For 0 ≤ p ≤ 1 and for any positive a , b ; let I ( p ) = ( a + b ) p , J ( p ) = a p + b p , then

I ( p ) < J ( p ) in 0 , p 2 and l ( p ) > J ( p ) in p 2 , ∞

I ( p ) < J ( p ) in p 2 , ∞ and J ( p ) < I ( p ) in 0 , p 2

x ∈ ℝ : 6 + x - x 2 2 x + 5 ≥ 6 + x - x 2 x + 4 =

( - ∞ , - 4 ] ∪ - 5 2 , - 1

( - ∞ , - 4 ] ∪ [ - 2 , - 1 ]

Let N be the set of positive integers. For all n ∈ N , let f n = n + 1 1 / 3 - n 1 / 3 and A = n ∈ N : f n + 1 < 1 3 n + 1 2 / 3 < f n Then,

the complement of A in N is nonempty, but finite

A and its complement in N are both infinite

If x + 5 ≥ 10 , then

x ∈ ( - ∞ , - 15 ] ∪ [ 5 , ∞ )

x ∈ - ∞ , - 15 ∪ [ 5 , ∞ )

The equation x1logx132xx2x3x has solutions

MEDIUM

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The maximum value of

z = 7 x + 5 y

subject to

2 x + y \leq 100, 4 x + 3 y \leq 240, x \geq 0, y \geq 0

is

MEDIUM

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The maximum value of

2 x + y

subject to

3 x + 5 y \leq 26

and

5 x + 3 y \leq 30, x \geq 0, y \geq 0

is:

HARD

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The solution set of the rational inequality

\frac{x + 9}{x - 6} \leq 0

is

EASY

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

For

0 \leq p \leq 1

and for any positive

a, b;

let

I (p) = (a + b)^{p}, J (p) = a^{p} + b^{p},

then

MEDIUM

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The least positive integer

n

for which

\sqrt{n + 1} - \sqrt{n - 1} < 0.2

is

HARD

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

\{x \in ℝ : \frac{\sqrt{6 + x - x^{2}}}{2 x + 5} \geq \frac{\sqrt{6 + x - x^{2}}}{x + 4}\} =

MEDIUM

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The number of integers satisfying the inequality

|n^{2} - 100| < 50

is

EASY

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The solution of the inequality

|x^{2} - 4 x| < 5

is

HARD

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The least positive integer

n

for which

\sqrt[3]{n + 1} - \sqrt[3]{n} < \frac{1}{12}

is-

HARD

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

If

\frac{6 x^{2} - 5 x - 3}{x^{2} - 2 x + 6} \leq 4,

then the least and the highest values of

4 x^{2}

are

MEDIUM

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The number of non-negative integer solutions of the equations

6 x + 4 y + z = 200

and

x + y + z = 100

is

HARD

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The number of integral values of

m

for which the quadratic expression

(1 + 2 m) x^{2} - 2 (1 + 3 m) x + 4 (1 + m), x \in R

is always positive, is

MEDIUM

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

Let

N

be the set of positive integers. For all

n \in N,

let

f_{n} = {(n + 1)}^{1 / 3} - n^{1 / 3}

and

A = \{n \in N : f_{n + 1} < \frac{1}{3 {(n + 1)}^{2 / 3}} < f_{n}\}

Then,

MEDIUM

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

For all

n \in ℕ, n \in ℕ,

, if

1^{2} + 2^{2} + 3^{2} + \dots + n^{2} > x

, then

x =

MEDIUM

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

Let

x = [\frac{a + 2 b}{a + b}]

and

y = \frac{a}{b}

, where

a

and

b

are positive integers. If

y^{2} > 2

, then

EASY

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The solution of

\frac{6 x}{4 x - 1} < \frac{1}{2}

is

EASY

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

If

|x + 5| \geq 10

, then

EASY

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

The smallest prime number satisfying the inequality

\frac{2 n - 3}{3} \geq \frac{n - 1}{6} + 1

is

HARD

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

Let

a, b, c, d, e

be real numbers such that

a + b < c + d, b + c < d + e, c + d < e + a, d + e < a + b .

Then

MEDIUM

Mathematics>Algebra>Creating Equations>Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

If

5, 5 r, 5 r^{2}

are the lengths of the sides of a triangle, then

r

can not be equal to:

The equation ${(x + 1)}^{\log_{(x + 1)} (3 + 2 x - x^{2})} = (x - 3) |x|$ has solutions

Important Questions on Creating Equations

Learn and Prepare for any exam you want !

The equation x+1logx+1⁡3+2x-x2=x-3x has solutions

Important Questions on Creating Equations

Learn and Prepare for any exam you want !

The equation ${(x + 1)}^{\log_{(x + 1)} (3 + 2 x - x^{2})} = (x - 3) |x|$ has solutions