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A "solution'' of an inequality is a number which when substituted for the variable makes the inequality a true statement.

Consider the inequality

                 x - 2 > 5

When we substitute 8 for x, the inequality becomes 8-2 > 5. Thus, x=8 is a solution of the inequality. On the other hand, substituting -2 for x yields the false statement (-2)-2 > 5. Thus x = -2 is NOT a solution of the inequality. Inequalities usually have many solutions.

As in the case of solving equations, there are certain manipulations of the inequality which do not change the solutions. Here is a list of "permissible'' manipulations:

Rule 1. Adding/subtracting the same number on both sides.

Example: The inequality x-2>5 has the same solutions as the inequality x > 7. (The second inequality was obtained from the first one by adding 2 on both sides.)

 

Rule 2. Switching sides and changing the orientation of the inequality sign.

Example: The inequality 5-x> 4 has the same solutions as the inequality 4 < 5 - x. (We have switched sides and turned the ``>'' into a ``<'').

 

            Last, but not least, the operation which is at the source of all the trouble with inequalities:

Rule 3a. Multiplying/dividing by the same POSITIVE number on both sides.

 

Rule 3b. Multiplying/dividing by the same NEGATIVE number on both sides AND changing the orientation of the inequality sign.

Examples: This sounds harmless enough. The inequality  2x =< 6 has the same solutions as the inequality x =<  3 . (We divided by +2 on both sides).

The inequality -2x > 4 has the same solutions as the inequality x< -2. (We divided by (-2) on both sides and switched ">'' to "<''.)

But Rule 3 prohibits fancier moves: The inequality x² > x . DOES NOT have the same solutions as the inequality x > 1. (We were planning on dividing both sides by x, but we can't, because we do not know at this point whether x will be positive or negative!) In fact, it is easy to check that x = -2 solves the first inequality, but does not solve the second inequality.

 

            Only ``easy'' inequalities are solved using these three rules; most inequalities are solved by using different techniques.