Question 255117
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{{{abs(2x-9)<11}}}

Learn the ways to rewrite absolute value inequalities
without using absolute value bars:

"{{{EXPRESSION}}}" refers to whatever is between the absolute value bars.
"{{{N}}}" refers to whatever POSITIVE* number is on the right side.

1. {{{abs(EXPRESSION)<N}}} can be rewritten without absolute value bars as {{{-N<EXPRESSION<N}}}

2. {{{abs(EXPRESSION)<=N}}} can be rewritten without absolute value bars as {{{-N<=EXPRESSION<=N}}}

3. {{{abs(EXPRESSION)>N}}} can be rewritten without absolute value bars as {{{EXPRESSION<-N}}}{{{OR}}}{{{EXPRESSION>N}}}, 
and the word "{{{OR}}}" must be included.

4, {{{abs(EXPRESSION)>=N}}} can be rewritten without absolute value bars as {{{EXPRESSION<=-N}}}{{{OR}}}{{{EXPRESSION>=N}}},
and the word "{{{OR}}}" must be included.

Yours is case 1 but I thought I'd include the others so you could
solve other inequalities you'll be studying.

The "{{{EXPRESSION}}}" here is is what's between the absolute value bars,
and that is "{{{2x-9}}}" and "{{{N}}}" is {{{11}}}. 

{{{abs(2x-9)<11}}} can be rewritten without absolute value bars as {{{-11<2x-9<11}}}

You solve 

     -11 < 2x - 9 < 11

by getting x alone in the MIDDLE.  Begin by adding 9 to all
three sides:

     -11 < 2x - 9 < 11
      +9       +9   +9
     -----------------
      -2 < 2x     < 20

Then divide all three sides by 2 and
since 2 is not a negative number the
inequalities are not reversed.

     {{{(-2)/2 < (2x)/2 < 20/2}}}
     {{{-1<x<10}}}

The graph of that solution is

---------o===========================================o--------
-3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12 

and the interval notation for that is (-1, 10)

Edwin</pre>
* when N is not a positive number, there is no solution in
cases 1 and 2, and "all real numbers" in cases 3 and 4. 

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