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

The region represented by $|x - y| \leq 2$ and $|x + y| \leq 2$ is bounded by a

The number of elements in the set $\{n \in \mathbb{Z} : |n^2 - 10n + 19| < 6\}$ is $...........$

The solution set for the system of linear inequations $x+y \geq 1$,$7x+9y \leq 63$,$y \leq 5$,$x \leq 6$,$x \geq 0$,and $y \geq 0$ is represented graphically. Which of the following figures represents the correct solution set?

Let $2\sin^2 x + 3\sin x - 2 > 0$ and $x^2 - x - 2 < 0$ ($x$ is measured in radians). Then $x$ lies in the interval

Let $a$ and $b$ be two positive real numbers such that $a+2b \leq 1$. Let $A_1$ and $A_2$ be respectively the areas of circles with radii $ab^3$ and $b^2$. Then,the maximum possible value of $\frac{A_1}{A_2}$ is