An inequality is an relation that uses one of the following relationship operators:
|Less than or equal to|
|Not equal to; This could also be called less than or greater than|
|Greater than or equal to|
|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.
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.
|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.|
Inequalities are solved much the same was as equalities. Here is an example solving the inequality -x + 5 ≤ -5.
|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|
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