An inequality is an equation without an equal sign. Instead there is a less-than sign, greater-than sign, greater-than or equal to sign, etc. The statement n < 5 means n is less than 5, and 2n > 5, means that 2n is greater than 5. The same rules apply to inequalities as they do to equations, in that you can subtract or add any number to both sides of the inequality and still retain the relationship. Also, when you divide or multiply each side of the inequality by a positive number, then the relationship also holds. However if you multiply or divide each side of an inequality by a negative number, then the inequality is REVERSED. This is an important concept that you should know, because questions that require you to reverse an inequality may appear on the math assessment section of your GRE exam.
Multiplying or dividing an inequality by a negative number reverses the inequality, but multiplying or dividing by a positive number or doing any type of addition or subtraction on both sides of the inequality maintains the inequality relationship.
Here is an example of when the inequality does not change, because we do not multiply or divide by a negative number:
| Step 0 | Original Inequality, solve for n | 12n > 4 |
| Step 1 | Divide both sides of the inequality by 4 | 3n > 1 |
| Step 2 | Divide both sides by 3 | n > 1/3 |
Here is an example of when the inequality does change, because we multiply by a negative number:
| Step 0 | Original Inequality, solve for x | -12x + 3 < 7 |
| Step 1 | Subtract 3 from both sides | -12x < 4 |
| Step 2 | Divide both sides by -4, and we must CHANGE the inequality sign | 3x > -1 |
| Step 3 | Divide both sides by 3 | x > -1/3 |
Whenever you are given a question with an inequality sign, just follow the same rules as if you were solving an equation, but remember to reverse the inequality when you multiply or divide by a negative number.