The cost and revenue functions of a product are given by $C(x) = 20x + 4000$ and $R(x) = 60x + 2000$ respectively,where $x$ is the number of items produced and sold. The value of $x$ to earn profit is

If $x \in R$ and $1 \leq \frac{3x^2-7x+8}{x^2+1} \leq 2$,then the minimum and maximum values of $x$ are respectively.

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 $'B'$.

If $\frac{2x}{2x^2 + 5x + 2} > \frac{1}{x + 1}$,then

How many points having integer coordinates are there in the feasible region of the inequalities $3x + 4y \leq 12$,$x \geq 0$,and $y \geq 1$?

If $x$ is real and satisfies $x + 2 > \sqrt{x + 4}$,then