# Inequality

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

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

Inequality
Operator
DescriptionGraph
< 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.

EquationGraphExplanation
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.

StepInequalityExplanation
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
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

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

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

### 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.)