PreCalculus

7. Inequalities and intervals

For any two values, x and y, the Law of Trichotomy states that one and only one of the following statements can be true for real numbers:

x is greater than y, or
x is equal to y, or
x is less than y

Inequalities

An inequality is an algebraic statement that the magnitude of one expression is greater than or less than another. On the real number line, magnitude increases in the positive direction. That is, to the right. Below, the value of b is greater than a.

        a                b
--------+----------------+-------

It is equivalent to state that a is less than b. If a could also be equal to b, we could state that b is greater than or equal to a; or, equivalently, a is less than or equal to b. These relationships can be represented with symbols:

a < b    a is less than b
a ≤ b    a is less than or equal to b
b > a    b is greater than a
b ≥ a    b is greater than or equal to a

The symbols, < and >, define what is known as the sense of the inequality. Two inequalities have the same sense if the inequality symbols point in the same direction and the opposite sense if the inequality symbols point in the opposite direction.

Intervals

An interval is a subset of the real numbers. An interval begins at one number and ends at another number. Inequality statements can be combined to define an interval using a statement such as:

a < x < b

This statement means that the variable x can take any value that is greater than a and is less than b. It defines an open interval which is written as (a, b); neither a nor b is in the interval. A closed interval would be defined as:

a ≤ x ≤ b

This would be written as [a, b]; both a and b are in the interval. Intervals can be closed to the left and open to the right or conversely; these are called half open intervals or half closed intervals. Below is a table showing the possible combinations of intervals.

a < x < b      (a, b)
a < x ≤ b      (a, b]
a ≤ x < b      [a, b)
a ≤ x ≤ b      [a, b]

Intervals can also be defined as unbounded to the right or to the left:

a < x          (a, +∞)
a ≤ x          [a, +∞)
x < b          (-∞, b)
x ≤ b          (-∞, b]

An unbounded, or infinite limit must be open because the end point cannot be defined.

7.1. Transformations of inequality statements
7.2. Solving linear inequalities
7.3. Solving polynomial inequalities
7.4. Solving rational inequalities
7.5. Solved problems