Do I flip the sign if I subtract a larger number from both sides?

No. Adding or subtracting values from both sides never changes the direction of the inequality, even if it creates a negative result.

Does dividing by a negative flip the sign too?

Yes. Dividing by a negative number is exactly the same as multiplying by a negative fraction, so the same rule applies and the sign must flip.

What if the other side of the inequality is already negative?

The sign only flips based on the number you are multiplying or dividing by. If you multiply both sides by a positive number, the sign stays the same, even if the other numbers in the equation are negative.

Why does the inequality sign flip when you multiply by a negative?

When you multiply or divide both sides of an inequality by a negative number, you are reversing the direction of the number line. Because negatives act like a mirror, whatever was larger on the positive side becomes a larger negative, making it smaller in value. That is why the greater-than or less-than sign has to flip to keep the mathematical statement true.

The Mirror Analogy

Think of the number line like a street where positive numbers go east and negative numbers go west. Multiplying by a negative is like doing a U-turn. If house A is further east than house B, and you suddenly flip your perspective to face west, house A is now further behind you. In math terms, multiplying by a negative reverses the order of the numbers. What was a big positive becomes a big negative, which is actually a smaller number.

Visualizing on a Number Line

Let us look at a simple fact: $2 < 5$ . This is true because 5 is further to the right on the number line. If we multiply both sides by $-1$ , we get $-2$ and $-5$ . If we look at the number line now, $-5$ is further to the left than $-2$ . So, $-2$ is actually greater than $-5$ . To make the statement true, we must write $-2 > -5$ . The sign had to flip to preserve the truth of the relationship.

Where Students Slip Up

The most common mistake is flipping the sign when it is not needed. You only flip the sign when you multiply or divide both sides by a negative number. If you are just subtracting a number from both sides, or if you multiply by a positive number, the sign stays exactly the same. Another tricky spot is when the number you are dividing into is negative, but the number you are dividing by is positive. In that case, no flip is required.

Worked through

Solve the inequality: $-4x + 7 < 23$

First, subtract 7 from both sides to isolate the term with x. This gives $-4x < 16$ . Notice that we only subtracted, so the sign stays the same. Next, divide both sides by $-4$ to solve for x. Because we are dividing by a negative number, we must flip the less-than sign to a greater-than sign. This gives $x > -4$ . The solution is all numbers greater than $-4$ .

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Where this comes from: OpenStax Intermediate Algebra · Khan Academy Algebra 1: Solving Inequalities