Inequality Symbols and Graphs
Learning Outcomes
- Represent inequalities using an inequality symbol
- Represent inequalities on a number line
Inequality Symbols
One way to represent such a list of numbers, an inequality, is by using an inequality symbol:- [latex]{x}\lt{9}[/latex] indicates the list of numbers that are less than [latex]9[/latex]. Since this list is infinite, it would be impossilbe to list all numbers less than [latex]9[/latex].
- [latex]-5\le{t}[/latex] indicates all the numbers that are greater than or equal to [latex]-5[/latex].
- [latex]{x}\lt{9}[/latex]
- [latex]{9}\gt{x}[/latex]
- [latex]x\lt5[/latex] means all the real numbers that are less than 5, whereas;
- [latex]5\lt{x}[/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\gt{5}[/latex]. Note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5 which is easier to interpret than 5 is less than x.
Symbol | Words | Example |
---|---|---|
[latex]\neq [/latex] | not equal to | [latex]{2}\neq{8}[/latex], 2 is not equal to 8. |
[latex]\gt[/latex] | greater than | [latex]{5}\gt{1}[/latex], 5 is greater than 1 |
[latex]\lt[/latex] | less than | [latex]{2}\lt{11}[/latex], 2 is less than 11 |
[latex] \geq [/latex] | greater than or equal to | [latex]{4}\geq{ 4}[/latex], 4 is greater than or equal to 4 |
[latex]\leq [/latex] | less than or equal to | [latex]{7}\leq{9}[/latex], 7 is less than or equal to 9 |
Graphing an Inequality
Another way to represent an inequality is by graphing it on a number line: Below are three examples of inequalities and their graphs. Graphs are often helpful for visualizing information. [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]. [latex]{x}\geq{-3}[/latex]. This translates to all the real numbers on the number line that are greater than or equal to -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 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]Answer: 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].
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.Answer: 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.
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Solving One-Step Inequalities from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Graph Linear Inequalities in One Variable (Basic). Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.