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Well done to Minhaj, Amrit, Adithya, Sasi, Will, Charlie, Luke and Luke, Eric, Pablo, Julian, Johnny, Peter, Gabriel, and Manolis for their hard work on this problem!

Peter made the following observation:

I worked out that the range of numbers is one to three if you are talking about positive intergers.

Gabriel used a graphical method to solve the inequality.

Others used variations of the following method:

Square both sides. Multiply throughout by x. Rearrange to form the quadratic inequality:
$$x^2-14x+1< 0.$$
Use the quadratic formula to solve this inequality. From the graph of $y =x^2 -14x + 1$ we see that the solution is $7-4\sqrt 3 < x < 7+4\sqrt 3$ or approximately $0.072< x< 13.928.$

Click on the pdf links to read Eric's, Minhaj's, Manolis's and Sasi's solutions.