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### Solving Inequalities

```
Date: 08/31/98 at 16:07:50
From: Debbi Bross
Subject: Algebra 2 Compound Sentences wuth Inequalities

We have to graph:

0.3 - 0.7z > 0.4z - 3.0 and
4(z - 2) - 6 <= 15
```

```
Date: 09/01/98 at 13:00:12
From: Doctor Peterson
Subject: Re: Algebra 2 Compound Sentences wuth Inequalities

Hi, Debbi.

Inequalities really aren't much different from equations, but you do
have to be a little more careful.

If you've seen equations compared to balanced scales, where the left
side balances the right side, you can picture an inequality as an
unbalanced scale, tipped to one side or the other. You can do most of
the same things you can do to an equation, such as adding the same
amount to both sides, and the scale will still remain unbalanced in the
same direction. The only thing you have to watch out for is that when
you multiply or divide by a negative number, the inequality switches
over. For that reason, I generally avoid multiplying by a negative, and
just work around that if I need to.

Here's a picture of the scale flipping over when I multiply by -1. It's
like reversing gravity so that what went up (a negative "weight," maybe
a helium baloon, on each side) now comes down, because what was more
negative (that is, less) is now more positive (that is, bigger):

|
-3               |                1
\                |                /
\             |             /
\          |          /
A  \  -1   |    3  /  A
\    |    /
|
-3 < -1       |        3 > 1

0.3 - 0.7z > 0.4z - 3.0

First we want to gather the z's on one side. We can do that by
subtracting 0.4z from both sides, but then we would get a negative
coefficient of z, which I'd rather avoid. I'll go ahead and do that
anyway, so you can see what to do when that happens. Ordinarily I'd
move the z to the right instead, but if you don't notice that, things
will still work out.

0.3 - 1.1z > -3.0

Now we can get the non-z stuff out of the left side by subtracting 0.3
from both sides:

-1.1z > -3.3

Now here's where the one bit of magic comes in. Let's just multiply
everything by -1. Note that the inequality switches direction, as I
explained above:

1.1z < 3.3

Now we can divide each side by 1.1 (which is safely positive):

z < 3

Now to graph this, we just find the place where z = 3, and mark all
values to the left of that:

<===+===+===+===+===+===o---+---+--->
-2  -1   0   1   2   3   4   5

I used an "open circle" at 3 because z = 3 does not satisfy z > 3.

Now see if you can do the second problem. It's easier than this one.

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

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