How do you solve an absolute value equation?
describe all numbers x that are at a distance of 1/4 from the number −6. express this using absolute value notation.
create an absolute value inequality that has no solution?

The distance between two real numbers is the absolute value of their difference, . So the statement "all that are 1/4 away from -6" is captured in the equation

or

To solve, you can use the definition of absolute value: if is not negative, then ; otherwise, .

So there are two cases to consider:

1. Suppose . Then , and the equation reduces to

2. Suppose . Then and

, so the two solutions we found are correct because both and are indeed 1/4 away from -6.

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For the question regarding inequality, you know that the absolute value must return a non-negative number. In other words, there is no such that .

incomplete question. how do you expect anyone to you when you do not post the question in its entirety?

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove. 

an inaccurate proof leads to errors, or stating results that are not accurate.  which wil mess the probelm up you dont want this