In an algebra course, students are introduced to the concept of order on the real numbers. This is a relation between the real numbers, denoted by the symbol , which is often read as "greater than". We have the following 4 order axioms for real numbers:
Axiom 1: [Trichotomy] If are real numbers, then one and only one of the following is true: , , .
Axiom 2: [Transitivity] If are real numbers and , then .
Axiom 3: If are real numbers and , then .
Axiom 4: If are real numbers such that , then .
With this, we can formally define a positive and negative number. A positive number is a number that satisfies , while a negative number is a number that satisfies . By the trichotomy axiom, a real number is either positive, negative, or zero. However, this does not appear to tell us much. In particular, is 1 a positive number?
An important inequality is the trivial inequality which states that the square of any real number is non-negative. This is easily shown by considering Trichotomy cases:
If , then .
If , then multiplying both sides by gives , where we keep the sign since is positive.
If , then (Axiom 3) and multiplying both sides by gives (Axiom 4), and adding to both sides yields (Axiom 3).
With the trivial inequality, we now know that is a positive number.
Polynomial inequalities can be solved by understanding how the graph behaves. Because the graph is continuous (naively this means that the graph can be drawn without lifting off your pen), we only need to find the roots of the polynomial and test one value in each of the corresponding regions to determine if they are positive or negative.
Determine the region in which .
The proper approach would be to shift the terms to one side, and then factorize, to get .
This has roots and , so we test the 3 different regions ( i.e. , to conclude that the inequality is satisfied for .
Note: Similar to the equality case , the first instinct of many students is to simply take square roots on the inequality and conclude that . This likewise leads to the wrong answer.
Determine the region in which .
To avoid dividing by 0, we want to ensure that the denominator is never 0. As such, we have to exclude as a possibility. From Axiom 4, we can only multiply by a quantity with a fixed sign.
Since is negative when , we cannot simply multiply by . Instead, we can multiply by , which we know is always non-negative, to obtain
At this point, it is tempting to cancel the term on both sides, but this is equivalent to multiplying by . We may not do so, we choose to factorize instead. Shifting all the terms to one side and factorizing, we get
We easily understand the graph of this quadratic equation, and (by using the number line) can conclude that the inequality is satisfied when (Recall that is excluded) or .
Show that for all real values of .
If we tried to calculate the roots to the quadratic equation , since the discriminant is negative, we know that the roots are complex numbers. Hence there is only 1 region on the number line. Testing shows that the quantity is always positive.
Another approach is to rewrite by Completing the square. Since both terms are squares, they are non-negative, and hence their sum is non-negative (axiom 3).
Show that the product of 2 positive numbers is positive.
If and are positive numbers, then . Using axiom 4, (with ), we get that . Hence, the product is positive.