Solving Inequalities

Date: 9/8/95 at 11:18:47
From: Anonymous
Subject: Algebra 2 Honors

This is my first day in Algebra 2 Honors at Harbor High School in
Santa Cruz,California.  I am in the 10th grade.  Can you help me with this
problem?

"1 is less than or equal to (4 - 5K) is less than 9"
Thanks very much for your help.  Annie

Date: 9/13/95 at 15:42:45
From: Doctor Steve
Subject: Re: Algebra 2 Honors

Hi Annie, 

In my typing below I'll use <_ to mean less than or equal to and 
_> to mean greater than or equal to.

1<_ 4-5K < 9

You can do similar sorts of operations to inequalities as you can to
equations. For instance you can add or multiply a number to all parts 
of the inequality.

You just have to be careful about how it affects the inequality. For instance,
suppose we had 1<_ x <_ 8. If we multiply all the parts by -1, what happens? 
You can figure this out just by thinking for a bit.

If x is between 1 and 8 then -x is between -1 and -8. But -8 is smaller 
than -1 so something will have to change in the way the inequality is
written besides just adding negative signs if it's to make sense.  If we
don't change anything else we would get nonsense: -1<_ -x <_-8. How 
could it be both bigger than -1 AND smaller than -8?

So the correct way to write the result of multiplying by -1 would be:
-1_> -x _> -8

So, back to your problem.  We can estimate by looking at approximately
what K is, right?  This is always good to do so that your common sense can
check your calculations.  K can't be as big as 1 and it can't be smaller than
-1. Do you see why I say that?

So now you can start trying to isolate K.  I would first subtract 4
from each part and then divide by -5.

This would be easier if the inequality began with
-1 instead of 1.  Did you make a typo there?

Let me know if you can't finish this.

- Doctor Steve,  The Geometry Forum