Equations and inequalities with parameters

9000104504

Level: 
C
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{0\}\). \[\frac{1} {x-a} + 1 = \frac{1} {a}\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=1 & \emptyset \\ a\notin\{0,1\} & \left\lbrace\frac{a(a-2)}{a-1}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=1 & \mathbb{R}\setminus\{1\} \\ a\notin\{0;1\} & \left\lbrace\frac{a(a-2)}{a-1}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=1 & \mathbb{R} \\ a\notin\{0,1\} & \left\lbrace\frac{a(a-2)}{a-1}\right\rbrace \\\hline \end{array}\)

9000104505

Level: 
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{-3;3\}\). \[\frac{a-x} {a-3} - \frac{6a} {a^{2}-9} = \frac{x-3} {a+3} \]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a\notin\{-3;0;3\} & \left\lbrace\frac{a^2-9}{2a}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \mathbb{R} \\ a\notin\{-3;0;3\} & \left\lbrace\frac{a^2-9}{2a}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \mathbb{R}\setminus\{0\} \\ a\notin\{-3;0;3\} & \left\lbrace\frac{a^2-9}{2a}\right\rbrace \\\hline \end{array}\)

9000104502

Level: 
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{-1\}\). \[\frac{x} {a+1} = x - a\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \mathbb{R} \\ a\notin\{-1;0\} & \{a+1\} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \mathbb{R} \\ a\notin\{-1;0\} & \emptyset \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a\notin\{-1;0\} & \{a+1\} \\\hline \end{array}\)

9000104307

Level: 
B
Assuming \(a\in \left (0;2\right )\), solve the following inequality. \[ a\left (a - 2\right )x > 1 \]
\(\left (-\infty ; \frac{1} {a\left (a-2\right )}\right )\)
\(\left ( \frac{1} {a\left (a-2\right )};\infty \right )\)
\(\emptyset \)
\(\left \{ \frac{1} {a\left (a-2\right )}\right \}\)