Associated Topics || Dr. Math Home || Search Dr. Math

### Absolute Value (Real Numbers: Rational)

```
Date: 02/02/99 at 05:11:50
From: Laura Noack
Subject: Absolute Value (Real Numbers: Rational)

|1/(x-2)| >= 4

If x > 0

1/(x-2) >= 4
1 >= 4x-8
9 >= 4x
9/4 >= x

This part is correct.

If x < 0,

1/-(x-2) >= 4
1/4 <= -(x-2)
1 <= -4x + 8
-7 <= -4x
7/4 >= x

This is what I got for x < 0, but the answer is supposed to be
7/4 <= x, not 7/4 >= x. Can you also help me display the answer on a
real number line?

It should look like this:

(*)---------(*)
_________________
7/4         9/4
```

```
Date: 02/02/99 at 12:11:35
From: Doctor Rick
Subject: Re: Absolute Value (Real Numbers: Rational)

Do you know what you mean by x > 0 ? The correct condition is:

1/(x-2) > 0

that is, if what's between the absolute value signs is greater than
zero, you can remove the absolute value, since it has no effect in this
case. Further, note that you multiply both sides by (x-2) and the
direction of the inequality does not change. This is correct, but only
because

1/(x-2) > 0 => x > 2

so that the quantity you multiply by, (x-2), is positive.

From what I said about the first part, can you see where you went wrong
in the second part? The condition should be

1/(x-2) < 0

which implies that x < 2. In this case, as you say, you can replace the
absolute value with multiplication by -1. But the quantity that you
multiply by, -(x-2), is POSITIVE (check out that condition), and once
again you shouldn't change the direction of the inequality!

In case you're not clear on this, let me spell out the complete
solution.

(x > 2 and x <= 9/4) or (x > 2 and x >= 7/4)

The first () is what you get from the first part, and the condition
x > 2 (so that the quantity in the absolute value is positive) must be
kept as part of the solution. Likewise in the second (), the condition
that the quantity in the absolute value is negative must be kept as
part of the solution.

What happens at x = 2?

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search