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

which is the answer.

For a little bit more explanation on why you reverse the inequality 
sign, check out this answer: 

  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/