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We're asked to solve for p and we have the inequality here

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negative 3 p minus 7 is less than p + 9.

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So what we really want to do here is isolate the p on one side of this inequality.

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And preferably the left, that just makes it just a little easier to read.

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It doesn't have to be, but we just want to isolate the p.

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So a good step to that is to get rid of this p on the right hand side

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and the best way I can think of doing that is subtracting p from the right.

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But of course, if we want to make sure that this inequality is always going to be true,

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if we do anything to the right, we also have to do that to the left.

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So we also have to subtract p from the left.

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and so the left hand side, negative 3p minus p, that's negative 4p.

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And then we still have a minus seven up here, is going to be less than p minus p, those cancel out, it is

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less than 9.

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Now the next thing I'm in the mood to do is get rid of this negative 7 here,

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so that we can better isolate the p on the left hand side. So the best way I can think of

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to get rid of a negative 7 is to add 7 to it, then it will just cancel out to 0.

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So let's add 7 to both sides of this inequality.

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Negative 7 plus 7 cancels out, all we're left with is negative 4 p.

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On the right hand side we have 9 plus 7 equals 16, and it's still less than.

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Now the last step to isolate the p is to get rid of this negative 4 coefficient,

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and the easiest way I can think of to get rid of this negative 4 coefficient,

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is to divide both sides by negative 4. So if we divide this side by negative 4,

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these guys are going to cancel out, we're just going to be left with p.

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We also have to do it to the right hand side.

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Now there's one thing that you really have to rememeber, since this is an

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inequality, not an equation. If you're dealing with an inequality and you

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multiply both sides of an equation by a negative number, you have to swap the inequality.

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So in this case the less than becomes greater than, since we're dividing by a negative number.

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And so negative 4 divided by negative 4, those cancel out.

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We have p is greater than 16 divided by negative 4, which is negative four.

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And we can plot this solution set right over here, and then we can try out

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some values to help us feel good about the idea of it working.

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So let's say this is negative 5, negative 4, negative 3, negative 2

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negative one, zero... let me write that a little bit neater--

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and then we can keep going to the right, and so our solution is

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p is not greater than or equal, so we have to exclude negative 4.

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p is greater than negative 4, so all the values above that.

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So, negative 3.999999999 will work, negative 4 will not work.

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Now let's just try some values out to feel good that this really the solution set.

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So first let's try out when p is equal to negative 3.

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This should work. The way I've drawn it, this is in our solution set.

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p equals negative 3 is greater than negative 4.

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So let's try that out.

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We have negative 3 times negative 3. The first negative 3 is this one, and then we're saying

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p is negative 3. Minus seven is less than-- instead of putting p

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we're putting a negative 3.

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Should be less than negative 3 plus 9.

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Negative 3 times negative 3 is 9, minus seven, should be less than negative 3 plus 9

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is 6. 9 minus seven is 2. 2 should be less than 6, which of course, it is.

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So let's try a value that definitely should not work.

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So let's try negative 5, negative 5 is not in our solution set, so it should not work.

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So we have negative 3 times negative 5, minus seven. Let's see whether is it less than negative 5 plus

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9. Negative 3 times negative 5 is 15, minus 7. It really should not be less than

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negative 5 plus 9.

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So we're just seeing if p equals negative 5 works, 15 minus 7 is 8.

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And so we get 8 is less than 4, which is definitely not the case.

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So p equals negative 5 doesn't work, and it shouldn't work,

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because it is not in our solution set.

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And now if we really want to feel good about it,

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we can actually try this boundary point.

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Negative 4 should not work, but it should satisfy the related equation.

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When I talk about the related equation, negative 4 should satisfy negative 3 minus 7

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is equal to p plus 9. It will satisfy this, but it won't satisfy this,

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because when we get the same value on both sides, the same value is not

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less than the same value.

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So let's try it out, let's see if negative 4 at least satisfies the

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related equation. So if we get negative 3 times negative 4 minus 7, this should be equal to negative

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4 plus 9. So this is 12 minus seven should be equal to negative 4 plus 9. It should be equal to 5.

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And this of course, is true.

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5 is equal to 5.

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So it satisfies the related equation, but it should not satisfy this.

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If you put negative 4 for p here, and I encourage you to do so,

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actually we could do it over here instead of an equals sign,

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if you put it into the original in equality --let me delete all of that

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It really just becomes this. The original inequality is this right over here.

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If you have negative 4, you have less than, and then you'll get

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5 is less than 5, which is not the case.

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And that's good because we did not include that in the solution set.

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We put an open circle. If negative 4 was included, we'd fill that in.

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But the only reason we'd include negative four is if this was greater than or equal. So it's good that

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this does not work, because negative four is not part of our solution set.

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You can kind of view it as a boundary point.