Negatives in Absolute Value Expressions and Equations

Date: 10/15/2005 at 19:38:20
From: Amy
Subject: Negatives in absolute value expressions

I am trying to get the below expression to equal 0 but am not sure 
how to go about it:

|x - 3| - 4

|-1 - 3| - 4  <-- is the answer to this 0?

|-7 - 3| - 4  <-- is the answer to this 0?

|7 - 3| - 4 evaluates out to 0, correct?

I get confused by the negative before the first constant in an
absolute value expression.

Date: 10/15/2005 at 23:24:25
From: Doctor Peterson
Subject: Re: Negatives in absolute value expressions

Hi, Amy.

When you evaluate an expression containing an absolute value, you must
first evaluate the expression INSIDE the bars (as for parentheses),
and then take the absolute value (that is, chop off the sign!).  So

  |-1 - 3| = |-4| = 4     so   |-1 - 3| - 4 = 4 - 4 = 0

  |-7 - 3| = |-10| = 10   so   |-7 - 3| - 4 = 10 - 4 = 6

  |7 - 3| = |4| = 4       so   |7 - 3| - 4 = 4 - 4 = 0

To solve the equation

  |x - 3| - 4 = 0

you would first add 4 to both sides to get

  |x - 3| = 4

and then recognize that there are two numbers whose absolute value is
4, namely 4 itself, and -4. So either

  x - 3 = 4   from which x = 4 + 3 = 7

or

  x - 3 = -4  from which x = -4 + 3 = -1

Those are our two solutions.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/16/2005 at 09:01:47
From: Amy
Subject: Thank you (Negatives in absolute value expressions)

Thanks so much!  You explained it well and in an absorbable way.