23. Solution: (1) When $a = 1$, $f(x) = \left\{\begin{array}{l} 5 - 2x, & x \leq 1 \\ 3, & 1 < x < 4 \\ 2x - 5, & x \geq 4 \end{array}\right.$ ..... 3 marks
Thus the solution set of the inequality $f(x) < x$ is $(3, 5)$. ..... 5 marks
(2) $f(x) = |x - a| + |x - 4| \geq |(x - a) - (x - 4)| = |a - 4|$. ..... 6 marks
$\therefore |a - 4| \geq \frac{4}{a} - 1 = \frac{4 - a}{a}$. ..... 7 marks
When $a < 0$ or $a \geq 4$, the inequality clearly holds. ..... 8 marks
When $0 < a < 4$, $\frac{1}{a} \leq 1$, then $1 \leq a < 4$. ..... 9 marks
Thus the range of $a$ is $(-\infty, 0) \cup [1, +\infty)$. ..... 10 marks
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Given the function $f ( x ) = \left| x - a ^ { 2 } \right| + | x - 2 a + 1 |$ .
(1) When $a = 2$, find the solution set of the inequality $f ( x ) \geqslant 4$;
(2) If $f ( x ) \geqslant 4$ for all $x$, find the range of values of $a$.

[Elective 4-5: Inequalities]
Given $f(x) = 2|x - a| - a , \ a > 0$ .
(1) Solve the inequality $f(x) < x$ ;
(2) If the area enclosed by $y = f(x)$ and the coordinate axes is 2 , find $a$ .

106- The solution set of the inequality $|x|(2x - 5) \leq |x - 4|$ is which of the following?

• [(1)] $(1, 5)$
• [(2)] $(1 - \sqrt{6}\ ,\ 1 + \sqrt{6}\ )$
• [(3)] $(1,5) \cup (1+\sqrt{6}\ , +\infty)$
• [(4)] $(-\infty, 1-\sqrt{6}\ ) \cup (1, 5)$

The real number $x$ satisfies $$\frac{|x|^2 - |x| - 2}{2|x| - |x|^2 - 2} > 2$$ if and only if $x$ belongs to
(A) $(-2, -1) \cup (1, 2)$
(B) $(-2/3, 0) \cup (0, 2/3)$
(C) $(-1, -2/3) \cup (2/3, 1)$
(D) $(-1, 0) \cup (0, 1)$

8. The set of all real numbers $x$ for which $x ^ { 2 } - | x + 2 | + x > 0$ is
(A) $( - \infty , - 2 ) \cup ( 2 , \infty )$
(B) $( - \infty , - \sqrt { } 2 ) \cup ( \sqrt { } 2 , \infty )$
(C) $( - \infty , - 1 ) \cup ( 1 , \infty )$
(D) $( \sqrt { } 2 , \infty )$

Let $S = \{ x \in R : x \geq 0 \& 2 | \sqrt { x } - 3 | + \sqrt { x } ( \sqrt { x } - 6 ) + 6 = 0 \}$. Then $S$ :
(1) Contains exactly four elements
(2) Is an empty set
(3) Contains exactly one element
(4) Contains exactly two elements

Let $S$ be the set of all real roots of the equation, $3^{x}\left(3^{x} - 1\right) + 2 = \left|3^{x} - 1\right| + \left|3^{x} - 2\right|$, then
(1) contains exactly two elements.
(2) is a singleton.
(3) is an empty set.
(4) contains at least four elements.

The number of real solution(s) of the equation $x^{2} + 3x + 2 = \min\{|x - 3|, |x + 2|\}$ is:
(1) 1
(2) 0
(3) 2
(4) 3

$x | x + 4 | + 3 | x + 2 | + 10 = 0$ No. of real soln.

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
$$x = \frac{\mathrm{S}\left(\frac{\mathrm{T}}{\mathrm{V}} - \sqrt{\mathrm{U}}\right)}{\mathrm{V}}.$$
(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 integer $n$ satisfy $| 5 n - 21 | \geq 7 | n |$ . Select the correct options.
(1) $| 5 n - 7 n | \geq 21$
(2) $- 1 \leq \frac { 7 n } { 5 n - 21 } \leq 1$
(3) $7 n \leq 5 n - 21$
(4) $( 5 n - 21 ) ^ { 2 } \geq 49 n ^ { 2 }$
(5) There are infinitely many integers $n$ satisfying the given inequality

$x$ and $y$ satisfy $| 2 - x | \leq 6$ and $| y + 2 | \leq 4$.
What is the greatest possible value of $| x y |$ ?
A 16
B 24
C 32
D 40
E 48
F There is no greatest possible value.

Consider the two inequalities:
$$\begin{aligned} & | x + 5 | < | x + 11 | \\ & | x + 11 | < | x + 1 | \end{aligned}$$
Which one of the following is correct?
A There is no real number for which both inequalities are true.
B There is exactly one real number for which both inequalities are true.
C The real numbers for which both inequalities are true form an interval of length 1 .
D The real numbers for which both inequalities are true form an interval of length 2 .
E The real numbers for which both inequalities are true form an interval of length 3 .
F The real numbers for which both inequalities are true form an interval of length 4 .
G The real numbers for which both inequalities are true form an interval of length 5 .

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) $| 2 x + 4 | < | x + 4 |$

A weather forecaster made the following statement during a live broadcast on Sunday evening.
"In our city where the temperature has been 5 degrees throughout this week, the weather will suddenly warm up starting tomorrow and winter will give way to spring-like weather. On Monday afternoon, the air temperature throughout the city will have increased by 6 to 10 degrees compared to the previous day."
Based on this information, which of the following inequalities expresses the range of values that the temperature in the city could take on Monday afternoon?
A) $|x - 13| \leq 2$
B) $|x - 10| \leq 6$
C) $|x - 6| \leq 5$
D) $|x - 1| \leq 6$
E) $|x - 11| \leq 2$

A positive number A is shown on the number line as in the figure.
Then, on this number line; numbers whose distance from 0 is equal to half the distance of number A from 0 are marked.
If the distance from one of the marked numbers to number A is 6 units, what is the sum of the possible values of number A?
A) 15
B) 16
C) 18
D) 20
E) 21

An instructor at a parachute jumping course gives the following explanation to the trainees:
"When jumping from an airplane at a height of 800 meters from the ground, you need to open your parachute 400 to 500 meters after jumping from the airplane in order to land safely on the ground."
Accordingly, which of the following inequalities expresses the values that the height from the ground when the parachute opens can take in order to land safely?
A) $|x - 350| \leq 50$ B) $|x - 300| \leq 100$ C) $|x - 250| \leq 150$ D) $|x - 200| \leq 200$ E) $|x - 150| \leq 250$

In a store, the cost prices of a dishwasher, a washing machine, and a refrigerator are 18 thousand, 22 thousand, and $b$ TL, respectively. These products are sold together as a triple white goods package. This package is sold at twice the cost price of a refrigerator plus 30 thousand TL, the store makes a profit; when sold at three times the cost price of a refrigerator minus 20 thousand TL, the store makes a loss.
Accordingly, which of the following inequalities expresses all possible values of the cost price of a refrigerator?
A) $|b - 20000| < 10000$
B) $|b - 15000| < 5000$
C) $|2b - 15000| < 12000$
D) $|2b - 24000| < 17000$
E) $|3b - 16000| < 8000$