12. Let $f ( x ) = \ln ( 1 + | x | ) - \frac { 1 } { 1 + x ^ { 2 } }$. The range of $x$ for which $f ( x ) > f ( 2 x - 1 )$ holds is
A. $\left( \frac { 1 } { 3 } , 1 \right)$
B. $\left( - \infty , \frac { 1 } { 3 } \right) \cup ( 1 , + \infty )$
C. $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$
D. $\left( - \infty , - \frac { 1 } { 3 } \right) \cup \left( \frac { 1 } { 3 } , + \infty \right)$
II. Fill-in-the-Blank Questions: This section contains 4 questions, 5 points each, 20 points total

Let $\alpha$ denote a real number. The range of values of $| \alpha - 4 |$ such that $| \alpha - 1 | + | \alpha + 3 | \leq 8$ is
(A) $( 0,7 )$
(B) $( 1,8 )$
(C) $[ 1,9 ]$
(D) $[ 2,5 ]$.

Consider the real numbers $a$ and $b$ such that the equation in $x$
$$| x - 3 | + | x - 6 | = a x + b \tag{1}$$
has a solution.
Set the left side of (1) as $y = | x - 3 | + | x - 6 |$. This can be represented without using the absolute value signs in the following way.
$$\begin{array} { l l } \text { If } x < \mathbf { A } , & \text { then } y = - \mathbf { B } x + \mathbf { C } ; \\ \text { if } \mathbf { A } \leqq x < \mathbf { D } , & \text { then } y = \mathbf { E } ; \\ \text { if } \mathbf { D } \leqq x , & \text { then } y = \mathbf { F } x - \mathbf { G } . \end{array}$$
Next, let us consider the common point(s) of the graph of this function and the straight line $y = a x + b$ on the $x y$-plane. Then we see the following:
(i) If $a = 1$, then the range of the values of $b$ such that (1) has one or more solutions is
$$b \geqq \mathbf { H I } .$$
(ii) If $b = 6$, then the range of the values of $a$ such that (1) has two different solutions is
$$\mathbf { J K } < a < \mathbf { L }.$$

On a number line, there is the origin $O$ and three points $A ( - 2 ) , B ( 10 ) , C ( x )$, where $x$ is a real number. Given that the lengths of segments $\overline { B C } , \overline { A C } , \overline { O B }$ satisfy $\overline { B C } < \overline { A C } < \overline { O B }$, then the maximum range of $x$ is (8) $< x <$ (9).

Let $a , b$ be real numbers. It is known that the four numbers $- 3 , - 1, 4, 7$ all satisfy the inequality $| x - a | \leq b$ in $x$. Select the correct options.
(1) $\sqrt { 10 }$ also satisfies the inequality $| x - a | \leq b$ in $x$
(2) $3, 1 , - 4 , - 7$ satisfy the inequality $| x + a | \leq b$ in $x$
(3) $- \frac { 3 } { 2 } , - \frac { 1 } { 2 } , 2 , \frac { 7 } { 2 }$ satisfy the inequality $| x - a | \leq \frac { b } { 2 }$ in $x$
(4) $b$ could equal 4
(5) $a$ and $b$ could be equal

Let x be a real number with $| x | \leq 4$, and
$$2 x + 3 y = 1$$
What is the sum of the integer values of y that satisfy this equation?
A) - 1
B) 0
C) 1
D) 2
E) 3