2010008604
Level:
C
Find the sum of all the roots of the given equation.
\[ \bigl|2- |x+1|\bigr|=0 \]
\( -2\)
\( 2\)
\( -4 \)
\( 0 \)
Absolute value equations and inequalities
2010008603
Level:
B
Calculate for which values of the parameter \(t \in\mathbb{R} \) the given equation has no solution.
\[ |2-4x|=1-t\]
\( t\in (1 ;\infty) \)
\( t\in (-\infty;1) \)
\( t\in \left(\frac12 ;\infty\right) \)
\( t\in \left(-\infty;\frac12\right) \)
2010008601
Level:
B
Choose the equation which is, on the interval \( \left[ \frac12; 2 \right] \), equivalent to
\[ |2x-1|+2|x+5|=1-|2-x|. \]
\( (2x-1)+2(x+5)=1-(2-x) \)
\( (2x-1)+2(x+5)=1+(2-x) \)
\( (2x-1)-2(x+5)=1+(2-x) \)
\( -(2x-1)+2(x+5)=1+(2-x) \)
2010008505
Level:
C
Give the solution set of the equation \( \left| |2+x|-|x-4| \right|=6\).
\( (-\infty;-2] \cup [ 4 ;+\infty) \)
\( (-\infty;-4] \cup [ 2 ;+\infty) \)
\( [ -2 ;+\infty) \)
\( [ 4 ;+\infty) \)
2010008504
Level:
B
Give the number which satisfies the equation \( |x+2|-1=x+3\).
\(-3\)
\( -2 \)
\( 0 \)
\( 1 \)
2010008503
Level:
A
Choose the equation which has only one solution.
\( 5-|2x+2|-1=4 \)
\( 5-|2x+2|+1=5 \)
\( -|2x+2|+3=4 \)
\( -|2x+2|+1=0 \)
2010008502
Level:
C
Find all \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has more than two solutions.
\[ \Bigl| 1-|2-x|\Bigr|+1=t \]
\( t\in(1;2 ] \)
\( t\in\{2\} \)
\( t\in(2;\infty) \)
\( t \in \emptyset \)
2010008501
Level:
B
Find all \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has exactly two solutions.
\[ |x+2|+3=t-1 \]
\( t\in(4;\infty) \)
\( t\in(-2;\infty) \)
\( t\in\{4\} \)
\( \mathbb{R}\)
2000002205
Level:
A
Identify the solution set of the following inequality.
\[
|7x+1|+1>0
\]
\( \mathbb{R} \)
\( \emptyset \)
\( \mathbb{R} \setminus \left\{\frac{1}{7}\right\} \)
\( \mathbb{R} \setminus \left\{-\frac{1}{7}\right\} \)
2000002204
Level:
A
Identify the solution set of the following inequality.
\[
-|2x-100|< 0
\]
\( \mathbb{R} \setminus \{50\} \)
\( \mathbb{R} \)
\( \mathbb{R} \setminus \{-50\} \)
\( \{50\} \)