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## Algebra (all content)

How to solve and graph one-step inequalities. Created by Sal Khan and Monterey Institute for Technology and Education.

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• What if both sides of the inequality are negative? Does the sign switch? • When would would need to flip the direction of the inequality •  When you divide or multiply both sides of the inequality by a negative number. For example: -4x > 9 Here you have to divide both sides by a negative number, negative 4, so you carry out the division just like you would in a regular equality, but the only thing you do differently is you flip the inequality sign. So in this case it would be: x < -9/4. Hope this helps!
• • • Dylan,

You only need to switch the inequality sign when you multiply or divide both sides of the inequality by a NEGATIVE number. I find this sometimes makes more sense if we take all of the variables away and just look at numbers.

For example, 3 < 4, right? Well what happens if I multiply both sides by -2, and I DON'T remember to switch the signs? I get -6 < -8… but -6 ISN'T less than -8. -8 is Less than -6. So, since I multiplied by a negative number, if I switch the signs the statement becomes true again: -6 > -8.

Suppose I divide by a negative number? The same thing happens. Watch:

Let's say I start with 12 > 9. Clearly that is true, 12 IS greater than 9. But suppose I divide both sides by -3 and forget to switch the signs? Then I get:
-4 > -3. Well that's not true anymore. -4 ISN'T greater than -3, it's LESS than -3. That's why I have to switch the sign: -4 < -3.

You'll notice that this only happens when I multiply or divide by a NEGATIVE number. If I multiply or divide by a positive number, I don't need to do anything: 1 < 2 is a true statement. 1 IS less than 2. And if I multiply both sides by POSITIVE 4, I get 4 < 8, which is STILL a true statement.

I also don't need to switch the signs if I Add or subtract ANY numbers, positive or negative: 6 < 9 is true. And if I subtract 5 from both sides (or add -5) I get: 1 < 4. Still true, and I didn't need to switch the signs.

It can be harder to see this when there are a bunch of variables in the expression. But if you ever forget, just take the variables out and use numbers, you can check to see when you need to switch the inequality sign all by yourself just like we did above. It shouldn't take you long to prove to yourself that you only switch the signs when you multiply or divide by a negative number.

Does that help, Dylan?
• Can someone help here? When do you flip the sign? I never get these correct! • Why do you have to flip the sign when you divide by a negative? • Multiplying or dividing by a negative is like multiplying -1 and then multiplying or dividing a positive normally. Thus, I'll just explain why multiplying by -1 makes you flip the sign to explain all of it.

When you multiply by -1, you are flipping the numbers on both sides of the inequality over 0. Remember that -1 is greater than -2 because -1 is less left of 0 on the number line. If you take the negative of each side of the inequality, you have to flip the sign because lesser positive numbers before will become greater than greater positive numbers before. When you flip it over 0, you're switching the numbers places so that one number is left of the other [on the number line] when it was right of it [on the number line] before and vice versa. Let's do some examples to show you this.

A number with less absolute value will be less when positive, but more when negative.
1<2-->-1>-2
Positive numbers are always greater than negative numbers, just as negative numbers are always less than positive numbers
2>-1-->-2<1
A number with greater absolute value will be less when negative, but greater when positive.
-2<-1-->2>1

I hope this helps!  • • 