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preetamsaha
what is the range of x where ||x - 3| - 4| > 3?
A.( -∞, -4) or (2, 4) or ( 10, ∞)
B.( -∞, -4) or (-2, 4) or ( 10, ∞)
C.( -∞, -4) or ( 10, ∞)
D.( -∞, 4) or ( 10, ∞)
E.( -∞, -4) or (-2, 2) or ( 10, ∞)

Asked: what is the range of x where ||x - 3| - 4| > 3?

||x - 3| - 4| > 3
Case 1: |x-3|-4 >3
|x-3| >7
x > 10 or x < -4

Case 2: |x-3|-4 <-3
|x-3| < 1
2<x<4

Range of x = ( -∞, -4) U (2,4) U (10, ∞)

IMO A
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Assume |x-3| as y, therefore the equation ||x-3|- 4| > 3 translates to

|y - 4| > 3

y > 7 or y < 1

Case 1 : |x-3| > 7 => x > 10 or x < -4

Case 2 : |x-3| < 1 => 2 < x < 4

Combine the ranges in bold-

Option A
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We need to find the range of x where ||x - 3| - 4| > 3?

|A| > B
=> A > B or A < -B

We have two cases as we have ||x - 3| - 4| > 3
-Case 1: |x - 3| -4 > 3
=> | x - 3 | > 7
=> x - 3 > 7 or x - 3 < -7
=> x > 10 or x < -4

=> ( -∞, -4) or ( 10, ∞)
-Case 2: |x - 3| -4 < -3
=> | x - 3 | < 1
=> -1 < x -3 < 1
=> -1 + 3 < x < 1 + 3
=> 2 < x < 4
=> (2, 4)

So, Answer will be A
Hope it helps!

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