1003162903
Level:
C
Find the sum of all the roots of the given equation.
\[ \bigl| |x-4|-2\bigr|=0 \]
\( 8 \)
\( 12 \)
\( 4 \)
\( 6 \)
Absolute value equations and inequalities
1003162902
Level:
B
Calculate for which value(s) of the parameter \( t \) the given equation has no solution.
\[ |x+4|-2=t \]
\( t\in(-\infty;-2) \)
\( t\in(-\infty;-4) \)
\( t\in(-4;\infty) \)
\( t\in(-\infty;2) \)
1003162901
Level:
B
Calculate for which value(s) of the parameter \( t \) the given equation has at least one solution.
\[ |x-5|=t \]
\( t\in[0;\infty) \)
\( t\in[5;\infty) \)
\( t\in\{0\} \)
\( t\in(-\infty;5] \)
1003163005
Level:
B
Find all \( t \), \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has exactly two solutions.
\[ |x-t|+1=3 \]
\( t\in\mathbb{R} \)
\( t\in(-\infty;2) \)
\( t\in(2;\infty) \)
\( t\in(-2;\infty) \)
1003163004
Level:
C
Find the sum of all the roots of the given equation.
\[ \Bigl| |x-4|-9\Bigr|=0 \]
\( 8 \)
\( 18 \)
\( 6 \)
\( 4 \)
1003163003
Level:
B
How many roots has the given equation for \( x\in(-\infty;-3] \)?
\[ |3-2x|-3|x|=-8 \]
\( 1 \)
\( 2 \)
\( 0 \)
\( 3 \)
1003163002
Level:
C
Find all \( t \), \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has more than two solutions.
\[ \Bigl| |3-x|-3\Bigr|=t \]
\( t\in(0;3] \)
\( t\in\{0;3\} \)
\( t\in\{3\} \)
\( t\in[1;3] \)
1003163001
Level:
B
Find all \( t \), \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has exactly two solutions.
\[ |x|+t=-3 \]
\( t\in(-\infty;-3) \)
\( t\in\{-3\} \)
\( t\in(-3;\infty) \)
\( t\in\{-3;3\} \)
1003177805
Level:
C
Let \( x\in\mathbb{Z} \). Find the sum of all the solutions of the given inequality.
\[ \Bigl| \bigl| |x+1|+1 \bigr|+1\Bigr| \leq 3 \]
\( -3 \)
\( -2 \)
\( -4 \)
\( 0 \)
1003177804
Level:
C
Choose the solution set of the given inequality.
\[ \left| |x-5|+2\right| > 3 \]
\( (-\infty;4)\cup(6;\infty) \)
\( (-\infty;-4)\cup(6;\infty) \)
\( (6;\infty) \)
\( (-\infty;-5)\cup(6;\infty) \)