Inequality

Pronunciation: /ˌɪn ɪˈkwɒl ɪ ti/ ?

An inequality is an relation that uses one of the following relationship operators: [1]

Inequality
Operator
DescriptionGraph
<Less thanHollow circle on a number line with an arrow to the left.

<=
Less than or equal toSolid circle on a number line with an arrow to the left.

!=
#
Not equal to; This could also be called less than or greater thanHollow circle on a number line with an arrow to the left and to the right.

>=
Greater than or equal toSolid circle on a number line with an arrow to the right.
>Greater thanHollow circle on a number line with an arrow to the right.
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.

Graphing 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: Solid circle. If the variable can not be equal to the end point, draw a a hollow circle on the number line: Hollow circle. Then draw the lines representing the inequalities. Table 2 shows some examples.

EquationGraphExplanation
a < 5Number line with hollow circle on 5 with an arrow to the left.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 ≤ -3Number line with solid circle on -3 with an arrow to the left.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 ≥ 2Number line with solid circle on 2 with an arrow to the right.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 > -6Number line with hollow circle on -6 with an arrow to the right.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 ≤ 2Number line with hollow circle on -4, a solid circle on 2, and a line in between -4 and 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 ≥ 3Number line with solid circle on 1, a hollow circle on 3, an arrow going left from 1 and an arrow going right from 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.

Solving 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 ≤ -5Original 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 .
411 ≥ 10Substitute 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

More Information

  • McAdams, David. Complex Inequality. allmathwords.org. Life is a Story Problem LLC. 2009-10-26. http://www.allmathwords.org/article.aspx?lang=en&id=Complex%20Inequality.

Cite this article as:


Inequality. 2010-10-12. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/inequality.html.

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Revision History


2010-10-12: Opening paragraph incorrectly stated that an inequality is an equivalence relation. Changed to relation. (McAdams, David.)
2010-02-10: Added "References" (McAdams, David.)
2009-04-17: Fixed equations in example 3 (McAdams, David.)
2008-08-11: Fixed typographical errors (McAdams, David.)
2008-06-09: Initial version (McAdams, David.)

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