Spivak Absolute Value Problem (Prologue 9-v)

Question

I'm working on the following problem

Express the following with at least one less pair of absolute value signs

$$|(| \sqrt2 + \sqrt3| - |\sqrt5 - \sqrt7|)|$$

Now I can see that the expression can be written as $|\sqrt2 + \sqrt3 + \sqrt5 - \sqrt7|$ and that satisfies the requirement of the problem (ie I've eliminated at least one pair of absolute value signs), but the solution in the back of the book is $\sqrt2 + \sqrt3 + \sqrt5 - \sqrt7$ without the absolute value sign.

Is there a way to show that $\sqrt2 + \sqrt3 + \sqrt5 > \sqrt7$ ?

Answer 1

Note that $\sqrt{2}, \sqrt{3}, \sqrt{5} > 1$ but $\sqrt{7} < 3$.

Answer 2

Square both sides. The left is at least $2 + 3 + 5 = 10$ and the right is equal to $7$.

Explicitly:

$$(\sqrt{2} + \sqrt{3} + \sqrt{5})^2 = 2 + 3 + 5 + 2\sqrt{6} +2\sqrt{10} + 2\sqrt{15} > 10$$