1003162905
Level:
C
Choose the equation that is equivalent to \( \bigl| |2-x|-1 \bigr|=1 \) for \( x\in(3;\infty) \).
\( (x-2)-1=1 \)
\( -(x-2)-1=1 \)
\( -\bigl(1-(2-x)\bigr)=1 \)
\( 1-(x-2)=1 \)
Absolute value equations and inequalities
1003163002
Level:
C
Find all \( t \), \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has more than two solutions.
\[ \Bigl| |3-x|-3\Bigr|=t \]
\( t\in(0;3] \)
\( t\in\{0;3\} \)
\( t\in\{3\} \)
\( t\in[1;3] \)
1003163004
Level:
C
Find the sum of all the roots of the given equation.
\[ \Bigl| |x-4|-9\Bigr|=0 \]
\( 8 \)
\( 18 \)
\( 6 \)
\( 4 \)
1003177804
Level:
C
Choose the solution set of the given inequality.
\[ \left| |x-5|+2\right| > 3 \]
\( (-\infty;4)\cup(6;\infty) \)
\( (-\infty;-4)\cup(6;\infty) \)
\( (6;\infty) \)
\( (-\infty;-5)\cup(6;\infty) \)
1003177805
Level:
C
Let \( x\in\mathbb{Z} \). Find the sum of all the solutions of the given inequality.
\[ \Bigl| \bigl| |x+1|+1 \bigr|+1\Bigr| \leq 3 \]
\( -3 \)
\( -2 \)
\( -4 \)
\( 0 \)
1003187302
Level:
C
The equation \( \left| |2x-8|-4 \right|=4 \) has:
three real solutions
one real solution
four real solutions
two real solutions
1003187401
Level:
C
Give the solution set of the equation \( \left| |x-1|-|3-x| \right|=2 \).
\( (-\infty;1]\cup[3;+\infty) \)
\( (-\infty;1)\cup(3;+\infty) \)
\( \mathbb{R} \)
\( (1;3) \)
1003187403
Level:
C
How many solutions does the equation \( \left| |2x+6|-8\right|=4 \) have?
\( 4 \)
\( 2 \)
\( 0 \)
\( 6 \)
1003187410
Level:
C
How many integers are in the solution set of the given inequalities?
\[ \left|\sqrt{17}-2x\right| \leq 5 \text{ and } \left|2x-\sqrt{17}\right| \leq 5 \]
\( 5 \)
\( 4 \)
\( 6 \)
\( 7 \)
2000002203
Level:
C
Identify the solution set of the following inequality.
\[
|x + 2| + |x + 3| \leq 0
\]
\( \emptyset \)
\( \mathbb{R} \)
\( \mathbb{R} \setminus \{-3;-2\} \)
\( \{-3;-2\} \)