Prove that if c is a positive real number and x is any real number, then -c </= x </= c if and only if |x|</= C.
</= greater than or equal to
Thank you!
yeah. we often take this fact for granted around here. it's hard to come up with a rigorous proof. but here is a possible one for the reverse implication.
Recall that $\displaystyle |x| = \left \{ \begin {array}{cc} x & \mbox { if } x \ge 0 \\ -x & \mbox { if } x < 0 \end {array} \right.$
Thus, by the definition of $\displaystyle |x|$:
$\displaystyle |x| \le C$
$\displaystyle \implies x \le C$ or $\displaystyle -x \le C$
$\displaystyle \implies x \le C$ or $\displaystyle x \ge -C$
combining these two inequalities, we obtain:
$\displaystyle -C \le x \le C$
now try to go the other way. i would probably try to reverse this exact proof. so split the inequality into two, and work on each case by case (if x > 0 or x < 0)