$$\left. \begin{array} { l } x ( 3 - x ) > 0 \\ ( 2 x + 1 ) ( x - 2 ) < 0 \end{array} \right\}$$
If the solution set of the inequality system given above is the open interval $(\mathbf { a } , \mathbf { b })$, what is the difference $\mathbf { a - b }$?
A) - 2
B) 0
C) 1
D) $\frac { 1 } { 2 }$
E) $\frac { 3 } { 2 }$

$$-2 < x < 4$$
Given that, what is the greatest integer value that the expression $1 - x$ can take?
A) $-3$
B) $-2$
C) $-1$
D) 2
E) 3

Let x and y be real numbers with $-1 < y < 0 < x$. Which of the following statements are always true?
I. $x + y > 0$ II. $x - y > 1$ III. $x \cdot ( y + 1 ) > 0$
A) Only I
B) Only III
C) I and II
D) I and III
E) II and III

Given that $x < 0 < y$, I. $y - x ^ { - 1 }$ II. $x ^ { 2 } + y ^ { - 1 }$ III. $( x \cdot y ) ^ { - 1 }$ Which of these expressions have negative values?
A) Only I
B) Only II
C) Only III
D) I and III
E) II and III

For real numbers $x , y$ and $z$
$$x + y < 0 < x < y + z$$
Given this, which of the following orderings is correct?
A) $x < y < z$
B) $x < z < y$
C) $y < x < z$
D) $y < z < x$
E) $z < y < x$

a, b are real numbers and
$$\begin{aligned} & 0 < a < 3 a ^ { 2 } \\ & b - 1 = 6 a \end{aligned}$$
Given this, what is the smallest integer value that b can take?
A) 3
B) 4
C) 5
D) 6
E) 7

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

If Ahmet's salary is increased by half of Deniz's salary, then the sum of their salaries becomes 2 times Ahmet's initial salary.
If Ahmet's salary is A TL and Deniz's salary is D TL, what is the relationship between $A$ and $D$?
A) $5A = 8D$
B) $5A = 6D$
C) $4A = 5D$
D) $3A = 4D$
E) $2A = 3D$

Given that $| a | = 2 , | b | = 5$ and $| c | = 6$,
$$\begin{aligned} & \mathrm { c } < \mathrm { a } < \mathrm { b } \\ & \mathrm { a } \cdot \mathrm {~b} \cdot \mathrm { c } > 0 \end{aligned}$$
What is the sum $a + b + c$?
A) - 9
B) - 3
C) - 1
D) 1
E) 3

$$| x - 2 | \cdot | x - 3 | = 3 - x$$
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 $a$ 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

Let $f : \mathbf { R } \backslash \{ 0 \} \rightarrow \mathbf { R }$ with
$$f ( x ) = \frac { 2 } { x } - x + 1$$
For this function, which of the following is the set of all $x$ points such that $f ( x ) \in ( 0 , \infty )$?
A) $( - \infty , 0 )$
B) $( - 1 , \infty )$
C) $( 0,1 ) \cup ( 2 , \infty )$
D) $( - 2,0 ) \cup ( 2 , \infty )$
E) $( - \infty , - 1 ) \cup ( 0,2 )$

$$\frac { \sqrt { 2 - 2 x } } { \sqrt { 3 + 3 x } } = \frac { 1 } { 2 }$$
Given that this holds, what is x?
A) $\frac { 2 } { 7 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 4 } { 9 }$
D) $\frac { 5 } { 11 }$
E) $\frac { 7 } { 12 }$

For real numbers $\mathbf { a }$ and $\mathbf { b }$
$$b ^ { 2 } < a \cdot b < b - a$$
Given that, which of the following orderings is correct?
A) $a < 0 < b$ B) $b < 0 < a$ C) $0 < a < b$ D) $\mathrm { b } < \mathrm { a } < 0$ E) $a < b < 0$

Let $a , b , c$ be real numbers and $0 < b < 1$ such that
$$\begin{aligned} & a = b \cdot c \\ & a + c = b \end{aligned}$$
Given this, which of the following orderings is correct?
A) $a < b < c$
B) $a < c < b$
C) $b < a < c$
D) $c < a < b$
E) $c < b < a$

For real numbers $x$ and $y$,
$$\begin{aligned} & 3 < x < 12 \\ & \frac { x } { y } = \frac { 3 } { 2 } \end{aligned}$$
Given this, what is the sum of the integer values that $y$ can take?
A) 18
B) 21
C) 25
D) 28
E) 32

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) $x - y \leq 0$

Integers a and b satisfy the inequality
$$1 < a < b - a < 5$$
Accordingly, what is the sum of the values that b can take?
A) 11
B) 14
C) 15
D) 16
E) 18

On the real number line, numbers whose distance to point 2 is less than half the distance to point $-4$ form the solution set of which of the following inequalities?
A) $| x - 2 | < | x + 4 |$
B) $| x + 2 | < | x - 4 |$
C) $| 2 x - 4 | < | x + 4 |$
D) $| 2 x - 4 | < | x - 4 |$
E) $| 2 x + 4 | < | x + 4 |$

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

For real number $x$
$$- 3 < 2 x < 7$$
Accordingly, what is the sum of the integer values that the expression $5 - x$ can take?
A) 5 B) 10 C) 15 D) 20 E) 25

Three-digit natural numbers $ADB$, $ADC$, $DAA$, $DAD$ $$\begin{aligned}& \mathrm{ADB} < \mathrm{DAA} \\& \mathrm{DAD} < \mathrm{ADC}\end{aligned}$$ satisfy the inequalities.\ Accordingly, which of the following orderings is correct?\ A) A $=$ D $<$ B $<$ C\ B) C $<$ A $=$ B $<$ D\ C) D $<$ A $=$ B $<$ C\ D) B $<$ A $=$ D $<$ C\ E) C $<$ A $=$ D $<$ B

For nonzero real numbers $x$ and $y$, given that $y < x$ and $x ^ { 2 } < y ^ { 2 }$,\ I. $x \cdot y > 0$\ II. $x + y < 0$\ III. $\frac { 1 } { x } - \frac { 1 } { y } > 0$\ Which of the following statements are always true?\ A) Only I\ B) Only II\ C) I and II\ D) I and III\ E) II and III

$( x - 1 ) ^ { 2 } < | x - 1 | + 6$\ What is the sum of the integers $x$ that satisfy this inequality?\ A) 2\ B) 3\ C) 4\ D) 5\ E) 6

$\frac { 6 x + 1 } { ( x + 1 ) ^ { 2 } } > 1$\ Which of the following is the set of all real numbers that satisfy this inequality?\ A) $( - 1,4 )$\ B) $( - 1,6 )$\ C) $( 0,4 )$\ D) $( 0 , \infty )$\ E) $( 2 , \infty )$