Inequalities With Radicals

Algebra Level 1

True or false :

\quad For all real x x , the inequality x < x \sqrt x < x holds true.

True False

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3 solutions

Matheus Henrique
Apr 7, 2016

False because, for example:

Hung Woei Neoh
Apr 14, 2016

The inequality x < x \sqrt x < x only holds true for real values x > 1 x>1

Eg. x = 4 x = 4 = 2 < 4 x=4 \implies \sqrt x = \sqrt4 = 2 < 4

If x = 1 x=1 or x = 0 x=0 , we have x = x \sqrt x = x :

x = 1 x = 1 = 1 = x x=1\implies \sqrt x = \sqrt 1 = 1 = x

x = 0 x = 0 = 0 = x x=0\implies \sqrt x = \sqrt 0 = 0 = x

And for real values 0 < x < 1 0<x<1 , the inequality becomes x > x \sqrt x > x

Eg. x = 0.01 x = 0.01 = 0.1 > 0.01 x=0.01\implies \sqrt x = \sqrt {0.01} = 0.1 > 0.01

Lastly, for real values x < 0 x<0 , we will get a complex value for its square root

Eg. x = 4 x = 4 = 1 4 = 2 i x=-4\implies \sqrt x = \sqrt {-4} = \sqrt{-1}\sqrt{4} = 2i

Now, I do not know whether we can compare complex values with real values or not. Hopefully, someone else will answer this question.

Even without considering real values x < 0 x<0 , we already have sufficient evidence to prove that x < x \sqrt x < x does not hold true for all real x x .

Thus, the answer is False \boxed{\text{False}}

The inequality is false for all real x 1 x \le 1

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