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## Pre-algebra

### Course: Pre-algebra>Unit 10

Lesson 7: One-step inequalities

# One-step inequalities examples

Our discussion of linear inequalities begins with multiplying and dividing by negative numbers. Pay attention for the word "swap." Super important! Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• I don't understand why you flip the >
• A way to think about it is that if you have a positive number greater than another positive number, if you make the numbers negative, the first number will be LESS than the other number.

It's kind of hard to explain, but I will give you an example:
If you have the inequality 5 < 7,
If you multiply both sides by negative one and don't flip the sign, you will get:
-5 < -7, but that is not right. -5 is actually greater than -7, so you have to flip the sign to get -5 > -7

I hope this helps :)
• In , is there even a negative infinity? I thought infinity wasn't really real or be distinguished as negative or positive.
• It's better to treat negative infinity as a direction rather than a value perse. Instead of trying to quantify it, think that you are continuing to count, without stopping, in the negative direction.
• But why do you flip the sign?
• Basically, when you divide by a negative number you switch the sign to make the inequality true. Hope this helps :)
• When you divide x by a decimal like .5 you get 2x, but when you multiply it by a decimal like .5 you get 1/2 of x. Why is this?
• Division is the inverse of multiplication. If you divide by 1/2 it's the same thing as multiplying by 2/1. This rule holds for all fractional multiplication and division. The rule is when you turn the fraction upside down the you also switch divide/multiply and it's the same thing.

The same hold true when you convert the fractions into decimals. 1/2 = 0.5 and it's inverse 2/1 = 2. This means dividing by 0.5 is the same as multiplying by 2. When you turn the fraction upside down you also switch divide/multiply.

Folks like Sal know this so well they don't think it through, they just do it without thinking about it at all.
• At , Sal uses a set notation {x is a real number I x > or = -15}. My question is when would this be used instead of the seemingly simpler x > or = -15?
• With this sort of notation, you can show more complicated sets, like only even numbers, or only perfect squares. For something that simple though, yes, your way is simpler.
• Could you say (10/3,-infinity) instead of (-infinity,10/3)?
• No, the less value should always be on the left.
• If ∞ is not a real number, is it an imaginary number?
• There is a number system, invented by British combinatorial game theorist John Conway, called the surreal numbers.

The surreal numbers include the real numbers, along with a variety of infinite numbers and a variety of infinitesimal numbers that are positive and yet less than any positive real number! The combinatorial game called Blue-Red Hackenbush is a good model of the surreal numbers.

Have a blessed, wonderful day!
• where did you get the infinity
• Is there actually proof for why you flip the sign at ?
• i wonder what happen to these people
• I can see why you would need equalities, but why would you need inequalities in real life?