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If $a, b$ and $c$ are three real numbers, then which of the following is not true?

  1. $\mid a+b \mid\leq \mid a \mid+\mid b \mid$
  2. $\mid a – b \mid \leq \mid a \mid  + \mid b\mid$
  3. $\mid a-b \mid \leq  \mid a \mid -\mid b \mid$
  4. $\mid a-c \mid \leq \mid a-b \mid+\mid b-c \mid$
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Given that $a, b$ and $c$ are three real numbers, so  we can take $a = 1,b = 2$ and $c = 3$

Now verify each option one by one.

A.$\mid a+b \mid\leq \mid a \mid + \mid b \mid$

$\implies \mid 1 + 2 \mid\leq \mid 1 \mid + \mid 2 \mid$

$\implies 3 \leq 3\:\text{(True)}$

B.$\mid a – b \mid \leq \mid a \mid  + \mid b\mid$

$\implies\mid 1 – 2 \mid \leq \mid 1 \mid  + \mid 2 \mid$

$\implies 1 \leq 3\:\text{(True)}$

C.$\mid a – b \mid \leq  \mid a \mid – \mid b \mid$

$\implies \mid 1 – 2 \mid \leq  \mid 1 \mid – \mid 2 \mid$

$\implies 1 \leq -1\:\textbf{(False)}$

D.$\mid a-c \mid \leq \mid a-b \mid+\mid b-c \mid$

$\implies \mid 1-3 \mid \leq \mid 1-2 \mid+\mid 2-3 \mid$

$\implies \mid -2 \mid \leq \mid -1 \mid+\mid -1 \mid$

$\implies 2 \leq 2\:\text{(True)}$

So, the correct answer is $(C).$
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