An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. If there is an infinite collection of solutions to an inequality, we can’t list all of them.

There are three ways to write solutions to inequalities:

• using inequality notation
• as a graph
• as an interval

We’ll begin by reviewing inequality symbols and graphing inequalities on the number line, and then describe how to translate to interval notation.

Inequality Signs

The box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it is easy to get tangled up in inequalities; just remember to read them from left to right.

The inequality [latex]x>y[/latex] can also be written as [latex]{y}<{x}[/latex]. The sides of any inequality can be switched, as long as the inequality symbol between them is also reversed.

Graphing an Inequality

Inequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs. Graphs are a very helpful way to visualize information, especially when that information represents an infinite list of numbers!

[latex]x\leq -4[/latex]. This translates to all the real numbers on a number line that are less than or equal to -4. This includes -4, so we draw a closed dot at -4 on the number line. This is the endpoint of our solution set. All numbers less than -4 fall to the left of -4 on the number line, so we shade the portion on the number line to the left of -4. The graph of this inequality is shown below.

[latex]{x}\geq{-3}[/latex]. This translates to all the real numbers on the number line that are greater than or equal to -3. This includes -3 as well as all the numbers greater than -3. So we graph a closed dot at -3 and shade all the numbers to the right of -3.

Each of these graphs begins with a circle—either an open or closed (shaded) circle. This point is often called the end point of the solution. A closed, or shaded, circle is used to represent the inequalities greater than or equal to [latex] \displaystyle \left(\geq\right) [/latex] or less than or equal to [latex] \displaystyle \left(\leq\right) [/latex]. The point is part of the solution. An open circle is used for greater than (>) or less than (<). The point is not part of the solution.

The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex] \displaystyle x\geq -3[/latex] shown above, the end point is [latex]−3[/latex], represented with a closed circle since the inequality is greater than or equal to [latex]−3[/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]−3[/latex]. The arrow at the end indicates that the solutions continue infinitely.

Example

Graph the inequality [latex]x\ge 4[/latex]

Show Solution

We can use a number line as shown. Because the values for [latex]x[/latex] include [latex]4[/latex], we place a solid dot on the number line at [latex]4[/latex].

Then we draw a line that begins at [latex]x=4[/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to [latex]4[/latex].

This video shows an example of how to draw the graph of an inequality.

Represent Inequalities Using Interval Notation

Another commonly used, and arguably the most concise, method for describing inequalities and solutions to inequalities is called interval notation. With this convention, sets are built with parentheses or brackets, each having a distinct meaning. The solutions to [latex]x\geq 4[/latex] are represented as [latex]\left[4,\infty \right)[/latex]. This method is widely used and will be present in other math courses you may take.

The main concept to remember is that parentheses represent solutions greater than or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are [latex]\left[-2,6\right)[/latex], or all numbers between [latex]-2[/latex] and [latex]6[/latex], including [latex]-2[/latex], but not including [latex]6[/latex]; [latex]\left(-1,0\right)[/latex], all real numbers between, but not including [latex]-1[/latex] and [latex]0[/latex]; and [latex]\left(-\infty,1\right][/latex], all real numbers less than and including [latex]1[/latex]. The table below outlines the possibilities. Remember to read inequalities from left to right, just like text.

The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.

In the following video, you will see examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph.

Bounded Intervals: Width and Midpoint

An interval of the form [latex](a,b), (a,b], [a,b) \ \mathrm{or} \ [a,b],[/latex] where [latex]a[/latex] and [latex]b[/latex] are real numbers, is called a bounded interval. Such an interval does not extendd infinitely far in the positive or negative direction. The endpoints, [latex]a[/latex] and [latex]b[/latex], bound the interval. Another term for intervals of this form is a finite interval, because its width is finite.

The width of an interval of the form [latex](a,b), (a,b], [a,b) \ \mathrm{or} \ [a,b],[/latex] where [latex]a[/latex] and [latex]b[/latex] are real numbers, is the distance between its endpoints,

[latex]\mathrm{width} \ =b-a[/latex].

The midpoint of an interval of the form [latex](a,b), (a,b], [a,b) \ \mathrm{or} \ [a,b],[/latex] where [latex]a[/latex] and [latex]b[/latex] are real numbers, is the point which is the same distance from each of its endpoints,

[latex]\mathrm{midpoint} \ = \frac{a+b}{2}[/latex].