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
9000104505
Level:
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{-3,3\}\).
\[\frac{a-x} {a-3} - \frac{6a} {a^{2}-9} = \frac{x-3}
{a+3} \]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=0 & \emptyset \\
a\notin\{-3,0,3\} & \left\lbrace\frac{a^2-9}{2a}\right\rbrace
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=0 & \mathbb{R} \\
a\notin\{-3,0,3\} & \left\lbrace\frac{a^2-9}{2a}\right\rbrace
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=0 & \mathbb{R}\setminus\{0\} \\
a\notin\{-3,0,3\} & \left\lbrace\frac{a^2-9}{2a}\right\rbrace
\\\hline \end{array}\)
9000104304
Level:
B
Assuming \(a < 0\),
solve the following inequality.
\[
\frac{x}
{a}\geq 1
\]
\(\left (-\infty ,a\right ] \)
\(\left (-\infty ,a\right )\)
\(\left [ a,\infty \right )\)
\(\left (a,\infty \right )\)
9000104305
Level:
B
Assuming \(a > -1\),
solve the following inequality.
\[
\frac{2x}
{a + 1} - 1 < 0
\]
\(\left (-\infty , \frac{a+1}
{2} \right )\)
\(\left (-\frac{a+1}
{2} , \frac{a+1}
{2} \right )\)
\(\left \{\frac{a+1}
{2} \right \}\)
\(\left (\frac{a+1}
{2} ,\infty \right )\)
9000104307
Level:
B
Assuming \(a\in \left (0,2\right )\),
solve the following inequality.
\[
a\left (a - 2\right )x > 1
\]
\(\left (-\infty , \frac{1}
{a\left (a-2\right )}\right )\)
\(\left ( \frac{1}
{a\left (a-2\right )},\infty \right )\)
\(\emptyset \)
\(\left \{ \frac{1}
{a\left (a-2\right )}\right \}\)
9000104308
Level:
A
Assuming \(a = \frac{1}
{2}\),
solve the following inequality.
\[
2a^{2}x - 1 > ax
\]
\(\emptyset \)
\(\mathbb{R}\)
\(\left ( \frac{1}
{a\left (2a-1\right )},\infty \right )\)
\(\left (-\infty , \frac{1}
{a\left (2a-1\right )}\right )\)
9000034701
Level:
B
Identify a set of the values of the real parameter
\(m\) which
ensure that the equation
\[
\frac{m}
{x} - 8 = \frac{1}
{x} -\frac{m + 3}
{2}
\]
has solution \(x = 2\).
\(\left \{7\right \}\)
\(\left \{10\right \}\)
\(\left \{6\right \}\)
\(\left \{\frac{5}
{2}\right \}\)
9000034709
Level:
A
Consider equation
\[
p(2 - p)x = 4p
\]
with a real parameter \(p\).
Solve the equation for \(p = 2\).
\(\emptyset \)
\(\mathbb{R}\)
\(\left \{ \frac{4}
{2-p}\right \}\)
\(\mathbb{R}\setminus \left \{0\right \}\)
9000034702
Level:
B
Identify the set of the values of the real parameter \(d\) for which the following equation has no solution in \(\mathbb{R}\).
\[
x^{2} - 2dx + 2d^{2} - 9 = 0
\]
\((-\infty ,-3)\cup (3,\infty )\)
\((-3,3)\)
\((3,\infty )\)
\((-\infty ,-3)\)
9000034703
Level:
B
Identify the set of the values of the real parameter \(t\) for which the following equation has two different real roots.
\[
x^{2} + (t + 2)x + 1 = 0
\]
\((-\infty ,-4)\cup (0,\infty )\)
\((-\infty ,-4)\)
\((-4,0)\)
\((0,\infty )\)