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 Description  
 Inequality:  Any mathematical sentence that contains the symbols >(greater than), <(less than), <(less than or equal to), or >(greater than or equal to).
Inequalities

Inequalites can be solved using the following properties of order. Similar properties hold for >, <, and >.
  • If a < b, then a + c < b + c.
  • If a < b and c > 0, then ac < bc.
  • If a < b and c < 0, then ac > bc.
Solutions for inequalities in one variable are graphed on one axis, or a number line. Consider the following example.

3x - 4 < x + 12 =>

2x -4 < 12 =>

2x < 16 =>

x < 8

This result tells us that all the values less than 8 will satisfy this inequality. The set of solutions, graphed on a number line below, is indicated in red.



Solution sets can be checked by substituting individual solutions back into the initial inequality. If the inequality remains true, then the number substituted came from the solution set. If the inequality becomes false, the number substituted did not come from the solution set. For example, let's check to see if 9 is a solution for the given inequality.

3(9) - 4 < 9 + 12 =>

27 - 4 < 21 =>

23 < 21

This last inequality is false, so 9 does not belong to the solution set. And if we look at the number line above, we can see that 9 is indeed not part of the shaded solution set.

Let's check to see what 5 gives us:

3(5) - 4 < 5 + 12 =>

15 - 4 < 17 =>

11 < 17

This last inequality is true, so we know that 5 is a solution to the given inequality.