Solving absolute value inequalities
=
x
{
x
≤
0
x

x
<
x
0
.
x
can be thought of as the
distance
of
x
from 0 on the real number line.
Introductory question
: (a) Find the set of all real numbers
x
such that
≤
x
3, and (b) the set of all real numbers
x
such that
x
> 3.
Solution
:
(a)
≤
x
3
when
x
lies at a distance less than or equal to 3 units from 0 on the real number line.
Hence
≤
x
3
<=>
≤

3
x
≤
3.
The solution set of the inequality in interval notation is
[
]
,

3 3 .
(b)
x
> 3 when
x
lies at a distance greater than 3 from 0 on the real number line.
Hence
x
>3
<=>
<
x

3
or
x > 3.
The solution set of the inequality in interval notation is
∪
(
)
,
∞ 
3
(
)
,
3
∞
.
Note
: As an aid to understanding the solution to part (a), think of a dog on a lead of length 3 units tethered to the origin wandering to the
left and right along the number line. How far to the left and right can it wander?
The limited territory the dog can access is the solution set
for the inequality.
The following are
basic absolute value inequalities
and their solutions (for
a
> 0).
x
>
a
<=>
<
x

a
or
x
>
a
 (i)
Solution set:
∪
(
)
,
∞ 
a
(
)
,
a
∞
x
>
a
<=>
≤
x

a
or
x
>
a
 (ii)
Solution set:
(
,
∞ 
a
]
∪
[ ,
a
∞
)
<
x
a
<=>
<

a
x
<
a
 (iii)
Solution set:
(
)
,

a a
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≤
x
a
<=>
≤

a
x
≤
a
 (iv)
Solution set:
[
]
,

a a
Example 1
:
Solve for
x
:
≤
+
x
6
10.
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 Spring '08
 Uri
 Real Numbers, Inequalities, Complex number

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