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

Less than

(-∞, 4)

2x − 1 ≤ 7

2x ≤ 8     x ≤ 4

Less than or Equal to

(-∞, 4]

2x − 1 > 7

2x > 8     x > 4

Greater than

(4, ∞)

2x − 1 ≥ 7

2x ≥ 8     x ≥ 4

Greater than or Equal to

[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