SOLUTION: Solve the solution set for... |2(|x-3|)-7| <5

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Question 1139381: Solve the solution set for...
|2(|x-3|)-7| <5
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

abs%282%28abs%28x-3%29%29-7%29+%3C5+
which is sqrt%28%282%2Asqrt%28%28x+-+3%29%5E2%29+-+7%29%5E2%29%3C5
first find solutions for absolute value under sqrt
2%28abs%28x-3%29%29-7%29%3C5
2%28x-3%29-7%3C5
2x-6-7%3C5
2x-13%3C5
2x%3C5%2B13
2x%3C18
x%3C9
or
2%28-%28x-3%29%29-7%3C5
2%28-x%2B3%29-7%3C5
-2x%2B6-7%3C5
-2x-1%3C5
-5-1%3C2x
2x%3E-6
x%3E-3

then find abs value of

abs%282%28x-3%29-7%29+%3C5++
abs%2813+-+2x%29%3C5
%2813+-+2x%29%3C5
13+-+5%3C2x
8%3C2x
4%3Cx
or
-%2813+-+2x%29%3C5
-13+%2B2x%3C5
2x%3C5%2B13
2x%3C18
x%3C9

combined, solutions are:
-3%3Cx%3C2
4%3Cx%3C9

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Answer by ikleyn(52831) About Me  (Show Source):
You can put this solution on YOUR website!
.

    | 2*(|x-3|)-7 | < 5


is equivalent to


    -5 < 2*|x-3| - 7 < 5


is equivalent  (after adding 7 to each of 3 parts of the inequality)


    -5 + 7 < 2*|x-3| < 5 + 7


is the same as


    2 < 2*|x-3| < 12    


is equivalent to   (after dividing all the three terms of the inequality by 2)


    1 < |x-3| < 6.


The solutions to the last inequality are those values of x (those numbers or points in the number line) 
that are remoted from the point x= 3 farther than 1 unit and closer than 6 units.


Obviously, these points (numbers, solutions) are 


    -3 < x < 2   and/or  4 < x < 9.


ANSWER.  The solution set to the original inequality is the union  of two intervals  {-3,2)  and  (4,9):  (-3,2) U (4,9).

Solved, answered, explained and completed.