Solve the following inequality: $\frac{|x-1|}{x+2} < 1$.

In the first four examinations,each of $100$ marks,Ravi scored $94, 73, 72,$ and $84$ marks. If a final average greater than or equal to $80$ and less than $90$ is needed to obtain a final grade $'B'$ in a course,find the minimum marks that Ravi must obtain in the fifth examination to get grade $'B'$.

Find the set of all values of $x$ satisfying $\frac{x^4 - 4x^3 + 3x^2}{(x^2 - 4)(x^2 - 7x + 10)} \ge 0$.

$A$ solution of $8 \%$ boric acid is to be diluted by adding a $2 \%$ boric acid solution to it. The resulting mixture is to be more than $4 \%$ but less than $6 \%$ boric acid. If we have $640$ litres of the $8 \%$ solution,how many litres of the $2 \%$ solution will have to be added?

The solution set contained in $R$ of the inequation $3^x+3^{1-x}-4 < 0$ is:

The longest side of a triangle is $3$ times the shortest side and the third side is $2 \, cm$ shorter than the longest side. If the perimeter of the triangle is at least $61 \, cm$,find the minimum length of the shortest side. (in $, cm$)