The number of real roots of the equation $x | x | - 5 | x + 2 | + 6 = 0$, is
(1) 5
(2) 4
(3) 6
(4) 3

Let $f(x) = |2x^2 + 5x - 3|$, $x \in \mathbb{R}$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to:
(1) 5
(2) 2
(3) 0
(4) 3

The number of distinct real roots of the equation $| x + 1 | | x + 3 | - 4 | x + 2 | + 5 = 0$, 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 $x = \mathbf{X}$.

Consider the expression in $x$
$$P = | x - 1 | + | x - 2 | + | x - a | .$$
We are to find the range of real numbers $a$ such that the value of $P$ is minimized at $x = a$.
First, let us note that the inequality
$$| x - 1 | + | x - 2 | + | x - a | \geqq | x - 1 | + | x - 2 |$$
always holds, and is an equality in the case $x = a$.
When we set
$$y = | x - 1 | + | x - 2 | ,$$
we have
$$y = \begin{cases} - \mathbf { A } & x + \mathbf { B } \\ \mathbf { D } & ( x < \mathbf { C } ) \\ \mathbf { F } & ( \mathbf { C } \leqq x \leqq \mathbf { E } ) \\ \mathbf { G } & ( \mathbf { E } < x ) . \end{cases}$$
When we consider the graph of (1), we see that the minimum value of $y$ is $\mathbf { H }$ and $y$ takes the value $\mathbf{H}$ at every $x$ satisfying $\mathbf{I} \leqq x \leqq \mathbf{J}$.
Thus, for every $a$ satisfying $\mathbf{K} \leqq a \leqq \mathbf{L}$, the value of $P$ is minimized at $x = a$ and its value there is $\mathbf{M}$.

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

A point $P$ on a number line satisfies the condition that the distance from $P$ to 1 plus the distance from $P$ to 4 equals 4. How many such points $P$ are there?
(1) 0
(2) 1
(3) 2
(4) 3
(5) Infinitely many

$$f(x) = \sqrt{2-|x+3|}$$
Which of the following is the domain interval of the function?
A) $3 \leq x \leq 5$
B) $-1 \leq x \leq 5$
C) $-3 \leq x \leq 4$
D) $-3 \leq x \leq 0$
E) $-5 \leq x \leq -1$

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

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 $a$ 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

What is the sum of all natural numbers such that when divided by 6, the quotient and remainder are equal to each other?
A) 84
B) 91
C) 96
D) 105
E) 112

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

I. $f ( x ) = x - 1$ II. $g ( x ) = | x - 1 |$ III. $h ( x ) = \sqrt [ 3 ] { ( x - 1 ) ^ { 2 } }$ Which of the following functions do not have a derivative at the point $x = 1$?
A) Only I
B) Only II
C) I and II
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) $x - y \leq 0$

$$f ( x ) = \left| \frac { 2 x - 1 } { x - 1 } \right|$$
The graph of the function intersects its horizontal asymptote at the point (a, b).
Accordingly, what is the sum $a + b$?
A) $\frac { 5 } { 2 }$
B) $\frac { 7 } { 2 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 9 } { 4 }$

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 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 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) $- 8$