Why Does an Inequality Sign Flip When You Divide by a Negative?

When must you reverse the inequality sign?

Reverse an inequality sign when you multiply or divide both sides by a negative number. For example, −2x < 6 becomes x > −3 after dividing by −2. Adding or subtracting the same number on both sides does not reverse the sign.

Why the rule works

Start with 2 < 5. Multiplying both numbers by −1 gives −2 and −5. On a number line, −2 lies to the right of −5, so −2 > −5. Multiplication by a negative reflects the order of the numbers. The direction must therefore reverse.

Multiplying by a positive number preserves order. Multiplying by zero destroys the information: both sides become zero. Never divide by zero.

Worked example 1

Solve −3x + 4 ≤ 19.

1. Subtract 4 from both sides: −3x ≤ 15.
2. Divide both sides by −3 and reverse ≤ to ≥: x ≥ −5.
3. Test the boundary: x = −5 gives 19 ≤ 19, which is true.
4. Test a value inside the solution: x = 0 gives 4 ≤ 19, true.
5. Test a value outside: x = −6 gives 22 ≤ 19, false.

These checks confirm both the boundary and the direction.

Worked example 2

Solve 7 − 2x > 11. Subtract 7 to get −2x > 4. Dividing by −2 reverses the sign, so x < −2. The endpoint −2 is excluded because it produces equality, not a strict inequality. On a number line, use an open circle at −2 and shade to the left.

A common trap: dividing by an unknown

Do not divide an inequality by x unless you know whether x is positive or negative, and know that it is not zero. Otherwise, use cases or a sign analysis. The reversal rule depends on the sign of the quantity used for division, not just on whether the original expression contains a minus sign.

Practice with checks

• −4x < 12 gives x > −3. The test value 0 works; −4 does not.
• 5 − x ≥ 8 gives x ≤ −3. The boundary gives 8 ≥ 8.
• x/−2 ≥ 3 gives x ≤ −6. Multiplying by −2 reverses the direction.

If you want feedback, include your steps in the DoubtPay study space and choose attempt checking. A successful AI response uses one credit. Always test a value on each side of your proposed boundary.