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## Homework Statement

Let [itex]b \geq 0[/itex]. Prove that [itex]|a| \leq b \Leftrightarrow -b \leq a \leq b[/itex].

## Homework Equations

The definition for the absolute value function provided in the book is

[itex]|x| = \begin{cases} x, & \mbox{if } x \geq 0 \\ -x, & \mbox{if } x < 0 \end{cases}[/itex]

I have a relevant side question. Prior to this section, the book covers some symbolic logic and logical equivalence. Am I to understand this definition as an "and" or an "or" definition? Meaning,

[itex]|x| := (a \: \mbox{for} \: a \geq 0) \wedge (-a \: \mbox{for} \: a < 0)[/itex], or

[itex]|x| := (a \: \mbox{for} \: a \geq 0) \vee (-a \: \mbox{for} \: a < 0)[/itex]

## The Attempt at a Solution

(1) Left to right: Assume [itex]|a| \leq b[/itex].

[itex]|a|[/itex] is defined to be

[itex]|a| = \begin{cases} a, & \mbox{if } a \geq 0 \\ -a, & \mbox{if } a < 0 \end{cases}[/itex].

If [itex]a \geq 0[/itex], we have

[itex]|a| = a \leq b[/itex].

If [itex]a < 0[/itex], we have

[itex]|a| = -a \leq b[/itex], or equivalently,

[itex] a \geq -b[/itex].

From here, I conclude that [itex]-b \leq a \leq b[/itex], but I'm not sure if this would be accepted if this were actually assigned. I have a feeling that I should also show what happens when b = 0 and b > 0, but I'm not sure it's necessary.

(2) Right to left: Assume [itex]-b \leq a \leq b[/itex].

Suppose [itex]b = 0[/itex].

[itex]-0 \leq a \leq 0 \Rightarrow a = 0 \Rightarrow |a| = b[/itex]

Suppose [itex]b > 0[/itex].

- [itex]a = 0 \Rightarrow |a| = 0 \Rightarrow |a| < b[/itex]
- [itex]a > 0 \Rightarrow |a| = a > 0 \Rightarrow 0 < |a| < b \Rightarrow |a| < b[/itex]
- [itex]a < 0 \Rightarrow |a| = -a > 0 \Rightarrow |a| < b[/itex]

Therefore, for [itex]b \geq 0[/itex], [itex]|a| \leq b[/itex].

Thanks for any input!