9000104303
Level:
B
Assuming \(a < 3\),
solve the following inequality.
\[
ax - 3\geq 3x - a
\]
\(\left (-\infty ;-1\right ] \)
\(\left (-\infty ;-1\right )\)
\(\left (-1;\infty \right )\)
\(\mathbb{R}\)
Equations and Inequalities with Parameters
9000104401
Level:
A
Find a set of the values of the real parameter
\(a\) which
ensure that the following equation has no solution.
\[
a^{2}x + 2ax - 3a = 0
\]
\(\{ - 2\}\)
\(\{2\}\)
\(\{0\}\)
\(\{ - 3;1\}\)
9000104304
Level:
B
Assuming \(a < 0\),
solve the following inequality.
\[
\frac{x}
{a}\geq 1
\]
\(\left (-\infty ;a\right ] \)
\(\left (-\infty ;a\right )\)
\(\left [ a;\infty \right )\)
\(\left (a;\infty \right )\)
9000104502
Level:
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{-1\}\).
\[\frac{x} {a+1} = x - a\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=0 & \mathbb{R} \\
a\notin\{-1;0\} & \{a+1\}
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=0 & \mathbb{R} \\
a\notin\{-1;0\} & \emptyset
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=0 & \emptyset \\
a\notin\{-1;0\} & \{a+1\}
\\\hline \end{array}\)
9000104305
Level:
B
Assuming \(a > -1\),
solve the following inequality.
\[
\frac{2x}
{a + 1} - 1 < 0
\]
\(\left (-\infty ; \frac{a+1}
{2} \right )\)
\(\left (-\frac{a+1}
{2} ; \frac{a+1}
{2} \right )\)
\(\left \{\frac{a+1}
{2} \right \}\)
\(\left (\frac{a+1}
{2} ;\infty \right )\)
9000104306
Level:
A
Assuming \(a = 0\),
solve the following inequality.
\[
a\left (a - 1\right )x < 1
\]
\(x\in\mathbb{R}\)
\(x\in\mathbb{R}\setminus \{1\}\)
\(x\in\emptyset \)
\( x\in\left \{ \frac{1}
{a\left (a-1\right )}\right \}\)
9000034704
Level:
B
Solve the inequality
\[
ax - 2 > 0
\]
with a real unknown \(x\) and a
nonpositive real parameter \(a < 0\).
\(\left (-\infty ; \frac{2}
{a}\right )\)
\(\left (-\infty ;-\frac{2}
{a}\right )\)
\(\left (\frac{2}
{a};\infty \right )\)
\(\left (-\frac{2}
{a};\infty \right )\)
9000034706
Level:
A
Consider the inequality
\[
px^{2} - 2x + 2 > 0
\]
with the real parameter \(p\).
Solve this inequality for \(p = 0\).
\((-\infty ;1)\)
\((-\infty ;-1)\)
\((-1;\infty )\)
\((1;\infty )\)
9000034708
Level:
A
Consider the equation
\[
2x^{2} + 5px + 2 = 0
\]
with the real parameter \(p\).
Solve the equation for \(p = -\frac{4}
{5}\).
\(\left \{1\right \}\)
\(\left \{-1\right \}\)
\(\left \{0\right \}\)
\(\emptyset \)
9000034710
Level:
A
Solve the following equation with a real parameter
\(t\), assuming
\(t\neq - 1\) and
\(t\neq 1\).
\[
x(t^{2} - 1) = t - 1
\]
\(\left \{ \frac{1}
{t+1}\right \}\)
\(\emptyset \)
\(\mathbb{R}\)
\(\left \{0\right \}\)