1003187403
Level:
C
How many solutions does the equation \( \left| |2x+6|-8\right|=4 \) have?
\( 4 \)
\( 2 \)
\( 0 \)
\( 6 \)
Absolute value equations and inequalities
1003187402
Level:
B
For a given parameter \( a\in\mathbb{R} \) the number \( \pi-\sqrt[3]5-\sqrt2+7 \) represents one of the solutions of the equation \( |2x|=2a^2 \). Which of the following numbers is also the solution of this equation?
\( \sqrt2-7+\sqrt[3]5-\pi \)
\( \sqrt{7-\pi+\sqrt[3]5+\sqrt2} \)
\( \sqrt[3]5-\pi-\sqrt2-7 \)
\( \sqrt{\pi-\sqrt[3]5-\sqrt2+7} \)
1003187401
Level:
C
Give the solution set of the equation \( \left| |x-1|-|3-x| \right|=2 \).
\( (-\infty;1]\cup[3;+\infty) \)
\( (-\infty;1)\cup(3;+\infty) \)
\( \mathbb{R} \)
\( (1;3) \)
1003047107
Level:
B
Decide which of the following intervals ensures that the expressions in all the absolute values of the given equation are positive on this interval.
\[ |2x-1|+|2x|=|1-x|-|x+4| \]
\( (0.5;1) \)
\( (-4; 1) \)
\( (1;\infty) \)
\( (-4; 0) \)
1003047106
Level:
B
Decide which of the following intervals ensures that the expressions in all the absolute values of the given equation are negative on this interval.
\[ |5-x|-|x+3|=|x+1| \]
No such interval exists.
\( (-\infty;-3) \)
\( (-\infty;-1) \)
\( (-1; 5) \)
1003047105
Level:
B
Decide which of the following intervals ensures that the expressions in all the absolute values of the given equation are positive on this interval.
\[ |x+3|+|2x|=|x-1| \]
\( (1;\infty) \)
\( (-3;1) \)
\( (-3;\infty) \)
\( (-\infty;-3) \)
1003047104
Level:
B
Decide which of the following intervals ensures that the expressions in all the absolute values of the given equation are positive on this interval.
\[ |2x+4|+|-x+1|=2x \]
\( (-2;1) \)
\( (-2;0) \)
\( ( 0;1) \)
\( (1;\infty) \)
1003047103
Level:
B
Find the set of the zero points of all the absolute values from the equation.
\[ |-x+1|=|3x+9|-|2x|\]
(The zero point is the value of \( x \) for which the absolute value equals zero.)
\( \{ -3;0;1 \} \)
\( \{ -3;-1;0 \} \)
\( \{ -3;0 \} \)
\( \{ -3;-1;3 \} \)
1003047102
Level:
B
Find the sum of the zero points of all the absolute-value expressions of the equation.
\[ |x-3|=|x+9|-|-2+x| \]
(The zero point is the value of \( x \) for which the absolute-value expression equals zero.)
\( -4 \)
\( -8 \)
\( 14 \)
\( 10 \)
1003047101
Level:
B
Choose the equation which is, on the interval \( [-2; 3] \), equivalent to
\[ |-x+3|+|x-7|=|x+2|. \]
\( (-x+3)-(x-7)=(x+2) \)
\( -(-x+3)+(x-7)=(x+2) \)
\( (-x+3)+(x-7)=-(x+2) \)
\( (-x+3)-(x-7)=-(x+2) \)