Question

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

Answer 1

There are no real solutions for the given relation.

Explanation:

The absolute value of anything is `>=0`

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

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

Answer 2

No solutions for `x`

Explanation:

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.