9000026406
Level:
B
Assuming \(x\in (-3;2)\),
write the following equation in the form which does not contain an absolute value.
\[
|x + 3| = |x - 2|
\]
\(x + 3 = -x + 2\)
\(x + 3 = x - 2\)
\(- x - 3 = -x + 2\)
\(- x - 3 = x + 2\)
B
9000026407
Level:
B
Assuming \(x\in \left [ \frac{1}
{2};\infty \right )\),
write the following equation in the form which does not contain an absolute value.
\[
|x + 1| + |2x - 1| = 3
\]
\(x + 1 + 2x - 1 = 3\)
\(x + 1 - 2x - 1 = 3\)
\(x + 1 - 2x + 1 = 3\)
\(- x - 1 + 2x - 1 = 3\)
9000026409
Level:
B
Consider the following equation.
\[
|2x - 4| = 5x - 7
\]
Solving the equation on the intervals where it is possible to evaluate the absolute
value we get equations on partial subintervals as follows.
\[\begin{aligned}
\text{for }x &\in (-\infty ;2)\colon &\text{for }x &\in [ 2;\infty )\colon & & & &
\\ - 2x + 4 & = 5x - 7 &2x - 4 & = 5x - 7 & & & &
\\ - 7x & = -11 & - 3x & = -3 & & & &
\\x & = \frac{11}
{7} &x & = 1 & & & &
\end{aligned}\]
Find the solution set of the original equation.
\(\left \{\frac{11}
{7} \right \}\)
\(\left \{\frac{11}
{7} ;1\right \}\)
\(\left \{1\right \}\)
\(\emptyset \)
9000024809
Level:
B
Find the solution set of the following inequality.
\[
\sqrt{x + 3} > x - 3
\]
\([ -3;6)\)
\( (1;6)\)
\([ -3;3] \)
\( (-\infty ;1)\cup (6;+\infty )\)
9000025804
Level:
B
In the following list identify a true statement on the function
\(f\).
\[
f(x) = (x + 1)(x + 2)(x - 3)
\]
The function \(f\)
is positive on \(I_{1} = (-2;-1)\)
and \(I_{2} = (3;\infty )\).
The function \(f\)
is an increasing function (in its whole domain).
The function is decreasing only on
\(I = (-1;3)\).
The function is decreasing on \(I_{1} = (-\infty ;-2)\)
and \(I_{2} = (3;\infty )\).
9000024806
Level:
B
In the following list identify the interval which is a subset of the solution set of
the following inequality.
\[
\sqrt{x^{2 } + 2x - 3} > x + 2
\]
\((-\infty ;-3] \)
\(\left (-\frac{7}
{2};+\infty \right )\)
\((1;+\infty )\)
\((-\infty ;-2)\)
9000025610
Level:
B
Identify a quadratic equation which is solved by a graphical method in the
picture.
\(x^{2} - 6x + 9 = 0\)
\(x^{2} + 9x - 3 = 0\)
\(x^{2} - 9x - 3 = 0\)
\(x^{2} + 6x + 9 = 0\)
9000024801
Level:
B
In the following list identify an inequality which does not have a solution.
\(\sqrt{2x - 3} < -6\)
\(\sqrt{x^{2 } - 3x} > 5\)
\(\sqrt{1 + x^{2}} > -10\)
\(\sqrt{2x^{2}} < 4\)
9000024804
Level:
B
How many solutions does the inequality
\[
\sqrt{x + 17} > x - 3
\]
have in the set \(\mathbb{N}\)?
Seven solutions in \(\mathbb{N}\).
No solution in \(\mathbb{N}\).
Five solutions in \(\mathbb{N}\).
More than seven solutions in \(\mathbb{N}\).
9000022808
Level:
B
Find all the real values of \(x\)
which ensure that the following expression is negative.
\[
-x^{2} + 4x - 4
\]
\(x\in \mathbb{R}\setminus \{2\}\)
none \(x\)
with this property
\(x = 2\)
\(x\in \mathbb{R}\)