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### Absolute Value Equations

```Date: 10/30/2002 at 16:53:48
From: Abel Dallas
Subject: Absolute value equations

What is the answer to 'the absolute value of x-3 plus the absolute
value of 3-x equals 12' ?
```

```
Date: 11/03/2002 at 16:56:07
From: Doctor Ian
Subject: Re: Absolute value equations

Hi Abel,

Let's look at that:

|x-3| + |x+3| = 12

The first term tells you how far x is away from 3. For example, when
x is 4,

|4 - 3| = |1| = 1

And when x is 2,

|2 - 3| = |-1| = 1

Similarly, the second term tells you how far x is away from -3.

So the in words, the equation is saying:

What is x, if the distance from 3, plus the distance from -3,
is 12?

Let's think about how that would look on a number line.  Suppose we
want to test the value x = -7:

|------------10---------------|
x
|----4------|

<--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->
-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9

For this value of x, the total distance is 14.  Let's substitute that
into the expression, and see what it says:

|-7 - 3| + |-7 + 3| = |-10| + |-4|

= 10 + 4

= 14

So the picture and the equation seem to be saying the same thing.

The symmetry of the situation suggests that there will be two values
of x such that the sum of the distances to -3 and 3 will be 12. Try
to find them, and let me know what you come up with.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
Middle School Algebra

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