Inequalities
An algebraic inequality is an inequality in which two members are linked by one of these signs:
| < | less than | 2x − 1 < 7 |
| ≤ | less than or equal to | 2x − 1 ≤ 7 |
| > | greater than | 2x − 1 > 7 |
| ≥ | greater than or equal to | 2x − 1 ≥ 7 |
The solution of an inequality is the set of values which verifies the variable in the inequality.
The solution of the inequality can be expressed by:
1. A graphical representation.
2. An interval.
2x − 1 < 7
2x < 8 x < 4
(-∞, 4)
2x − 1 ≤ 7
2x ≤ 8 x ≤ 4
(-∞, 4]
2x − 1 > 7
2x > 8 x > 4
(4, ∞)
2x − 1 ≥ 7
2x ≥ 8 x ≥ 4
[4, ∞)
Properties of Inequalities
If the two members of an inequality are added or subtracted by the same number, the resulting inequality is equivalent to that given.
3x + 4 < 5 3x + 4 − 4 < 5 − 4 3x < 1
If the two members of an inequality are multiplied or divided by the same positive number, the resultant inequality is equivalent to the given one.
2x < 6 2x : 2 < 6 : 2 x < 3
If the two members of an inequality are multiplied or divided by the same negative number, the resultant inequality changes sense and it is equivalent to the given one.
−x < 5 (−x) · (−1) > 5 · (−1) x > −5
