2000001201
Level:
B
Find all \(x \in \mathbb{R}\) such that the given relation is true.
\[ |x|=x\]
\( x \in [ 0;\infty) \)
\( x \in (-\infty;0] \)
\( x \in\mathbb{R}\)
No such \(x\) exists.
\( x \in \{0\} \)
Absolute value
Absolute Value Expressions II
Submitted by ladislav.foltyn on Mon, 04/15/2019 - 22:11Question:
Evaluate the following expressions for given $x$ and choose the correct answer.
Absolute Value Expressions I
Submitted by ladislav.foltyn on Fri, 02/15/2019 - 15:00Question:
\footnotesize
Evaluate absolute value expressions:
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 \( (0;+\infty) \)?
\( 3|x|+12 \)
\( |3x-12| \)
\( |3x|-|-12| \)
\( 3|x-4| \)
1003187105
Level:
B
Let \( a \) be a positive real number. Then \( |x| \geq a \)
if and only if \( x \geq a \) or \( x \leq -a \).
if and only if \( x \geq a \).
if and only if \( x \leq -a \).
if and only if \( x > 0 \).
1003187104
Level:
B
Let \( a \) be a positive real number. Then \( |x| \leq a \)
if and only if \( -a \leq x \leq a \).
if and only if \( x \leq a \).
if and only if \( x \geq -a \).
if and only if \( x < 0 \).
1003187103
Level:
C
Find the relation that does not hold for any \( x \), \( y\in\mathbb{R} \).
\( \left| |x|-|y| \right| > |x+y| \)
\( |xy|=|x| |y| \)
\( \left|\frac xy \right|=\frac{|x|}{|y|}\text{, } y\neq0\text{ .} \)
\( \left| (xy)^2 \right|=|xy|^2=(xy)^2 \)
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.
1003187101
Level:
C
Which of the following relations is correct for all \( x \), \( y\in\mathbb{R} \)?
\( |x+y| \leq |x|+|y| \)
\( |x+y|=|x|+|y| \)
\( |x-y| < |x|-|y| \)
\( |x-y|=|x|-|y| \)
1003187304
Level:
B
How many solutions does the equation \( \left| |x-4|-2\right|+2=0 \) have?
\( 0 \)
\( 2 \)
\( 4 \)
\( 6 \)