The least integral value $\alpha$ of $x$ such that $\frac{x - 5}{x^2 + 5x - 14} > 0$ satisfies:

$\left\{x \in R: \frac{14 x}{x+1}-\frac{9 x-30}{x-4} < 0\right\}$ is equal to

In a manufacturing company,the cost and revenue functions of a product are given by $C(x) = 500 + \frac{5}{2}x$ and $R(x) = 3x$,where $x$ is the number of items produced and sold. How many items must be sold to realize no profit or loss? Is there any profit?

For $x$ and $y$ satisfying $|x|+|y|=|x-3|+|y-2|$,which of the following is correct?

The set of all real values of $x$ satisfying the inequalities $x^2-4x+3 > 0$ and $x^2-2x-8 \leq 0$ is

The longest side of a triangle is twice the shortest side and the third side is $3 \text{ cm}$ more than the shortest side. If the perimeter of the triangle is at least $51 \text{ cm}$, find the minimum length of the shortest side. (in $\text{ cm}$)