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EDIT: I forgot to mention the second condition: $a \leq c \leq b$.

I'm taking a first-year undergraduate calculus course and I'm faced with this problem. I'm not sure about how to proceed without using examples. I realize that $|c|$ might be greater than $b$ or $-a$ depending on the sign of each term.

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    $\begingroup$ You should add some context, as the claim is false as stated. Consider $a=b=0$ and $c=1$. $\endgroup$ – 211792 Sep 8 '16 at 18:38
  • $\begingroup$ Is there some known relationships between $a, b$ and $c$? $\endgroup$ – the_candyman Sep 8 '16 at 18:38
  • $\begingroup$ As you have it, the $c$ is not related to $a$ or $b$ in any way, so there is no reason why the inequality should hold in general. For example, take $c=10, b = 1, a = -2$. Is something missing in your problem statement? $\endgroup$ – avs Sep 8 '16 at 18:39
  • $\begingroup$ @avs Right idea, but incorrect counterexample. $\endgroup$ – 211792 Sep 8 '16 at 18:40
  • $\begingroup$ Thanks, @AustinC. Corrected. $\endgroup$ – avs Sep 8 '16 at 18:43
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If $c \geq 0$:

$$|c| \leq b \leq \max(b, -a)$$

If $c < 0$:

$$|c| \leq -a \leq \max(b, -a)$$

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