Why Do We Flip the Inequality Sign When Multiplying or Dividing by a Negative? | Math with Mr. J

[Music] welcome to math with mr j [Music] in this video i'm going to cover why we flip the inequality sign when we multiply or divide both sides by a negative and simply put we flip the sign to make the inequality true so we're going to go through a few examples in order to better understand why we flip the inequality sign when we multiply or divide both sides by a negative so let's jump into number one where we have two is less than five so we're going to see what happens when we multiply or divide both sides by a negative and compare that to when we multiply or divide both sides by a positive let's do negative first and see if the inequality is still true so 2 is less than 5 but let's multiply both sides by negative 1 so a negative number so two times negative one and five times negative one and we'll end up with negative two is less than negative five which is not a true inequality negative two is not less than negative 5. so we needed to flip the inequality sign to make that true so for example here we should have negative 2 flip the sign negative 5. so negative 2 is greater than negative 5. now that inequality is true not flipping the sign we get an inequality that is not true and again this is only if you multiply or divide both sides by a negative let's see what happens if we multiply both sides by a positive so let's multiply both sides by a positive two and we'll get 2 times 2 is 4 is less than 5 times 2 is 10. so when we multiplied both sides by a positive the inequality stayed true it's just when we multiply or divide by negatives that we need to flip that sign so let's move on to number two where we have a one-step inequality that we will need to divide both sides by a negative so we'll see how that impacts flipping the inequality sign and why we need to so we have negative 2 times x is less than 8 and we're going to start by plugging in a number that will make this inequality true so a solution let's plug in 2 here so 2 for x let's pretend x equals 2. so negative 2 times 2 is negative 4 and that's less than 8. so we ended up with negative 4 is less than 8 which is true 2 is a solution to that inequality now let's solve this one step inequality to get to the point where we have all the solutions that will be correct for x so i'm going to rewrite our inequality down here negative 2x is less than 8. so we need to isolate x get it by itself in order to solve this inequality so we're multiplying x by negative two we need to do the inverse or opposite so that would be dividing by negative two so we need to divide by negative two in order to cancel those negative twos out now whatever we do to one side we have to do to the other in order to keep everything balanced so we need to divide by negative 2 on the other side as well so our x is now isolated and we have x is less than negative four we're not going to flip the sign here so x is less than negative four so any number less than negative four would be a solution to that inequality so let's see we tested out a solution earlier this 2 right here and it ended up being a solution well is 2 less than negative 4 no so according to this answer right here 2 would not be a correct solution but earlier we proved that 2 was a correct solution let's see plug something else in for x and see if it works so again x has to be less than negative four so let's do negative five that's less than negative four and see what happens so negative two times negative five and it has to be less than eight well negative two times negative five gives us a positive 10. positive 10 is less than 8 well that's not true so that would not be a solution so we needed to flip that inequality sign to make the inequality true we're actually going to get the opposite of the correct solutions we're going to get all the incorrect solutions if we were to use this right here and not flip the sign so let's flip the sign and i'm going to come over to the left in this box here so we had negative 2 times x is less than 8. so divide both sides by negative 2 and we end up with x flip the sign negative four so x has to be greater than negative four so that two which we proved was correct will now work two is greater than negative four so that gives us all the correct solutions for that inequality so simply put again we flip the sign to make the inequality true if we don't flip it we're going to get incorrect solutions solutions that do not work for the inequality so i hope that helped thanks so much for watching until next time peace