How do you solve #abs(x-6)< -12#?

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Answer 1 · 2018-05-05T13:53:09

There are no real solutions for the given relation.

The absolute value of anything is #>=0#

Therefore, specifically #abs(x-6) >= 0#

So #abs(x-6)# can not be #< -12#

Answer 2 · 2018-05-05T14:12:43

No solutions for #x#

First of all there are some rules to be taken in mind while solving modulus inequalities.

One of the basic rules is that : If #x# #in# #R# then #|x|# #!=# #-a# where #a# #in# #R#.
This means simply that #|#anything#|# cannot be negative.

This is because #mod# converts every number inside, be it positive or negative, to positive. Just like squaring...

For eg. #|5|# = 5
and #|-5|# = 5

So even if #x# is negative #|x|# will always be positive.

Now coming back to the question, we have :

#|x-6|# < -12

That means #|x-6|# is lesser than -12 or simply is negative, which is absurd as #mod# of anything cannot be negative.

That means #x# doesn't have any Real solutions (Solutions that are counted in Real numbers)

#:.# #x# has no solutions.