Solutions

(i)

$$\begin{aligned} \sqrt{7} > \sqrt{5} & \implies \sqrt{7} - \sqrt{5} > 0 \\ & \implies \sqrt{2} + \sqrt{3} + (\sqrt{7} - \sqrt{5}) > 0 \\ & \implies | \sqrt{2} + \sqrt{3} - \sqrt{5} + \sqrt{7} | = \sqrt{2} + \sqrt{3} - \sqrt{5} + \sqrt{7} > 0 \end{aligned}$$

NB. We us the proposition that the square root function is
strictly increasing.

(ii)

$$\begin{aligned} |a + b| \leq |a| + |b| & \implies |a + b| - |a| - |b| \leq 0 && \text{Triangle inequality} \\ & \implies |a| + |b| - |a + b| \geq 0 \\ & \implies -(|a + b| - |a| - |b|) \geq 0 \\ & \implies ||a + b| - |a| - |b|| = -(|a + b| - |a| - |b|) \geq 0 \end{aligned}$$

(iii) For this exercise, one might be tempted to treat multiple cases
for the variables, and try to prove that a certain pair/pairs of
absolute values can be removed.

There is an easier way, however. What happens when we
substitute our sum $a + b$ with an arbitrary variable. In other
words, let $d = a + b$. Then,

$$\begin{aligned} |a + b| + |c| - |a + b + c| & \implies |d| + |c| - |d + c| \\ & \implies |d + c| \leq |d| + |c| \\ & \implies 0 \leq |d| + |c| - |d + c| \\ & \implies ||a + b| + |c| - |a + b + c|| = |a + b| + |c| - |a + b + c| \geq 0 \end{aligned}$$

(iv)

$$\begin{aligned} |{ (x - y) }^{2}| = { (x - y) }^{2} = x^{2} - 2xy + y^{2} && \text{Perfect square} \end{aligned}$$

(v)

$$\begin{aligned} |( \sqrt{2} + \sqrt{3} - (-(\sqrt{5} - \sqrt{7})) )| \end{aligned}$$