MIE 2016/Day 1/Problem 2

Problem 2

The following system has $k$ integer solutions. We can say that:

$\begin{cases}\frac{x^2-2x-14}{x}>3\\\\x\leq12\end{cases}$

(a) $0\leq k\leq 2$

(b) $2\leq k\leq 4$

(c) $4\leq k\leq6$

(d) $6\leq k\leq8$

(e) $k\geq8$

$\frac{x^2-2x-14}{x}>3$

$\frac{x^2-2x-14}{x}-\frac{3x}{x}>0$

$\frac{x^2-5x-14}{x}>0\begin{cases}x^2-5x-14>0\\x>0\end{cases}$

$\left.\begin{array}{l}x > 0\\x > 7\\x < -2\end{array}\right\}x\in(-2,0)\cup(7,+\infty)$

Adding the other interval we get

$x\in(-2,0)\cup(7,12]$

If $k$ is the number of integer solutions, then $k=7$ . $\boxed{D}$