Problem 9
Express each of the following with at least one less pair of absolute value signs.
(i) \[ \begin{align*} |\sqrt{2} + \sqrt{3} - \sqrt{5} + \sqrt{7}| \\ \end{align*} \] Because $\sqrt{5} < \sqrt{7}$ the whole expression becomes positive hence: \[ \begin{align*} \sqrt{2} + \sqrt{3} - \sqrt{5} + \sqrt{7} \\ \end{align*} \]
(ii) \[ \begin{align*} |(|a + b| - |a| - |b|)| \\ \end{align*} \]
- When $a, b >= 0$ then $|a + b| >= |a| - |b|$.
- When $b < 0$ then $|a - |b|| >= |a| - |b|$.
- The same is true when $a < 0$.
In conclusion that means: \[ \begin{align*} |a + b| - |a| - |b| \\ \end{align*} \]
(iii) \[ \begin{align*} |(|a + b| + |c| - |a + b + c|) \\ \end{align*} \]
- When $(a + b), c >= 0$ then $|a + b| + |c| = |a + b + c|$.
- When $(a + b) < 0$ then $|a + b| + |c| >= |-|a + b| + c|$.
- When $c < 0$ then $|a + b| + |c| >= |a + b - |c||$.
In conclusion that means: \[ \begin{align*} |a + b| + |c| - |a + b + c| \\ \end{align*} \]
(iv) \[ \begin{align*} |x^2 - 2xy + y^2| &= |(x - y)^2| \\ &= (x - y)^2 \\ \end{align*} \]
(v) \[ \begin{align*} |(|\sqrt{2} + \sqrt{3}| - |\sqrt{5} - \sqrt{7}|)| &= |(\sqrt{2} + \sqrt{3} - |\sqrt{5} - \sqrt{7}|)| \\ &= |(\sqrt{2} + \sqrt{3} - (\sqrt{7} - \sqrt{5}))| \\ &= |(\sqrt{2} + \sqrt{3} - \sqrt{7} + \sqrt{5})| \\ &= \sqrt{2} + \sqrt{3} + \sqrt{5} - \sqrt{7} \\ \end{align*} \]
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