Let $[ t ]$ denote the greatest integer $\leq t$ and $\{ t \}$ denote the fractional part of $t$. Then integral value of $\alpha$ for which the left hand limit of the function $f ( x ) = [ 1 + x ] + \frac { \alpha ^ { 2 [ x ] + \{ x \} } + [ x ] - 1 } { 2 [ x ] + \{ x \} }$ at $x = 0$ is equal to $\alpha - \frac { 4 } { 3 }$ is $\_\_\_\_$

The sum of the squares of all the roots of the equation $x ^ { 2 } + | 2 x - 3 | - 4 = 0$, is
(1) $3 ( 3 - \sqrt { 2 } )$
(2) $6 ( 3 - \sqrt { 2 } )$
(3) $6 ( 2 - \sqrt { 2 } )$
(4) $3 ( 2 - \sqrt { 2 } )$

Consider the following equation in $x$
$$|ax - 11| = 4x - 10, \tag{1}$$
where $a$ is a constant.
(1) Equation (1) can be rewritten without using the absolute value symbol as
$$\begin{aligned} & \text{when } ax \geqq 11, \quad \text{then } (a - \mathbf{N})x = \mathbf{O}; \\ & \text{when } ax < 11, \quad \text{then } (a + \mathbf{P})x = \mathbf{QR}. \end{aligned}$$
(2) When $a = \sqrt{7}$, the solution of equation (1) is
$$x = \frac{\mathrm{S}}{\mathrm{T}} - \sqrt{\mathrm{U}}.$$
(3) Let $a$ be a positive integer. When equation (1) has a positive integral solution, we have $a = \mathbf{W}$, and that solution $x = \mathbf{X}$.

Let a be a real number. The distance of a from 1 on the number line is $a + 4$ units.
Accordingly, what is $|a|$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 2 }$
D) $\frac { 7 } { 3 }$
E) $\frac { 8 } { 3 }$

For real numbers $\mathbf { x }$ and $\mathbf { y }$
$$\begin{aligned} & y - x = 1 \\ & y - | x - y | = 2 \end{aligned}$$
Given this, what is the sum $\mathbf { x } + \mathbf { y }$?
A) 5
B) 6
C) 7
D) 8
E) 9

$$| x - 2 | \cdot | x - 3 | = 3 - x$$
What is the sum of the $\mathbf { x }$ values that satisfy this equation?
A) - 3
B) - 2
C) 0
D) 2
E) 4

For real numbers x and y
$$\begin{aligned} & 2 x = 7 - | y | \\ & y = \frac { | x | } { 3 } \end{aligned}$$
Given that, what is the sum $\mathbf { x } + \mathbf { y }$?
A) 12 B) 10 C) 8 D) 6 E) 4

For non-zero real numbers $x$ and $y$
$$\begin{aligned} & | x \cdot y | = - 2 x \\ & \left| \frac { y } { x } \right| = 3 y \end{aligned}$$
the following equalities are given.
Accordingly, what is the sum $x + y$?
A) $\frac { 3 } { 2 }$ B) $\frac { 5 } { 2 }$ C) $\frac { 5 } { 3 }$ D) $\frac { 7 } { 3 }$ E) $\frac { 5 } { 6 }$

For integers $x$ and $y$,
$$| x - 3 | + | 2 x + y | + | 2 x + y - 1 | = 1$$
the equality is satisfied. Accordingly, what is the sum of the values that $y$ can take?
A) $- 12$
B) $- 11$
C) $- 10$
D) $- 9$
E) $- 8$