Inequalities
As we start this section, you should recall the basic rules which involve order (these are BR10, BR11, and BR12), that were introduced in the Real Numbers section. (These should start becoming quite familiar to you now.)We note the following:
 a < b implies that b > a implies that ba > 0
 a b implies that b a implies that ba > 0 or b = a.
Here are some rules for ordering real numbers. Try proving these yourself. If you run into difficulties, proofs or hints to the proofs are given for some.
Let a, b, c and d be real numbers:
 1)
 a < 0 and b < 0 implies that
ab > 0. Hint  2)
 a < b and b < c implies that
a < c. Proof  3)
 a < b implies that
a + c < b + c. Hint  4)
 a < b and c > 0 implies that
ac < bc Hint  5)
 a < b and c < 0 implies that
ac > bc. Multiplying by a negative number reverses the inequality.  6)
 a > 1 implies that
a^{2} > a. Hint  7)
 0 < a < 1 implies that
a^{2} < a  8)
 0 a < b implies that
a^{2} < b^{2} Proof  9)
 0 a, 0 b, and
a^{2} < b^{2} implies thata < b. Proof
Now that we have all of these rules, we can start solving inequalities. This is done by manipulating the inequality into a form that has the variable on one side and has a real expression on the other side of the inequality. For instance, if the variable is x and the real expression is represented by a then the final form of the inequality is one of the following:

x > a
x a
x < a
x a
When the inequalities involve absolte values, you must be very careful with the
use of the words and. and or . They end
up giving quite different results. Take a look
here to see what we mean. When we have the
absolute values, the and. condition applies
when we have < or signs and the
or. condition applies when we have >
or signs. So we have:
 x < a
 means a < x < a (which is the same as
a < x andx < a). x a means a x a (which is the same as a x andx a)  x > a
 means x < a or x > a
 x a
 means x a or x a
We can also have inequalities which involve polynomials. To solve these, we usually manipulate the inequality so 0 is on one side of the inequality then we factor the other side. Here are some examples.