| Inequality: Any mathematical sentence that contains the symbols >(greater than), <(less than), <(less than or equal to), or >(greater than or equal to).
Inequalites can be solved using the following properties of
order. Similar properties hold for >, <, and
Solutions for inequalities in one variable are graphed on one
axis, or a number line. Consider the following example.
- 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.
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.