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

May 5, 2018

There are no real solutions for the given relation.

#### Explanation:

The absolute value of anything is $\ge 0$

Therefore, specifically $\left\mid x - 6 \right\mid \ge 0$

So $\left\mid x - 6 \right\mid$ can not be $< - 12$

May 5, 2018

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 $| x |$ $\ne$ $- a$ 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)

$\therefore$ $x$ has no solutions.