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let's say that we want to solve the inequality x squared plus 3x is greater than 10 we want to figure out all of the X's that would satisfy this inequality well I encourage you to pause this video now and and I'll give you a hint try to manipulate it the way that you would have if this was a quadratic equation but then as you get to the end try to reason through it because it's the the reasoning might departure a little bit from what you are used to so I'm assuming you've given a go at it so the first thing that we might want to do just to get into a form that we're more comfortable with is subtract 10 from both sides if we subtract 10 from both sides then on the left hand side we're going to have x squared plus 3x minus 10 is still going to be greater than if we just if we add or subtract the same thing to both sides it won't change the inequality but it's now going to be greater than 0 10 minus 10 is 0 now this gets us into a form that we're more used to seeing quadratic expressions in and our temptation if this was an equal sign right over here we'd want to factor this thing so let's just try to factor it here too and see what happens so we're going to factor it we're going to think of two numbers whose product is negative 10 and whose sum is positive through it we've had a lot of practice doing this if you think about the factors of 10 it's 1 2 5 and 10 2 & 5 seem tempting because their difference is 3 so if you have positive 5 and negative 2 that seems to work out positive 5 and negative 2 their product is negative 10 their sum is positive 3 so we could write this as we can rewrite this as X X plus 5 X plus let me do that in that yellow color so you see where this 5 is coming from X plus 5 times X minus 2 times X minus 2 is going to be greater than 0 now this was an equality here we would say okay well how do we get this equal 0 one of these two things are going to be equal to or if either of these things we're equal to 0 this entire expression would be equal to 0 because 0 anything is zero we don't have an equality here we have a greater than symbol so let's think about how we could reason through this and I'll do a little bit of an aside here if I were to tell you that two numbers a and B and if I were to take their product a times B and if someone were to tell you that that product is greater than zero what do we know about a and B well we know that they have to have the same sign they're either both positive a positive times a positive is going to be a positive or they're both going to be negative a negative times a negative is a positive is going to be greater than zero so we know the same thing here so we could let me write it down so we know we know either we know either either a is greater than zero a is greater than zero and B is greater than 0 and B is greater than zero so either both of them are positive or or both of them are negative or a is less than zero and B is less than 0 B is less than zero so we apply that same logic here you could view this X plus five is a you could view this X minus two as our B of the product of two things the product is greater than zero that means that either both of these expressions are positive or they are both negative so let's write that down so this tells us that either either I'll write it this way so either both of these expressions are positive so either X plus 5 X plus 5 is greater than 0 and X minus 2 is greater than 0 and X minus 2 is greater than 0 so either this or or let me write it this way or X plus 5 or they're both negative or X plus 5 is less than 0 and X minus 2 is less than 0 X minus 2 is less than zero so now let's think about eat all of these inequalities independently but let's maintain this logic here of the and and the or so let's look at this either they're both positive so if both of these expressions are positive what do we know about X well if you subtract 5 from both sides of this inequality you get X is greater than negative 5 and if you add 2 to this inequality both sides of that inequality you're going to get X is greater than 2 so if X is greater than negative 5 and X is greater than 2 what do we know about X well any X that's greater than 2 is going to be greater than negative 5 so we could just simplify this right over here to be to say that X is greater than 2 so all of this this is equivalent to saying well X is greater than 2 because clearly anything that is greater than 2 will satisfy that and both of these things have to be true negative 4 for example X equaling of 4 it would satisfy this inequality but not this one and so the and would break down negative 4 does not satisfy both of these in order to satisfy both of these you essentially have to satisfy that one so this expression simplified to that now what about this or this what about this statement right over here well X plus 5 less than 0 subtract 5 from both sides that is X is less than negative 5 and add 2 to both sides of that equality you that inequality you get X is less than 2 now if X is less than negative 5 and X is less than 2 what do we know about X well that just means that X has to be less than negative 5 X has to be less less than negative 5 if it's less than negative 5 it's definitely going to be less than 2 and we got to remind ourselves that we have this over here and that's essentially describing the solution set for this quadratic inequality here X is going to be greater than Z X is going to be greater than 2 or X is going to be less than negative 5 and we could actually plot the solution set on a number line so this is our number line right over here and let's say that this is zero this is let's say that's one two right over here this is negative 1 negative 2 negative 3 negative 4 negative 5 that's negative 5 so X could be greater than 2 greater than 2 not greater than or equal to so I'm going to put an open circle here so it could be greater than 2 or it could be less than negative 5 not less than or equal to so I'm going to put an open circle here and so it could be less than that so X can be either X could be any number it could be negative 6 negative 6 would satisfy this you can verify that negative 6 squared is 36 plus 36 plus negative 18 which is going to be 18 which is greater than 10 or you could have say a positive 3 would work 3 squared is 9 plus another 9 is going to be 18 which once again it is greater than 10