If x,y,z are positive numbers such that x+y+z=1, prove xy+yz+zx-\dfrac{9}{4}xyz\leq \dfrac{1}{4}

Let z,y,z >0 such that xy+yz+zx+2xyz=1, show that

\dfrac{1}{4x+1}+\dfrac{1}{4y+1}+\dfrac{1}{4z+1} \geq 1

Let a,b,c be real numbers such that ab+bc+ac=1. Prove the inequality

\Big( a+\dfrac{1}{b} \Big) ^2+\Big( b+\dfrac{1}{c} \Big) ^2+\Big( c+\dfrac{1}{a} \Big) ^2\geq 16

Given a,b are positive real numbers such that

a+b= \sqrt{4+2\sqrt{3} }

and

\dfrac{1}{a}+\dfrac{1}{b} = \sqrt{4-2\sqrt{3} }

If x = a^2+b^2, find the value of x+\dfrac{4}{x}

Which point (x, y) is a solution to the given inequality y < −4x + 4 in

the xy-plane?

If (x-3)(x-7)(x-12)<0 , solve for real x in the inequality

Show that a^4+b^4+c^4≥ 2b^2+b^2c^2+c^2a^2≥(a+b+c)abc

If x \geq 0, y \geq 0 such that x^2 +\dfrac{y^2}{2} =1, find the maximum value of x\sqrt{1+y^2} .

If a,b,c are positive real numbers such that \dfrac{\sqrt{2}b-2c }{a}=1, show that b^2\geq 4ac

Given a,b,c are positive real numbers such that

a+b+c = \sqrt{10+\sqrt{19} }

and

\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} = \sqrt{10-\sqrt{19} }

If x = a^2+b^2+c^2, find the value of x+\dfrac{9}{x}

If x,y,z>0, and x+y+z=1, then prove xy+yz+zx≥9xyz.

If m,n are negative real numbers such that y = \dfrac{x}{(x-m)(x-n)} , find the maximum value of y in the domain of x>0

If three sides of a triangle are less than 1 iIn length, show that its area is less than \dfrac{\sqrt{3}}{4}

Show the following inequality

1-\dfrac{1}{6n^2}-\dfrac{1}{6n^3} <\displaystyle\sum_{k=1}^n \sqrt[3]{ n^3+k} <1-\dfrac{1}{6n^2}

Solve the inequality with square root

\sqrt{x+2}-\sqrt{x-1}>\sqrt{2x-3}

for real values of x