Equations and inequalities with parameters

2010008408

Level:

Solve the following equation with unknown

x

and a real parameter

a \in R ∖ {1}

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

\begin{array}{cc} Parameter & Solution set \\ a = 2 & \emptyset \\ a \notin {1; 2} & \frac{a - a^{2}}{2 - a} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 2 & R \\ a \notin {1; 2} & \frac{a - a^{2}}{2 - a} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 2 & R \\ a \notin {1; 2} & \emptyset \end{array}

9000021706

Level:

Find all the values of the parameter

k

so that the solution of the following equation is bigger than

10

3 x - 18 = \frac{10 x - 4 k}{2}

k \in (19; \infty)

k \in {9}

k \in (- \infty; 1)

k \in (9; \infty)

9000021707

Level:

Find all the values of the parameter

k

so that the solution of the following equation is positive.

2 k x + k = 4 x + 3

k \in (2; 3)

k > 0

k \in (3; \infty)

k \in (- \infty; 3)

9000034706

Level:

Consider the inequality

p x^{2} - 2 x + 2 > 0

with the real parameter

p

. Solve this inequality for

p = 0

(- \infty; 1)

(- \infty; - 1)

(- 1; \infty)

(1; \infty)

9000034707

Level:

Consider the equation

x^{2} (1 - q) + 2 x + 1 + q = 0

with the real parameter

q

. Solve the equation for

q = 3

{- 1; 2}

{1}

{- 2}

\emptyset

9000034708

Level:

Consider the equation

2 x^{2} + 5 p x + 2 = 0

with the real parameter

p

. Solve the equation for

p = - \frac{4}{5}

{1}

{- 1}

{0}

\emptyset

9000034709

Level:

Consider equation

p (2 - p) x = 4 p

with a real parameter

p

. Solve the equation for

p = 2

\emptyset

R

{\frac{4}{2 - p}}

R ∖ {0}

9000034710

Level:

Solve the following equation with a real parameter

t

, assuming

t \neq - 1

and

t \neq 1

x (t^{2} - 1) = t - 1

{\frac{1}{t + 1}}

\emptyset

R

{0}

9000104302

Level:

Assuming

a = 0

, solve the following inequality.

2 a x + 4 a < 1

R

\emptyset

(\frac{1 - 4 a}{2 a}; \infty)

(- \infty; \frac{1 - 4 a}{2 a})

9000104306

Level:

Assuming

a = 0

, solve the following inequality.

a (a - 1) x < 1

x \in R

x \in R ∖ {1}

x \in \emptyset

x \in {\frac{1}{a (a - 1)}}

Equations and inequalities with parameters

2010008408

9000021706

9000021707

9000034706

9000034707

9000034708

9000034709

9000034710

9000104302

9000104306

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