Inequalities with variables on both sides

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 just a little 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 7, or this minus 7 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 7 is to add 7 to it. Then it will 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 equals 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. We 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, not an equation. If you're dealing with an inequality and you multiply or divide both sides of an 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 by negative 4-- those cancel out. We have p is greater than 16 divided by negative 4, which is negative 4. And 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 3, negative 2, negative 1, 0. Let me write that a little bit neater. 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 4. p is greater than negative 4, so all the values above that. So negative 3.9999999 will work. Negative 4 will not work. And let's just try some values out to feel good that this is really the solution set. So first let's try out when p is equal to negative 3. This should work. The way I've drawn it, this is in our solution set. p equals negative 3 is greater than negative 4. So let's try that out. We have negative 3 times negative 3. The first negative 3 is this one, and then we're saying p is negative 3. Minus 7 should be less than-- instead of a p, we're going to putting a negative 3. Should be less than negative 3 plus 9. Negative 3 times negative 3 is 9, minus 7 should be less than negative 3 plus 9 is 6. 9 minus 7 is 2. 2 should be less than 6, which, of course, it is. Now let's try a value that definitely should not work. So let's try negative 5. Negative 5 is not in our solution set, so it should not work. So we have negative 3 times negative 5 minus 7. Let's see whether it is less than negative 5 plus 9. Negative 3 times negative 5 is 15, minus 7. It really should not be less than negative 5 plus 9. So we're just seeing if p equals negative 5 works. 15 minus 7 is 8. And so we get 8 is less than 4, which is definitely not the case. So p equals negative 5 doesn't work. And it shouldn't work, because that's not in our solution set. And now if we really want to feel good about it, we can actually try this boundary point. Negative 4 should not work, but it should satisfy the related equation. When I talk about the related equation, negative 4 should satisfy negative 3 minus 7 is equal to p plus 9. It'll satisfy this, but it won't satisfy this. Because when we get 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 4 at least satisfies the 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. It 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 equals sign, if you put it into the original inequality-- let me 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. 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'd include negative 4 is if this was greater than or equal. So it's good that this does not work, because negative 4 is not part of our solution set. You can kind of view it as a boundary point.