Problem 10

Express each of the following without absolute value signs, treating various cases separately when necessary.

(i) \[ \begin{align*} |a + b| - |b| \\ \end{align*} \]

• When $a, b >= 0$ then $a + b - b = a$.
• When $|a| > |b|$, $a >= 0$ and $b < 0$ (or vis versa) then $(a + b) - b = a$.
• When $|a| < |b|$, $a >= 0$ and $b < 0$ (or vis versa) then $(-b - a) - b = -a - 2b$.

(ii) \[ \begin{align*} |(|x| - 1)| \\ \end{align*} \]

• When $x = 0$ then $1$.
• When $x > 0$ then $|x - 1|$. Further $x < 1$ then $1 - x$ else $x - 1$.
• When $x < 0$ then $|-x - 1| = |-(x + 1)| = x + 1$.

(iii) \[ \begin{align*} |x| - |x^2| &= |x| - x^2 \\ \end{align*} \]

• When $x = 0$ then $0$.
• When $x > 0$ then $x - x^2$.
• When $x < 0$ then $-x - x^2$.

(iv) \[ \begin{align*} a - |(a - |a|)| \\ \end{align*} \]

• When $a = 0$ then $0$.
• When $a > 0$ then $a$.
• When $a < 0$ then $a - (a + a) = -a$.

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