Algebra / Linear Inequalities
Inequality sign reversal
When solving inequalities, multiplying or dividing by a negative number reverses the inequality sign. This is a common source of wrong answers.
Least you need to know
- Reverse the sign only when multiplying or dividing by a negative.
- Adding or subtracting does not reverse the sign.
- Check a sample value if the direction seems uncertain.
Key notation
- < — Less than
- \ge — Greater than or equal to
Worked example
For $-4x>12$, divide by -4 and reverse the sign to get $x<-3$.
Common mistakes
- Forgetting the reversal gives the opposite solution set.
- Distributing a negative can hide the reversal step later.
How to recognize it
- A negative coefficient is attached to the variable.
- The answer choices differ only by inequality direction.
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