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we're asked to solve for P and we have the inequality here negative 3p minus 7 is less than P plus 9 so what we really want to do is isolate the P on one side of this inequality and preferably the left that just makes it a little bit easier to read it doesn't have to be but we just want to isolate the P so a good step to that is to get rid of this P on the right hand side and the best way I can think of doing that is subtracting P from the right but of course if we want to make sure that this inequality is always going to be true if we do anything to the right we also have to do that to the left so we also have to subtract P from the left and so the left-hand side negative 3p minus P that's negative 4p and then we still have a minus 7 up here is going to be less than P minus P those cancel out it is less than 9 now the next thing I'm in the mood to do is get rid of this negative seven or this minus seven here so that we can better isolate the P on the left hand side so the best way I can think of to get rid of a negative seven is to add 7 to it then we'll just cancel out to 0 so let's add 7 to both sides of this inequality negative 7 plus 7 cancel out all we're left with is negative 4p on the right hand side we have 9 plus 7 is 16 and it's still less than now the last step to isolate the P is to get rid of this negative 4 coefficient and the easiest way I can think of to get rid of this negative 4 coefficient is to divide both sides by negative 4 so if we divide this side by negative 4 these guys are going to cancel out we're just going to be left with P but you also have to do it to the right hand side now there's one thing that you really have to remember since this is an inequality this is not an equation if you're dealing with an inequality and if you multiply or divide both sides of the equation by a negative number you have to swap the inequality so in this case the less than becomes greater than since we're dividing by a negative number and so negative 4 divided negative 4 those cancel out we have P is greater than 16 divided by negative 4 which is negative 4 now we can plot this solution set right over here and then we can try out some values to help us feel good about the idea of it working so let's say this is negative 5 negative 4 negative three negative two one negative one I should say zero let me write that a little bit neater negative one zero and then we can keep going to the right and so our solution is P is not greater than or equal so we have to exclude negative for P is greater than negative four so all the values above that also negative three point nine nine nine nine nine nine nine nine will work negative four will not work now let's just try some values out to feel good that this is really the solution set so first let's try out let's try out when P is equal to negative three this should work the way I've drawn it this is in our solution set P equals negative three is greater than negative four so let's try that out we have negative three times negative three this is the first negative three is this one and then we're saying P is negative three minus seven should be less than instead of a P we're going to put a negative three should be less than negative three plus nine negative three times negative three is nine minus seven should be less than negative three plus 9 is six nine minus seven is two 2 should be less than six of which of course it is now let's try a value that definitely should not work so let's try negative five negative five is not in our solution set so it should not work so we have negative three times negative five minus seven let's see whether it is less than negative five plus nine negative three times negative five is fifteen minus seven it should really should not be less than negative five plus nine so we're just seeing if P equals negative five works fifteen minus 15 minus seven is eight and so we get eight is less than four which is definitely this is definitely not the case so P equals negative five doesn't work and it shouldn't work because that's not in our solution set and now if we want to feel really good about it we can actually try this boundary point negative four should not work but it should satisfy the related equation when I talk about the related equation negative four should satisfy negative three minus seven is equal to P plus nine it'll satisfy this but it won't satisfy this because when we get to the same value on both sides the same value is not less than the same value so let's try it out let's see whether negative for at least satisfies the D related equation so if we get negative 3 times negative 4 minus 7 this should be equal to negative 4 plus 9 so this is 12 minus 7 should be equal to negative 4 plus 9 should be equal to 5 and this of course is true 5 is equal to 5 so it satisfies the related equation but it should not satisfy this if you put negative 4 for P here and I encourage you to do so actually we could do it over here instead of an equal sign if you put it into the original inequality let me do delete all of that it really just becomes this the original inequality is this right over here if you put negative 4 you have less than less than and then you get 5 is less than 5 which is not the case and that's good because we did not include that in the solution set we put an open circle if negative 4 was included we would fill that in but the only reason why we would include negative 4 is if this was greater than or equal so it's good that this does not work because 4 negative 4 is not part of our solution set you can kind of view it as a boundary point