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### Solving Inequalities

```
Date: 12/10/96 at 03:56:26
From: Anonymous
Subject: Algebra

Dr.Math,

When you are doing a question like 6(m-4)-2(m+2) < 7(m-4)-6, what are
you supposed to do? I'm in grade 9 and I always get stuck on these
type of questions. My mind goes blank. When do you reverse the sign?
When do you divide? Why do you divide? When do you subtract?

Thanks,  Kelly
```

```
Date: 12/10/96 at 14:46:05
From: Doctor Wilkinson
Subject: Re: Algebra

Solving an inequality like this means reducing it to a simple
inequality for m.  That is,  something like m < 27 or m > -1.  So it's
very similar to solving an equation for m, and you can do a lot of the
same things.  For example, you can combine terms on either side of the
equation and you can add or subtract the same thing from both sides of
the equation.  It's when you get to multiplication and division that
you have to be a little more careful.

So let's get started.  First of all, we certainly don't want m buried
in parentheses, so let's multiply out on both sides:

6m - 24 - 2m - 4 < 7m - 28 - 6

Now let's combine terms on both sides:

4m - 28 < 7m - 34

Now we can add 28 to both sides:

4m < 7m - 6

And we can subtract 7m from both sides:

-3m < -6

Now we would like to divide both sides by -3. But we have come to
something that is different for inequalities. You can multiply or
divide both sides by something which is greater than zero, but if you
want to multiply or divide by something which is less than zero, you
must reverse the inequality. So when you divide both sides by -3,
you need to change the '<' to a '>'.  This gives

m > 2

For a little bit more explanation on why you reverse the inequality

http://mathforum.org/dr.math/problems/polse8.html

I hope this helps!

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra

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