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Lesson 3: 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.

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• I don't understand why you flip the >
• Because he changed it to a negative. yes, 1<2 but -1>-2
• 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
• from Buzz Lightyear of course
(1 vote)
• I can see why you would need equalities, but why would you need inequalities in real life?
• Inequalities are everywhere in real life.
-- Your earnings need to be greater than or equal to your expenses.
-- You are on an elevator. Somewhere it has a sign for the maximum occupancy for that elevator. So, the total weight of its passengers must be less than the maximum for the elevator to work properly.
-- Many devices / appliances are designed to work within certain temperature ranges. For example, Apple recommends that iPhones only be operated in environments where the temperature is between 32 degrees and 95 degrees Fahrenheit.
-- Many materials used in manufacturing have tolerances for height, stress limits, etc. All of these are inequalities.

Hope this helps.