Equations and inequalities with parameters

9000104308

Level:

Assuming

a = \frac{1}{2}

, solve the following inequality.

2 a^{2} x - 1 > a x

\emptyset

R

(\frac{1}{a (2 a - 1)}; \infty)

(- \infty; \frac{1}{a (2 a - 1)})

9000104309

Level:

Assuming

a = - 1

, solve the following inequality.

a^{2} x - 1 < a - a x

\emptyset

R

R ∖ {- 1}

R ∖ {- 1; 0}

9000104401

Level:

Find a set of the values of the real parameter

a

which ensure that the following equation has no solution.

a^{2} x + 2 a x - 3 a = 0

{- 2}

{2}

{0}

{- 3; 1}

9000104402

Level:

Find a set of the values of the real parameter

a

which ensure that the following equation has no solution.

2 a^{2} x - a x - 2 a = - 1

{0}

{\frac{1}{2}}

{- \frac{1}{2}}

{- \frac{1}{2}; \frac{1}{2}}

9000104403

Level:

Find a set of the values of the real parameter

a

which ensure that the following equation has infinitely many solutions.

3 a^{2} x - 2 a x + 4 = 6 a

{\frac{2}{3}}

{- \frac{2}{3}}

{0}

{0; \frac{2}{3}}

9000104404

Level:

Find a set of the values of the real parameter

a

which ensure that the following equation has infinitely many solutions.

a^{2} x + 1 = a^{2} + a x

{1}

{- 1; 1}

{0}

{- 1}

9000104405

Level:

Find a set of the values of the real parameter

a

which ensure that the following equation has a unique solution.

a^{3} x + 3 = 3 a^{2} x + a

R ∖ {0; 3}

{0}

{0; 3}

R ∖ {3}

9000104501

Level:

Consider equation

\frac{x - 3}{a} = \frac{a - x}{3} + 2

with an unknown

x \in R

and a real parameter

a \in R ∖ {0}

. Identify a statement which is not true.

For

a \in {- 3; 0}

we have

x = \frac{1}{a + 3}

For

a \notin {- 3; 0}

we have

x = a + 3

a = - 3

, then the equation has infinitely many solutions.

9000104502

Level:

Solve the following equation with unknown

x

and a real parameter

a \in R ∖ {- 1}

\frac{x}{a + 1} = x - a

\begin{array}{cc} Parameter & Solution set \\ a = 0 & R \\ a \notin {- 1; 0} & {a + 1} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & R \\ a \notin {- 1; 0} & \emptyset \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & \emptyset \\ a \notin {- 1; 0} & {a + 1} \end{array}

9000104505

Level:

Solve the following equation with unknown

x

and a real parameter

a \in R ∖ {- 3; 3}

\frac{a - x}{a - 3} - \frac{6 a}{a^{2} - 9} = \frac{x - 3}{a + 3}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & \emptyset \\ a \notin {- 3; 0; 3} & {\frac{a^{2} - 9}{2 a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & R \\ a \notin {- 3; 0; 3} & {\frac{a^{2} - 9}{2 a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & R ∖ {0} \\ a \notin {- 3; 0; 3} & {\frac{a^{2} - 9}{2 a}} \end{array}

Equations and inequalities with parameters

9000104308

9000104309

9000104401

9000104402

9000104403

9000104404

9000104405

9000104501

9000104502

9000104505

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