Inequalities and Absolute Value

Date: 10/17/97 at 03:46:24
From: Lesley Trupiano
Subject: Solving Inequalities with Absolute Value Signs

I will use parentheses for absolute value signs. 

1. 3(s) - 2 >7

   and 

2. 6 - (2 - p) < 4

Date: 10/17/97 at 09:59:50
From: Doctor Chita
Subject: Re: Solving Inequalities with Absolute Value Signs

Hi Lesley:

Solving inequalities can be tricky. First of all, you have to 
understand the definition of absolute value to see why there are 
always two equations to solve. 

The definition of absolute value says that 

     |x| = x, if x >=0 
or 
     |x| = -x if x < 0

This definition may look very puzzling, but if you think about 
it for a while, it makes sense. Remember, the absolute value of 
a number can never be negative.

For example, if x = 3, then |x| = 3, because 3 > 0. However, 
if x = -3, then |x| = -(-3) = 3, because -3 < 0. In either case, 
|x| is not negative.

Here are two examples.

1) 5|x| + 3 > 12

   First, isolate |x| by subtracting 3 from both sides. You don't 
   have to reverse the inequality sign.

   5|x| > 9   Divide both sides by 5: again, no need to reverse the 
   inequality sign.

   |x| > 9/5

   Now, to drop the absolute value sign, represent the two conditions 
   of the definition:

   x > 9/5, if x >=0, or -x > 9/5, if x < 0.

   Simplify the second inequality by multiplying both sides by -1, 
   reversing the inequality sign:

   x > 9/5 or x < -9/5.

   Examine the two statements. Is there at least one number that 
   satisfies both conditions at the same time? That is, can you have 
   a number that is greater than 9/5 AND less than -9/5? If not, then 
   the solution is two nonintersecting sets on either side of |9/5|.

   <===============o-------------o================>
           x     -9/5           9/5       x

(2) Here's another example. Use the definition to rewrite the absolute 
    value expression as two equations and solve each equation:

    |x + 2| < 5

    (x + 2) < 5 and -(x + 2) < 5

    x < 3  and   x + 2 > -5        (multiply both sides by -1 
                                   and reverse the sign)
    x < 3  and    x > -7

    In this case, x is a number that lies between 3 and -7, so the 
    solution is the intersection of the two sets.

    <-------------o================o----------->
                  -7      x        3

I hope these two examples help you understand how to solve your 
equations. Let us know if you need more help.

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/