An inequality compares two expressions, identifying one as more, less, or simply different than the other. We will review the mathematical symbols used to represent various relationships between two quantities. Values of a variable which make a statement true are called the solution set. There may be a few or infinitely many numbers which satisfy an inequality. We will also review ways to represent these solution sets.

The symbol “<” means “less than.”  For example, 3 < 5.  The symbol “>” means “greater than.”  You can think of the inequality as an arrow which points to the smaller number. Or, you may have learned to think of the inequality as a hungry fish’s mouth, about to eat the larger number.

Inequality Symbols

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.

For example, [latex]2<7[/latex] is true, as is [latex]7>2[/latex].  So the statement [latex]2<x[/latex] can also be written [latex]x>2[/latex].  Any real number larger than [latex]2[/latex] is a solution to the inequality.

The expressions on each side of an inequality can be interchanged as long as the inequality symbol between them is also reversed.

Graphing an Inequality

Another way to represent an inequality is by graphing it on a real number line:

Consider the inequality [latex]x\leq -4[/latex]. This translates to all the real numbers on a number line that are less than or equal to [latex]4[/latex].

Consider the inequality [latex]{x}\geq{-3}[/latex]. This translates to all the real numbers on the number line that are greater than or equal to [latex]-3[/latex].

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 end point is part of the solution. An open circle is used for greater than (>) or less than (<). The end point is not part of the solution. When the end point is not included in the solution, we often say we have strict inequality rather than inequality with equality.

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.

Example

Write an inequality describing all the real numbers on the number line that are strictly less than [latex]2[/latex]. Then draw the corresponding graph.

Show Solution

We need to start from the left and work right, so we start from negative infinity and end at [latex]2[/latex]. We will not include either because infinity is not a number, and the inequality does not include [latex]2[/latex].

Inequality: [latex]x\lt2[/latex]

To draw the graph, place an open dot on the number line first, and then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity.

Note

Sometimes we are only interested in integer values that satisfy an inequality.  Suppose [latex]x[/latex] represents the number of cars a person owns and we are told [latex]x < 2[/latex].  Since no one can own a negative number of cars, our solutions set would be the nonnegative integers which are less than [latex]2[/latex].  Then our possible solutions are [latex]0[/latex] and [latex]1[/latex].  We would graph these individual points by plotting solid dots at these values.