When do you flip the inequality sign

The direction of an inequality changes only when the operation reverses the order of the numbers involved. Multiplying or dividing by a negative number swaps the order of the real line, so the sign must flip to keep the statement true. For example, if $a < b$ and you multiply both sides by $-1$, the inequality becomes $-a > -b$. This rule applies to any inequality symbol: <, >, \le, or \ge.

When the direction changes

Consider the inequality $-3x > 9$. To isolate $x$, divide both sides by $-3$. Because you are dividing by a negative, you must flip the sign, giving $x < -3$. Check the result: choose $x = -4$; then $-3(-4) = 12$ which is indeed greater than 9, confirming the flipped direction is correct. If you had forgotten to flip, you would have obtained $x > -3$, which fails the original statement.

Situations that require flipping the inequality sign:

• Multiplying both sides by a negative number
• Dividing both sides by a negative number

Adding or subtracting a quantity, whether positive or negative, never changes the inequality direction. For instance, $5 < 7$ remains true after adding $-3$ to both sides: $2 < 4$. Subtracting works the same way: $5 < 7$ becomes $3 < 5$ after subtracting 2. These operations shift the whole number line without reversing order, so the sign stays the same.

Procedure to solve a linear inequality:

1. 1Simplify each side by combining like terms.
2. 2Isolate the variable using addition or subtraction.
3. 3If you need to multiply or divide by a negative, flip the inequality sign.
4. 4Write the solution in interval notation if required.

Effect of common operations on inequality direction: