The Sign Flip Rule

The Sign Flip Rule | CodeSignal Learn

Introduction

Welcome to lesson three of Solving One-Step Inequalities! You are now halfway through this course, and your toolkit has been growing steadily. In the first two lessons, we solved inequalities using addition, subtraction, multiplication, and division — and every time, the inequality sign stayed exactly where it was. Now we face the one situation where that changes: multiplying or dividing by a negative number. This lesson introduces the sign-flip rule, explains why it works, and gives you a clear way to apply it every time.

What Changes with Negative Numbers?

3 < 5

Why Negation Reverses Order

3

Dividing Both Sides by a Negative Number

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Recognizing When to Flip

With three types of operations now in your toolkit, a simple mental checklist keeps things clear. Before solving, ask one question: Is the number I am multiplying or dividing by negative? Operation Flip the sign? Add or subtract (any number) No Multiply or divide by a positive No Multiply or divide by a negative Yes That single check covers every one-step inequality you will encounter. If the coefficient on the variable is negative, you will divide by a negative and flip. If the coefficient is positive, you divide normally and keep the sign.

Operation	Flip the sign?
Add or subtract (any number)	No
Multiply or divide by a positive	No
Multiply or divide by a negative	Yes

Let's put the rule to work. Solve $-4x > 20$ :

Divide both sides by $-4$ .
Because $-4$ is negative, flip the inequality sign from $>$ to $<$ .

\frac{-4x}{-4} \;<\; \frac{20}{-4}

x < -5

To verify, pick a value less than $-5$ , such as $x = -6$ . Substituting into the original gives $-4(-6) = 24$ , and $24 > 20$ . ✓

Here is one more. Solve $-3x \leq 15$ :

\frac{-3x}{-3} \;\geq\; \frac{15}{-3}

x \geq -5

The $\leq$ became $\geq$ because we divided by the negative number $-3$ . Every strict symbol ( $<$ , $>$ ) swaps with its strict counterpart, and every inclusive symbol ( $\leq$ , $\geq$ ) swaps with its inclusive counterpart.

The same rule applies when we multiply to clear a negative divisor. Consider:

\frac{x}{-2} \geq 7

To isolate $x$ , we multiply both sides by $-2$ . Since $-2$ is negative, we flip the sign:

x \leq 7 \times (-2)

x \leq -14

Let's check with $x = -16$ : $\frac{-16}{-2} = 8$ , and $8 \geq 7$ . ✓

Another example: solve $\frac{x}{-5} < 3$ . Multiply both sides by $-5$ and flip:

x > 3 \times (-5)

x > -15

Whether you are dividing or multiplying, the trigger is the same: if the number you apply to both sides is negative, the inequality sign reverses.

Imagine you are running a science experiment where a liquid cools at a steady rate of $6$ degrees Celsius per minute, starting from $0°C$ . Your lab protocol states the temperature must remain above $-30°C$ . If $t$ represents the number of minutes the cooling has been running, the temperature at time $t$ is $-6t$ . Setting up the inequality:

-6t > -30

Divide both sides by $-6$ and flip the sign:

t < 5

The experiment can run for any time less than 5 minutes before the liquid drops to the safety threshold. Without the sign flip, you would have gotten $t > 5$ , which would suggest longer cooling keeps the temperature higher — clearly wrong. This is a great reminder that the flip is not just a classroom rule; it protects real-world reasoning.

Introduction

What Changes with Negative Numbers?

Why Negation Reverses Order

Dividing Both Sides by a Negative Number

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Multiplying Both Sides by a Negative Number

A Real-World Example: Monitoring Temperature

Recognizing When to Flip

Conclusion and Next Steps