Algebra Of Inequalities
Inequalities involving real numbers are particularly important. There are four types of inequalities, or ordering relations, that are important when dealing with real numbers. They are (together with their symbols) "less than" (<), "less than or equal to" (≤), "greater than" (>), and "greater than or equal to" (≥). In each case the symbol points to the lesser of the two expressions being compared. Since, by convention, mathematical expressions and statements are read from left to right, the statement x + 2 < 6 is read "x plus two is less than six," while 6 > x + 2 is read "six is greater than x plus two." Algebraically, inequalities are manipulated in the same way
that equalities (equations) are manipulated, although most rules are slightly different.
The rule for addition is the same for inequalities as it is for equations:
for any three mathematical expressions, call them A, B, and C, if A > B then, A + C > B + C.
That is, the truth of an inequality does not change when the same quantity is added to both sides of the inequality. This rule also holds for subtraction because subtraction is defined as being addition of the opposite or negative of a quantity.
The multiplication rule for inequalities, however, is different from the rule for equations. It is: for any three mathematical expressions, call them A, B, and C, if A < B, and C is positive, then AC < BC, but if A < B, and C is negative, then AC > BC.
This rule also holds for division, since division is defined in terms of multiplication by the inverse.