9000078508
Level:
B
Assuming \(x\in (1;\infty )\),
simplify the following expression.
\[
3x -|2x + 1| + |x - 1|
\]
\(2x - 2\)
\(4x - 2\)
\(2x + 2\)
\(2x\)
Absolute value
9000078510
Level:
B
Assuming \(x\in (6;11)\),
simplify the following expression.
\[
3|x - 11|- 2|6 - x|
\]
\(- 5x + 45\)
\(5x - 45\)
\(x - 45\)
\(x - 21\)
9000081408
Level:
B
For \(x\in \mathbb{R}^{-}\) consider
expressions \(|x|\),
\(|- x|\),
\(-|x|\) and
\(- x\).
Which of these attains only negative values?
\(-|x|\)
\(|x|\)
\(|- x|\)
\(- x\)
1003049203
Level:
C
Identify which of the statements is false.
\( \forall a\text{, }b\in\mathbb{R}\colon |a+b|=|a|+|b| \)
\( \forall a\text{, }b\in\mathbb{R}\colon |a\cdot b|=|a|\cdot|b| \)
\( \forall a\in\mathbb{R}\text{, }b\in\mathbb{R}\setminus\{0\}\colon|\frac ab|=\frac{|a|}{|b|} \)
\( a\in\mathbb{R}\colon |a|=|-a| \)
1003187101
Level:
C
Which of the following relations is correct for all \( x \), \( y\in\mathbb{R} \)?
\( |x+y| \leq |x|+|y| \)
\( |x+y|=|x|+|y| \)
\( |x-y| < |x|-|y| \)
\( |x-y|=|x|-|y| \)
1003187102
Level:
C
For \( x \), \( y\in\mathbb{R} \) consider \( |x+y|=|x|+|y| \).
The equality holds if and only if the sign of \( x \) and \( y \) is the same.
The equality does not hold for any \( x \) and \( y \).
The equality holds if and only if \( x \) and \( y \) are all positive.
The equality holds if and only if \( x \) and \( y \) are both nonpositive.
1003187103
Level:
C
Find the relation that does not hold for any \( x \), \( y\in\mathbb{R} \).
\( \left| |x|-|y| \right| > |x+y| \)
\( |xy|=|x| |y| \)
\( \left|\frac xy \right|=\frac{|x|}{|y|}\text{, } y\neq0\text{ .} \)
\( \left| (xy)^2 \right|=|xy|^2=(xy)^2 \)
1003187106
Level:
C
Consider expressions \( |3x-12| \), \( 3|x|+12 \), \( |3x|-|-12| \), \( 3|x-4| \). Which of the expressions has the greatest value if we substitute any \( x \) from the interval \( (0;+\infty) \)?
\( 3|x|+12 \)
\( |3x-12| \)
\( |3x|-|-12| \)
\( 3|x-4| \)
9000081406
Level:
C
For \(x\in \mathbb{R}\) find the correct
relationship between \(|x|\)
and \(|- x|\).
\(|x| = |- x|\)
\(|x| > |- x|\)
\(|x| < |- x|\)
None of them. The answer depends on the particular value of
\(x\).
9000081407
Level:
C
For \(x,y\in \mathbb{R}\) find the correct
relationship between \(|x - y|\)
and \(|y - x|\).
\(|x - y| = |y - x|\)
\(|x - y| > |y - x|\)
\(|x - y| < |y - x|\)
None of them. The answer depends on the particular values of
\(x\),
\(y\).