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?
In the previous lesson, we saw that multiplying or dividing both sides of an inequality by a positive number keeps the direction unchanged. Positive scaling preserves order, so the inequality sign stays put.
Negative numbers, however, behave differently. Take the true statement 3<5 and multiply both sides by −1. We get −3 and −5. Is −3<−5? Not at all — on the number line, −3 is to the right of −5, so −3>−5. The inequality has reversed. Let's explore why this happens before we start solving.
Why Negation Reverses Order
Dividing Both Sides by a Negative Number
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Picture the numbers 3 and 5 on a number line. The number 3 sits to the left of 5, confirming that 3<5. When we multiply both by −1, each number reflects across zero like a mirror. The 3 lands at −3, and the 5 lands at −5.
Notice that 5, being farther from zero on the positive side, ends up farther from zero on the negative side. That places −5 to the left of −3, making it the smaller value. The original left-to-right order has flipped.
This reflection works the same way with any negative multiplier, not just −1. For example, multiplying both sides of 3<5 by −2:
3×(−2)=−6and5×(−2)=−10
Since −6>−10, the order reversed again. Whenever we multiply or divide both sides of an inequality by a negative number, we must flip the inequality sign to keep the statement true.
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 <.
−4−4x<−420x<−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≤15:
−3−3x≥−315x≥−5
The ≤ became ≥ because we divided by the negative number −3. Every strict symbol (<, >) swaps with its strict counterpart, and every inclusive symbol (≤, ≥) swaps with its inclusive counterpart.
Multiplying Both Sides by a Negative Number
A Real-World Example: Monitoring Temperature
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.
Conclusion and Next Steps
In this lesson, we learned the sign-flip rule: whenever we multiply or divide both sides of an inequality by a negative number, the inequality sign must reverse direction. This is not an arbitrary trick to memorize — it reflects the fact that multiplying by a negative mirrors values across zero on the number line, swapping their order. By checking whether the multiplier or divisor is negative, you can confidently decide when to flip.
Time to sharpen this skill! The practice exercises ahead will ask you to spot when the flip is needed, trace through each step of a sign-reversing division, and solve inequalities independently. You will also explain the reasoning behind the rule in your own words using a concrete numerical example, reinforcing the why behind the technique.
The same rule applies when we multiply to clear a negative divisor. Consider:
−2x≥7
To isolate x, we multiply both sides by −2. Since −2 is negative, we flip the sign:
x≤7×(−2)x≤−14
Let's check with x=−16: −2−16=8, and 8≥7. ✓
Another example: solve −5x<3. Multiply both sides by −5 and flip:
x>3×(−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.
