A common sense solution using mental arithmetic does exactly the same thing as the algebraic solution shown by the other tutor.
Comparing the solution below to that algebraic solution can help you understand how the algebra works.
After the number is tripled and increased by 1, the result is between -20 and 10. So after it was tripled but before it was increased by 1, the result was between (-20-1) = -21 and (10-1) = 9.
Then, if the original number after being tripled was between -21 and 9, the original number was between (-21/3) = -7 and (9/3) = 3.
Answer: the original number was between -7 and +3.
(