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

If x + 5 ≥ 10 , then

x ∈ ( - ∞ , - 15 ] ∪ [ 5 , ∞ )

x ∈ - ∞ , - 15 ∪ [ 5 , ∞ )

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

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

( - ∞ , - 4 ] ∪ - 5 2 , - 1

( - ∞ , - 4 ] ∪ [ - 2 , - 1 ]

MEDIUM

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Find the sum of all the real solutions of the inequality $\frac{(x^{2} + 2) (\sqrt{x^{2} - 16})}{(x^{4} + 2) (x^{2} - 9)} \leq 0$ .

50% studentsanswered this correctly

Important Questions on Quadratic Expressions

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

For all

n \in ℕ, n \in ℕ,

, if

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

, then

x =

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

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

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The smallest prime number satisfying the inequality

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

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The number of non-negative integer solutions of the equations

6 x + 4 y + z = 200

and

x + y + z = 100

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The least positive integer

n

for which

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

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

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

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The solution set of the rational inequality

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

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

|x + 5| \geq 10

, then

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The least positive integer

n

for which

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

is-

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

\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>Quadratic Expressions>Algebraic Inequalities

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>Quadratic Expressions>Algebraic Inequalities

5, 5 r, 5 r^{2}

are the lengths of the sides of a triangle, then

r

can not be equal to:

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

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

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

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

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The solution of the inequality

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

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

β

satisfies the equation

x^{2} - x - 6 > 0

, then a value exists for

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The solution of

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

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The number of integers satisfying the inequality

|n^{2} - 100| < 50

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

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

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

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

Find the sum of all the real solutions of the inequality x2+2(x2-16)x4+2x2-9≤0.

Important Questions on Quadratic Expressions

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Find the sum of all the real solutions of the inequality $\frac{(x^{2} + 2) (\sqrt{x^{2} - 16})}{(x^{4} + 2) (x^{2} - 9)} \leq 0$ .