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Inequalities and Absolute Value

```
Date: 8/28/96 at 20:50:45
From: Brian Hicks
Subject: Inequalities and Absolute Value

Hi Dr Math!

I have a few problems with inequalities and absolute values...
What would the solution set be to something like this...

| X | > X

For this inequality, I had  X > X  or X < -X  How could this result be
possible?

And...

| X + 2 | - X >= 0     On this problem, my solution set was
{ X : 2>_0 or X < -X }

48 | 7K-30 |           Where K = 14.  My solution to this one
was 48 | 68 | But, I cannot distribute
regularly right?  How would I solve this?

Brian
```

```
Date: 8/29/96 at 14:21:57
From: Doctor Jodi
Subject: Re: Inequalities and Absolute Value

Hi there! Let me go through each of your problems and try to give you
some suggestions.

> | X | > X

Since x > x is NEVER true, you can eliminate that as a "solution":  no
matter what the value of x, it will never be greater than itself.  So
this problem could also be written x < -x.  For any value less than
zero, x will be less than its opposite.  Does that make sense?

> | X + 2 | - X >= 0     On this problem, my solution set was
>                             { X : 2>_0 or X < -X }

When you're doing problems like this one, you'll want to remember when
the solutions apply.  Here's how I'd do this problem:

|x + 2| - x >= 0

|x + 2| >= x

Now you need to divide this problem into two parts:

|x + 2| is positive or zero:
----------------------------

Since |x + 2| is the same as x + 2, you can remove the abs. value sign
without changing the rest of the problem:

x + 2 >= x
2 >= 0

Since 2 is ALWAYS greater than zero, this means that if |x + 2| is
non-negative the statement

|x + 2| - x >= 0

is true.  So the final answer set must include all numbers greater
than or equal to -2.

Now there's the second part of the problem:
|x + 2| is negative:
--------------------
This happens when x is less than -2.

The absolute value sign will change |x + 2| to its opposite, - (x + 2)
(aka -x -2 ), so

|x + 2| >= x
-(x + 2) >= x
-x - 2  >= x
-2  >= 2x
-1  >= x

Thus, this statement is true for any x that's less than or equal to -1
and less than -2.  This means it's true for all numbers less than -2.

Combining the first and the second parts of the problem, we see that
the equation holds for all real numbers.

-------------------------------
> 48 | 7K-30 |

The absolute value of 68 is 68, so now multiply 48 * 68 to get the

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

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