Inequality

Pronunciation: /ˌɪn.ɪˈkwɒl.ɪ.ti/ Explain

An inequality is an relation that uses one of the following relationship operators:

Inequality Operator	Description	Graph
<	Less than
≤ <=	Less than or equal to
≠ != #	Not equal to; This could also be called less than or greater than
≥ ≥	Greater than or equal to
>	Greater than
Table 1: Inequality operators

A compound inequality has more than one inequality operator.

Inequalities can be solved like equalities with one important difference: If the inequality is multiplied by a negative number, < changes to > and > changes to <. Take the true equation -16 < 5 Now multiply both sides by -1, but don't change '>' to '<'. -1 · -16 < -1 · 5 ⇒ 16 < -5 But 16 > -5! When multiplying an inequality by a negative number, always switch '<' to '>' and switch '>' to '<'. Equals and not equals do not change.

How to Graph One Variable Inequalities

When graphing one variable inequalities on a number line, start with the end point(s). If the variable can be equal to the end point, draw a solid circle on the number line: . If the variable can not be equal to the end point, draw a a hollow circle on the number line: . Then draw the lines representing the inequalities. Table 2 shows some examples.

Equation	Graph	Explanation
a ≤ 5		Since 5 is the endpoint, the circle is on 5. The inequality is '≤' so the circle is hollow. Since a ≤ 5, the arrow goes to the left.
a ≤ -3		Since -3 is the endpoint, the circle is on -3. The inequality is '≤' so the circle is solid. Since a ≤ -3, the arrow goes to the left.
t ≥ 2		Since 2 is the endpoint, the circle is on 2. The inequality is '≥' so the circle is solid. Since a ≥ 2, the arrow goes to the right.
t > -6		Since -6 is the endpoint, the circle is on -6. The inequality is '>' so the circle is hollow. Since t > -6, the arrow goes to the right.
-4 < r ≤ 2		Since -4 and 2 are the endpoints, the circles are on -4 and 2. The inequality for -4 is '<' so that circle is hollow. The inequality for 2 is '≤' so that circle is solid.
1 < r or r ≥ 3		Since 1 and 3 are the endpoints, the circles are on 1 and 3. The inequality for 1 is '<' so that circle is hollow. The inequality for 3 is '≥' so that circle is solid.
Table 2: Graphs of one variable inequalities.

How to Solve One Variable Inequalities

Inequalities are solved much the same was as equalities. Here is an example solving the inequality -x + 5 ≤ -5.

Step	Inequality	Explanation
1	-x + 5 ≤ -5	Original equation
2	-x + 5 - 5 ≤ -5 - 5 ⇒ -x ≤ -10	Subtract 5 from both sides
3	-1 · -x ≥ -1 · -10 ⇒ x ≥ 10	Multiply both sides by -1. Since the inequality is being multiplied by a negative number, change the ≤ to ≥.
4	11 ≥ 10	Substitute 11 in for x in the original equation to check the work. Since 11 ≥ 10, 11 is a solution to the inequality.
Table 3: Solving an inequality

References

1. McAdams, David E.. All Math Words Dictionary, inequality. 2nd Classroom edition 20150108-4799968. pg 98. Life is a Story Problem LLC. January 8, 2015. Buy the book

More Information

• McAdams, David E.. Complex Inequality. allmathwords.org. Life is a Story Problem LLC. 10/26/2009. https://www.allmathwords.org/en/c/complexinequality.html.

Cite this article as:

McAdams, David E. Inequality. 4/23/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/i/inequality.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

4/23/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
10/12/2010: Opening paragraph incorrectly stated that an inequality is an equivalence relation. Changed to relation. (McAdams, David E.)
2/10/2010: Added "References". (McAdams, David E.)
4/17/2009: Fixed equations in example 3. (McAdams, David E.)
8/11/2008: Fixed typographical errors. (McAdams, David E.)
6/9/2008: Initial version. (McAdams, David E.)