7. For $x \in R$, define the sign function $\operatorname { sgn } x = \left\{ \begin{array} { c } 1 , x > 0 \\ 0 , x = 0 \\ - 1 , x < 0 \end{array} \right.$, then
A. $ \{ x | = x | \operatorname { sgn } x \mid \}$
B. $ \{ x | = \operatorname { sgn } | x \mid \}$
C. $ \{ x | = x | \operatorname { sgn } x \}$
D. $ \{ x \mid = x \operatorname { sgn } x \}$

6. The sign function is defined as $\operatorname{sgn} x = \begin{cases} 1, & x > 0, \\ 0, & x = 0, \\ -1, & x < 0. \end{cases}$ Let $f(x)$ be an increasing function on $\mathbf{R}$, and $g(x) = f(x) - f(ax)$ where $a > 1$. Then
A. $\operatorname{sgn}[g(x)] = \operatorname{sgn} x$
B. $\operatorname{sgn}[g(x)] = -\operatorname{sgn} x$
C. $\operatorname{sgn}[g(x)] = \operatorname{sgn}[f(x)]$
D. $\operatorname{sgn}[g(x)] = -\operatorname{sgn}[f(x)]$

For real numbers $a$ and $b$, it is known that $( | a | - a ) ( | b | + b ) > 0$.
Accordingly, I. $a + b < 0$ II. $a - b < 0$ III. $\mathrm { a } \cdot \mathrm { b } < 0$ Which of the following statements are always true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

Real numbers $x$ and $y$ satisfy the equality
$$\| x | + | y | | = | x + y |$$
Accordingly, which of the following inequalities is always true?
A) $x \cdot y \geq 0$
B) $x \cdot y \leq 0$
C) $x + y \geq 0$
D) $x + y \leq 0$
E) $x - y \leq 0$

For nonzero real numbers $x$, $y$, and $z$ whose absolute values are distinct from each other, $$\begin{aligned}| x + y | & = | x | - | y | \\| y + z | & = | y | + | z |\end{aligned}$$ the following equalities are satisfied.
Given that $x > 0$,\ I. $\frac { x } { x + y } < 1$\ II. $\frac { y } { y + z } < 1$\ III. $\frac { z } { x + z } < 1$\ Which of the following statements are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and III\ E) II and III