An inequality sign flips in two main situations: when you multiply or divide both sides by a negative number, and when you swap the two sides of the inequality. Those are the core rules, but a few other operations (reciprocals, logarithms with certain bases) also reverse the direction. Here’s each case explained so you can recognize it every time.
Multiplying or Dividing by a Negative Number
This is the rule that comes up most often in algebra. Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. A “less than” becomes a “greater than,” and vice versa.
A quick example shows why. Start with a true statement: 2 < 5. Now multiply both sides by −1. You get −2 and −5. On the number line, −2 sits to the right of −5, so −2 is actually greater than −5. If you kept the original “less than” sign, you’d have −2 < −5, which is false. Flipping to −2 > −5 keeps the statement true.
The same logic applies to division, because dividing by a number is the same as multiplying by its reciprocal. Dividing both sides by −3 is the same as multiplying by −1/3, so the sign flips.
This comes up constantly when solving inequalities. If you have −4x > 12 and you divide both sides by −4 to isolate x, the sign reverses: x < −3.
Why Negative Multiplication Reverses Order
Think of the number line. Positive numbers increase to the right, and multiplying by a negative number reflects every point across zero. A number that was farther right lands farther left after the reflection, and a number that was closer to zero stays closer. That reflection swaps the order of any two points. If point A was to the left of point B before the reflection, it ends up to the right of point B afterward. That geometric flip is exactly why the inequality sign reverses.
Swapping Both Sides
If you move the left side to the right and the right side to the left, the sign flips. This is straightforward: if 3 < 7, then 7 > 3. The relationship between the numbers hasn’t changed; you’re just reading it from the other direction. In algebra, if you have 10 < 2x and you’d rather write x on the left, you rewrite it as 2x > 10.
Taking Reciprocals
Flipping both sides of an inequality to their reciprocals (replacing a value with 1 divided by that value) reverses the inequality sign, as long as both sides have the same sign. This works whether both sides are positive or both sides are negative.
For example, 2 < 5 becomes 1/2 > 1/5, which checks out (0.5 is greater than 0.2). The reason is intuitive: among positive numbers, bigger denominators make smaller fractions. So the originally larger side becomes the smaller side after you take reciprocals.
Be careful when the two sides have different signs. If one side is positive and the other is negative, taking reciprocals doesn’t flip the sign, because a positive reciprocal is still greater than a negative reciprocal.
Absolute Value Inequalities
Absolute value problems don’t flip the sign in the usual sense, but they do create a second inequality that points in the opposite direction. When you have |f(x)| < a (where a is positive), you’re saying f(x) is less than a units away from zero. That splits into two conditions joined by “and”: f(x) < a, and f(x) > −a. The second condition has the reversed sign.
When the inequality goes the other way, |f(x)| > a, you’re saying f(x) is more than a units from zero. That splits into f(x) > a or f(x) < −a. Notice that the “less than” part points toward the negative value. Students sometimes trip up here because the direction of the sign changes when you remove the absolute value bars and write the negative case.
Logarithms With a Base Between 0 and 1
This one shows up in more advanced courses. When you apply a logarithm to both sides of an inequality, the sign stays the same if the base is greater than 1. But if the base is between 0 and 1, the sign flips.
The reason comes down to how the logarithmic function behaves. A logarithm with a base greater than 1 (like base 2, base 10, or the natural log) is an increasing function: larger inputs produce larger outputs, so the order is preserved. A logarithm with a base between 0 and 1 (like base 1/2 or base 0.3) is a decreasing function: larger inputs produce smaller outputs, which reverses the order. The same principle that makes multiplying by a negative flip the sign applies here. Any operation that reverses the ranking of values will reverse the inequality.
When the Sign Does NOT Flip
Adding or subtracting any number, positive or negative, never flips the inequality sign. If you add −5 to both sides, you’re shifting both values the same distance in the same direction on the number line. Their order doesn’t change. This is a common point of confusion: students sometimes flip the sign when they subtract a negative number or add a negative number, but that’s not necessary. The flip only happens with multiplication and division by a negative.
Multiplying or dividing both sides by a positive number also keeps the sign the same. The sign only reverses when the multiplier or divisor is negative.
Squaring both sides is trickier and doesn’t follow a single rule. If both sides are positive, squaring preserves the direction (2 < 3 becomes 4 < 9). If both sides are negative, squaring reverses it (−3 < −2 becomes 9 > 4, because the bigger negative number produces the bigger square). If one side is negative and the other positive, you can’t apply a blanket rule. In practice, it’s safest to consider cases rather than relying on a simple flip-or-don’t-flip shortcut when squaring.
A Quick Reference
- Multiply or divide by a negative number: flip the sign.
- Swap the left and right sides: flip the sign.
- Take reciprocals (both sides same sign): flip the sign.
- Apply a log with base between 0 and 1: flip the sign.
- Add or subtract anything: sign stays the same.
- Multiply or divide by a positive number: sign stays the same.
- Apply a log with base greater than 1: sign stays the same.