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Solving Inequalities with a Balance Scale Analogy

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

Let's do your first problem:

    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:

       -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
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Associated Topics:
High School Basic Algebra
Middle School Algebra

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