1003162906
Level:
B
Find the number of the roots of the equation \( |x+3|+|x+2|=6 \).
\( 2 \)
\( 1 \)
\( 4 \)
\( 3 \)
Absolute Value Equations and Inequalities
1003162905
Level:
C
Choose the equation that is equivalent to \( \bigl| |2-x|-1 \bigr|=1 \) for \( x\in(3,\infty) \).
\( (x-2)-1=1 \)
\( -(x-2)-1=1 \)
\( -\bigl(1-(2-x)\bigr)=1 \)
\( 1-(x-2)=1 \)
1003162904
Level:
C
Find the sum of all the roots of the equation \( \bigl| |1-x|-2 \bigr|=1 \) for \( x\in(1,\infty) \).
\( 6 \)
\( 4 \)
\( 1 \)
\( 2 \)
1003162903
Level:
C
Find the sum of all the roots of the given equation.
\[ \bigl| |x-4|-2\bigr|=0 \]
\( 8 \)
\( 12 \)
\( 4 \)
\( 6 \)
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.
\[ \Bigl| |3-x|-3\Bigr|=t \]
\( t\in(0,3] \)
\( t\in\{0,3\} \)
\( t\in\{3\} \)
\( t\in[1,3] \)