Math 1 · Lesson 6 · part 3 of 6

Why the gator turns around

Take the sentence x < 5 : x is somewhere to the left of 5.

Why the gator turns around

Take the sentence x < 5: x is somewhere to the left of 5. Now take the opposite of everything. The opposite of 5 is −5. And the opposite of any number that was left of 5 lands to the right of −5. Try it with x = 2, which is left of 5. Its opposite, −2, is right of −5. So the sentence about the opposites has to be −x > −5. The gator turned around.

Taking the opposite of a number is multiplying it by −1. So multiplying both sides of an inequality by −1 is a mirror: it flips the whole number line over, and a number that was on the small side is now on the big side. The same happens with any negative number, because it is a positive number times −1. That is the whole reason for the rule below.

The one new rule. When you multiply both sides of an inequality by a negative number, turn the sign around: < becomes >, and ≤ becomes ≥. Multiplying by a positive number leaves the sign alone. Adding any number, positive or negative, leaves the sign alone.

Start with a true sentence

2 < 6

Two is less than six. The gator eats the 6.

Multiply both sides by 3

6 < 18

Still true. A positive multiplier keeps the sign.

Multiply both sides by −3

−6 > −18

Now −6 is the bigger number, so the gator has to face it. A negative multiplier flips the sign.

Where to go next

Next: The three steps, one rule addedSolve an inequality exactly the way Lesson 5 solved an equation. All the partsSolving Inequalities: Gator Eats Greater