The instruction to “flip the sign” can feel like an arbitrary rule. A number line gives it a reason. Multiplying by a negative number reverses the order of numbers along that line.
Begin with a statement you can see: 2 < 5. Multiply both numbers by −1. The results are −2 and −5. On the number line, −2 lies to the right of −5, so the true statement is now −2 > −5. Keeping the original less-than sign would make the statement false.
Use the same idea with an unknown
Consider −3x < 12. To isolate x, divide both sides by −3. Division by a negative number also reverses the inequality, giving x > −4.
Check a value that should work. If x = 0, the original statement becomes 0 < 12, which is true. Check a value outside the solution: if x = −5, it becomes 15 < 12, which is false. At x = −4, both sides equal 12, so the strict inequality does not include that boundary.
Notice which operation causes the change
Adding or subtracting the same number on both sides does not reverse the order. Multiplying or dividing by a positive number also keeps the direction. The reversal is associated with multiplying or dividing both sides by a negative number.
- Write the operation beside the line of working.
- Ask whether the number used is positive or negative.
- Reverse the symbol only when the operation requires it.
- Substitute a sample solution into the original inequality.
For ≤ and ≥, the equality part remains included when the direction reverses. For example, dividing −2x ≥ 6 by −2 gives x ≤ −3.
Be careful when dividing by an expression whose sign is unknown. You cannot decide the direction without considering its possible values, and division by zero is not allowed. The number-line explanation is more dependable than a rule memorised without those conditions.