (what is common to both) of the 2 inequalities which is
.
Let us solve the second possibility:
and
So we have
and
using the same logic as before, our solution here is
. Convince yourself why!
Now read back a bit see what word we have between our 2 possible cases. Yes that's right, the word "or". What does "or" mean formally? The word "or" is a disjunctive, this means that unlike "and" where we just take the intersection, for "or" we must take the union of the 2 cases. But what does union mean? It makes means we take 'everything'. Eg,
or
is simply every single real number!
So connecting our final answer through the word "or" we have
(solution of first case) or
(solution of second case). More formally we can write this as:
. Basically I just translated the solution we got in English into maths language. We use
to represent a set of numbers, since our solution is not just 1 number, it's a whole group of numbers (duh cause it's an inequality!). The : literally reads "such that" and
means "or" in maths language. So read
that out aloud to yourself, what do you get? You should get: "The solution is the set of numbers of
such that
is greater than 2 or
is less than -4"
Another way of solving this inequality is simply to sketch the graph of the quadratic and see what parts of the quadratic is lies above the x axis and find the
values that correspond to it. You should get the same answer as above.
I'll leave it here for tonight because this guide will be quite long, I'll finish it tomorrow or something, taking an early night
locked until it's finished.
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