Let's look at some specific inequalities and perform operations to both sides to determine if the inequality still holds.
| Inequality |
Operation |
Left Side |
Right Side |
Inequality still true? |
| \(-2\lt5\) |
add \(3\) to both sides |
\(1\) |
\(8\) |
\(1\lt8\) |
True |
| \(-2\lt5\) |
subtract \(3\) from both sides |
\(-5\) |
\(2\) |
\(-5\lt2\) |
True |
| \(-8\gt-12\) |
multiply both sides by \(2\) |
\(-16\) |
\(-24\) |
\(-16\gt-24\) |
True |
| \(-8\gt-12\) |
divide both sides by \(2\) |
\(-4\) |
\(-6\) |
\(-4\gt-6\) |
True |
| \(-8\lt-6\) |
multiply both sides by \(-2\) |
\(16\) |
\(12\) |
\(16\lt12\) |
False |
| \(8\gt-12\) |
divide both sides by \(-2\) |
\(-4\) |
\(6\) |
\(-4\gt6\) |
False |
When solving inequalities, you can add or subtract the same number from each side of the inequality and the inequality will remain true.
You can multiply or divide both sides of the inequality by the same positive number and the inequality will remain true.
However, when you multiply or divide both sides of an inequality by the same negative number, the resulting inequality becomes false.
To make the inequality true again, reverse the direction of the inequality symbol.