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Video transcript

we've got the inequality negative x times the expression 2x minus 14 is greater than or equal to 24 so I encourage you to pause this video now and think about what the solution set to this inequality would actually be and actually plot the solution set on a number line so I'm assuming you've given a go at it so now let's just try to simplify this a little bit so on the left hand side we could distribute we could distribute this negative x and so if we did that we would get negative 2 negative 2x squared negative times a negative is positive plus 14 X is greater than or equal to 24 now I'm going to put the I'm going to subtract 24 from both sides just so that we just have a 0 here and then we could think about factoring what we have here on the left so we have negative 2x squared plus 14x I'm going to subtract 24 from both sides so minus 24 is greater than or equal to I've subtracted 24 from the right as well so that's going to be greater than or equal to 0 now I don't like I don't like having this negative 2 out front so what I want to do is I want to divide the select hand side by negative 2 but I can't just divide the left hand side only by negative 2 I have to divide the right hand side by negative 2 as well and anytime I multiply or divide both sides of an inequality by a negative number it's going to flip the inequality so if I divide both sides by negative 2 I'm going to be left with x squared positive x squared minus so I'm dividing by negative 2 so minus 7x plus 12 and now since I divided by negative 2 I'm going to flip this inequality is less than or equal to 0 divided by negative 2 is 0 so that simplified things a good bit and now let's see if we can factor this quadratic expression so two numbers whose product is positive 12 so that means they're gonna have the same sign and whose and whose sum is negative 7 so they have the same sign and their sum is negative 7 that tells us that they're both going to be negative and let's see negative 3 negative 4 seem to fit the their product is positive 12 their sum is negative 7 so we could write this as X minus 3 times X minus 4 is going to be less than or equal to 0 so now this is the point that we're going to do a little bit of interesting logic if the product of two things is less than is less than or equal to 0 what does that tell us tell what do we know about it well it that tells us that either either one or both of them is 0 or they have different signs the only way that you're going to get less than 0 is if one is positive and the other is negative or one is negative and the other is positive so or one is negative two is that they have different signs so let's write that down let's write that down so either either X minus 3 X minus 3 is less than or less than either X minus 3 is less than or equal to 0 and and X minus 4 is greater than or equal to 0 X minus 4 is greater than or equal to 0 so notice that this one is non positive this one is non-negative they're either equal to 0 or they are have different signs so that's one situation or or the other way around or X minus 3 is non-negative it's greater than or equal to 0 and X minus 4 is non positive X minus 4 is less than or equal to 0 once again there are either 0 or different signs that's all I'm doing with this little little logic work right over here so what are these what is this what does this simplify to so X minus 3 less than or equal to 0 add 3 to both sides you get X is less than or equal to 3 and and X minus 4 is greater than or equal to 0 if you add 4 to both sides of this you get X is greater than or equal to 4 so what values of X are going to be less than or equal to 3 and greater than or - for well anything that's less than or equal to three is not going to be greater than or equal to four and anything that's greater than or equal to four is not going to be less than or equal to three so there's no X there's no X value that can satisfy this situation right over here there's no X value that will result in this one being negative and this one being or this one being non positive and this one being not negative so let's go to this one right over here so if we add three to both sides we get X is greater than or equal to three and and we get adding 4 to both sides X is less than or equal to 4 now does this make sense that something could be greater than or equal to 3 and less than or equal to 4 sure for example well 3 is greater than or equal to 3 and it's less than or equal to 4 4 is greater than or equal to 3 and it's less than or equal to 4 and anything in between so we can plot the solution set here so this is actually all that matters because this one there's no situation which that would have been true so this is the only thing this is the only thing that's going to make this or part true this part is always going to be false so if we wanted to make the solution set it would look something like this so if this is our possible values of X so let's say that that is 0 so this is 1 2 3 & 4 3 & 4 3 & 4 X could be greater than or equal to 3 so it's greater than or equal to 3 but it's also let it also has to be less than or equal to 4 so we can't just go past 4 also less than or equal less than since it's less than or equal we can color in these dots less than or equal to 4 so anything in this range including 3 & 4 that's why we circled in the dots this would satisfy this equation here and if you wanted to think about it visually hey you know we know that this type of thing we know that this type of thing or this type of thing this type of thing right over here these are these are parabolas so how would that relate to this little solution set that we just thought about right over here well if you look at let's let's just look at one of these let's say we went to let's say we went to this forum right over here so all everything we did this is just another way of thinking about negative 2 x squared plus 14 X minus 24 is greater than or equal to 0 so this is right over here we have a negative coefficient on the x squared term that's going to be a downward-opening parabola so when is that greater than or equal to 0 so if if we thought about the downward-opening parabola it might look something like this it might look something like this if we're now thinking in two dimensions and this is the this is the if you think of this as the y axis right over here so when is that greater than or equal to zero well it's greater than or equal to zero it's above the x axis in this range for X right over here so that's one way of thinking about it if we thought about it from the point of view not of that parabola not of that parabola with this parabola right over here when is x squared minus 7x plus 12 less than or equal to zero well this is going to be an upward-opening parabola so it has a positive coefficient here so this parabola might look something like this might look something like this when is it less than or equal to zero well once again once again it's less than or equal to zero in that same range