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Inequalities

You know that 2 is a larger number than 1, and you know that ‐2 is a smaller number than ‐1, so how do you show both of these statements to be true?

While an equation is a demonstration of two quantities equal to one another, inequalities demonstrate two quantities not equal to one another. This includes algebraic expressions. You can show this relationship of inequality in four ways:

Less Than <

x < 3 means the the value of x is less than 3 but cannot equal 3

Less Than or Equal To ≤

x ≤ 3 means the the value of x can be any value less than 3 or the value of x can be 3

Greater Than or Equal To ≥

x ≥ 3 means the value of x is greater than or equal to 3; the value can be 3

Greater Than >

x > 3 means the value of x is greater than 3 but cannot equal 3

Keep in Mind

The definition of the inequality sign refers to the value on the left hand side of the sign
Think of the inequality sign as an open mouth that is always going towards the greater value (i.e. a person who is hungry will want the greater portion of food)
You can also remember less-than because the less-than sign (<) is shaped like a slanted “L.”

Solving Inequalities

The nice thing about solving algebraic inequalities is that the process is the same as balancing algebraic equations and solving for the correct value in algebraic equations.

The only difference is that whenever you multiply or divide both sides of the inequality by a negative number the direction of the sign switches.

Showing or graphing inequalities on a number line isn’t very different from showing a value on a number line.

The only difference is that an inequality will have many solutions.

Graphing Inequalities on a Number Line

A trick to remember which way to draw the arrow is to look at the inequality sign. The direction it is pointing creates an arrow for you.

Interval Notation Review

Another way to express the solutions to inequalities is with interval notation. Interval notation, like inequalities, expresses a range of numbers, and is written as (lower bound, upper bound).

For example, the inequality x≥1 is saying that x is every number greater than or equal to 1, from 1 to infinity (∞). We can express this as [1,∞).

The brackets [] represent a closed circle on the number line, and mean that the particular number is included, while parentheses () represent an open circle, where the particular number isn’t included. We always put parentheses next to infinity, because we can never actually arrive at infinity.

Compound Inequalities

Sometimes, algebraic inequalities are compared to more than one value or an algebraic inequality has more than one set of solutions (absolute value of an inequality).

These cases are called compound inequalities and can be represented in a series of inequality signs or two or more statements of inequalities joined by a conjunction. Compound inequalities are graphed in a similar manner to single inequalities, but will either have two rays or a solid line.

A solid line occurs when the algebraic expression is greater than the first value but less than the second value (which is greater than the first). All signs are "less than" signs (<) 1

< x ≤ 4

You may also see this written as (1,4) in interval notation.

Two rays occur when the algebraic expression is less than the first value but greater than the second value, (which is greater than the first). This may be written as

x < 1 and x ≥ 4 (two separate inequalities)

You may also see this written as (-∞, 1) ∪ [4,∞) in interval notation.