In mathematics, inequalities are used to compare the relative size of values. They can be used to compare integers, variables, and various other algebraic expressions. A description of different types of inequalities follows.

Strict Inequalities

A strict inequality is a relation that holds between two values when they are different. In the same way that equations use an equals sign, =, to show that two values are equal, inequalities use signs to show that two values are not equal and to describe their relationship. The strict inequality symbols are $<$ and $>$.

Strict inequalities differ from the notation $a \neq b$, which means that a is not equal to $b$. The $\neq$ symbol does not say that one value is greater than the other or even that they can be compared in size.

In the two types of strict inequalities, $a$ is not equal to $b$. To compare the size of the values, there are two types of relations:

1. The notation $a < b$ means that $a$ is less than $b$. 
2. The notation $a > b$ means that $a$ is greater than $b$.

The meaning of these symbols can be easily remembered by noting that the "bigger" side of the inequality symbol (the open side) faces the larger number. The "smaller" side of the symbol (the point) faces the smaller number.

The above relations can be demonstrated on a number line. Recall that the values on a number line increase as you move to the right. The following therefore represents the relation $a$ is less than $b$: 

$a < b$

$a$ is to the left of $b$ on this number line.

and the following demonstrates $a$ being greater than $b$: 

$a > b$

$a$ is to the right of $b$ on this number line.

In general, note that:

• $a < b$ is equivalent to $b > a$; for example, $7 < 11$ is equivalent to $11> 7$.
• $a > b$ is equivalent to $b < a$; for example, $6 < 9$ is equivalent to $9 > 6$.

Other Inequalities

In contrast to strict inequalities, there are two types of inequality relations that are not strict:

• The notation $a \leq b$ means that $a$ is less than or equal to $b$ (or, equivalently, "at most" $b$).
• The notation $a \geq b$ means that $a$ is greater than or equal to $b$ (or, equivalently, "at least" $b$).

Inequalities with Variables

In addition to showing relationships between integers, inequalities can be used to show relationships between variables and integers.

For example, consider $x > 5$. This would be read as "$x$ is greater than 5" and indicates that the unknown variable $x$ could be any value greater than 5, though not 5 itself. For a visualization of this, see the number line below:

$x > 5$

Note that the circle above the number 5 is open, indicating that 5 is not included in possible values of $x$.

For another example, consider $x \leq 3$. This would be read as "$x$ is less than or equal to 3" and indicates that the unknown variable $x$ could be 3 or any value less than 3. For a visualization of this, see the number line below:

$x \leq 3$

Note that the circle above the number 3 is filled, indicating that 3 is included in possible values of $x$.

Inequalities are demonstrated by coloring in an arrow over the appropriate range of the number line to indicate the possible values of $x$. Note that an open circle is used if the inequality is strict (i.e., for inequalities using $>$ or $<$), and a filled circle is used if the inequality is not strict (i.e., for inequalities using $\geq$ or $\leq$).

Solving Problems with Inequalities

Recall that equations can be used to demonstrate the equality of math expressions involving various operations (for example: $x + 5 = 9$). Likewise, inequalities can be used to demonstrate relationships between different expressions.

For example, consider the following inequalities:

• $x - 7 > 12$
• $2x + 4 \leq 25$
• $2x < y - 3$

Each of these represents the relationship between two different expressions.

One useful application of inequalities such as these is in problems that involve maximum or minimum values. 

Example 1

Jared has a boat with a maximum weight limit of 2,500 pounds. He wants to take as many of his friends as possible onto the boat, and he guesses that he and his friends weigh an average of 160 pounds. How many people can ride his boat at once?

This problem can be modeled with the following inequality:

$160n \leq 2500$

where $n$ is the number of people Jared can take on the boat. To see why this is so, consider the left side of the inequality. It represents the total weight of $n$ people weighing 160 pounds each. The inequality states that the total weight of Jared and his friends should be less than or equal to the maximum weight of 2,500, which is the boat's weight limit.

There are steps that can be followed to solve an inequality such as this one. For now, it is important simply to understand the meaning of such statements and cases in which they might be applicable.