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### Inequalities and Absolute Value: Examples with Solution Sets that Do and Do Not Intersect

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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
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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/
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Associated Topics:
High School Basic Algebra

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