Solve linear inequalities involving addition and subtraction A1.2.2.2

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[00:00:00]in the next few videos we'll be solving and graphing inequalities this first one's a word problem Tyler was sure that he didn't score less than 80 on his algebra test write an inequality to describe Tyler's possible score graph the inequality so if he was sure that he didn't score less than 80 that means that he's scored 80 or better so 80 or above for the graph remember that we will put our cutoff value in the middle so that's the 80 we need at least one value larger and one value smaller this has an equal to as part of it so that's a colored endpoint and then a score of 81 would be a true statement for showing that he did not score less than 80 but a 79 would not work so I shade only to the right let's look at a few more examples next we have X plus 5 is greater than or equal to 3 approach this the same way as you would solving an equation you're trying to isolate the variable so I want to get the X by itself and I'm going to be using inverses again the inverse of adding 5 would be subtracting 5 so I'm going to subtract 5 from both sides 5 minus 5 is 0 and X plus 0 is just X so it's almost like those X's on the Left cancel out on the right 3 minus 5 is negative 2 so then when I go to graph this my middle point is the negative 2 I need at least one value larger and one value smaller than negative 2 there's an equal to sign because negative 2 is a solution negative 2 is greater than or equal to 2 that's true so that's colored in and then I would test the negative 3 and the negative 1 the negative 3 is false so I do not shade on that side but the negative 1 is true so I do shade on that side similarly the inverse of subtracting 4 is adding 4 so to solve this next one I would add 4 to both sides negative 4 plus 4 is 0 which eliminates the the numerical value on the right hand side negative 2 plus 4 is 2 you can leave your answer like this but it might be better to rewrite it as an equivalent inequality 2 is greater than n is equivalent to n is less than 2 those are the same statement just written differently that makes it a little bit easier I think to graph on your number line so there's no inequality which means 2 is not a solution 2 is not less than 2 1 is less than 2 so I do shade on the side with the 1 three is not less than two so I don't cheat on the side with the three now the next thing I'd like you to do is think just a bit about what happens when we multiply or divide rather than add or subtract so the two examples we've done so far involved isolating the value of the variable either by adding or subtracting so what we're going to do is we're going to think of a value like 4 is less than 5 that we know that that's a true statement now the next thing we want to do is say let's maybe multiply both sides by 2 so 4 times 2 is 8 and 5 times 2 is 10 and we get 8 is less than 10 that's still a true statement next let's try going back to the original problem 4 is less than 5 and let's divide both sides by 1 if I divide both sides by 1 4 divided by 1 is 4 5 divided by 1 is 5 that is still a true statement 4 is less than 5 however now let's look at multiplying both sides by negative 2 I'm going back to this original problem 4 times negative 2 is negative 8 5 times negative 2 is negative 10 8 is not less than negative 10 that's actually a false statement and let's do one more here let's this time divide both sides by negative 1 4 divided by negative 1 would be negative 4 5 divided by negative 1 would be negative 5 and we get negative 4 is less than negative 5 that's also a false statement so what you'll notice is if you multiply or divide by a negative number you're going to get a false statement so to help out with that what we need to do is anytime we multiply or divide by a negative number we need to switch the inequality sign so that it remains a true statement so in that last example where we started with 4 is less than 5 and we multiplied both sides by negative 2 so that was a negative 8 and a negative 10 we can't leave the less than sign in there because that's false we have to switch it to a greater than sign