Strategies for Inequalities

I) Properties of absolute Functions :

a) |x| =x if x '>=' 0
|x| = -x if x '<' 0 * Remember the pattern...|expression| '<'>' -a
b) |x| = sqrt(x^2)
c) |x| '>' a -------> x '<' -a or x '>' a
d) |x| '<' a -------> -a '<' x '<' a a') |quantity| '>'A: quantity '>' A OR quantity '<'-A b') |quantity| '<' A: -A '<' quantity '<' A in the extremely unlikely event that A is negative, the b') has no solution at all, and the ai) will be solved by any value of x.

1)     DO NOT DIVIDE BY VARIABLES. Factor the variables for equality.

2)     Whenever you encounter the inequality as :

x/y ‘>’ 1

Then 2 possibilities take place:

a)     If y is + ve, then x has to be +ve and x’>’y’>’0

This can be further factored as x-y ‘>’ 0

b)     If y is - ve, then x has to be  -ve and x’<’y’<’0

This can be further factored as x-y ‘<’ 0

2^n and n^2:
watch the behavior of 2^n and n^2 as you get further and further away from 4. the pattern you'll observe is that 2^n begins to grow much, much faster than does n^2, making it clear that the two expressions won't be equal for any larger values.

- You can ADD TWO INEQUALITIES TOGETHER if the inequalities are BOTH "<" OR BOTH ">".