7.3 Inequalities (Inequations)
A statement that a quantity is either greater than or less than but not equal to the other. Instead of using an equal sign (=) as in an equation, these symbols are used : > (greater than ) and < (less than) or ³ (greater than or equal) and £ (less than or equal to).
Axioms and properties. For all real numbers a, b, and c
Axioms :1. Trichotomy: one of the relations a < b, a = b is
true.
2. Transitivity: If a > b, and b > c, then a > c. Also, If a < b, b < c, then a < c.
Fundamental properties :
1) If a > b, then a + c > b + c
This can be seen geometrically, since c shifts both a and b the same distance (to the right when c is positive and to the left when c is negative). Thus their relative positions are the same before and after shifting.
For example 4 > 2 so 4 + 1 > 2 + 1 and 4  1 > 2 1
2) If a > b and c > d then a + c > b + d
3) i) If a > b and c > 0 then a c > b c
ii) If a > b and c < 0 then a c < b c
Note : What (3 ii) shows is that when an inequality is multiplied (or divided) by a negative number, the inequality is reversed.
4) If a and b are either both positive or both negative and a > b, then
Solving inequalities. To solve an inequality means to find all solutions. This is done in the same way as solving equations. Except, if you multiply or divide both sides by a negative number, you must reverse the direction of the sign.
Example Solve  7 x + 4 < 18, x Î {2, 0, 1,3}. Graph the solution set.
Solution :  7 x + 4 < 18
 7 x < 18  4 i. e.  7 x < 14
Dividing throughout by  7 , we get
x >  2
\ Solution set = {0, 1, 3}
Note : 1) When graphing inequalities involving, only integers dots are used.
2) When graphing inequalities involving real numbers, lines, rays and dots are
used.
3) A dot is used if the number is included. A hollow dot is used if the number is not included.
