An inequality statement involving variables is either true or false depending on the values of the variables. When there is one variable, the solution of the statement is the set of values the variable may take such that the statement is true. There may be (sometimes infinitely) many possible solutions.
Inequality statements are equivalent if they have the same solution set. The solution set for an inequality is usually expressed as an interval. For example, the statements:
Are equivalent since the interval that contains the value of x for which the statements are true is the same:x + 1 < 2 2x < 2
An equality statement may be solved by transforming it into an equivalent statement for which the solution is obvious. The following operations transform an inequality statement into equivalent statements.(-∞, 1)
The same rule applies for the operators: ≤, >, ≥a < b a + c < b + c a - c < b - c
The same rule applies for the operators: ≤, >, ≥a < b a × c < b × c, where c ≠ 0 a ÷ c < b ÷ c, where c ≠ 0
Note that the direction or sense of the inequality operator switches as follows:a < b a × c > b × c, where c < 0 a ÷ c > b ÷ c, where c < 0
< becomes > ≤ becomes ≥ > becomes < ≥ becomes ≤