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## Multi-step inequalities

# Inequalities with variables on both sides

## Video transcript

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.