The absolute value of 2x-5 is less than or equal to 9. 
The answer is, x is less than or equal to -2, or x is 
greater than or equal to 7. 

I keep coming up with x is less than or equal to 7 

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Well, let's see:

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Rules for removing absolute values in inequalities
1.

|expression| < A 

becomes

-A < expression < +A

2.

|expression| > A 

becomes

expression < -A  OR expression > +A 

The rules for < amd > are the same as for < and >

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|2x - 5| < 9  uses rule 1:

-9 < 2x - 5 < 9
+5      +5   +5
---------------
-4 < 2x     < 14

-2 < x < 7

You are right.  If the problem had been

|2x - 5| > 9 we would have used rule 2:

2x - 5 < -9 OR 2x - 5 > 9
    2x < -4 OR     2x > 14
     x < -2 OR      x > 7    

Your book must have had a typo and this second
problem was what the author intended.

Edwin