Absolute Value Theorem

Question

When trying to prove the inequality

$$ |a +b| \leq |a| + |b| \text{, for any real numbers a and b} $$ I manage to use the absolute value definition to get to following inequality:

$$ -\big(|a|+|b|\big) \leq a + b \leq |a| + |b| $$

However, the text book leaps foward and states that:

$$ \Big\{-\big(|a|+|b|\big) \leq a + b \leq |a| + |b|\Big\} \leftrightarrow \Big\{ |a + b| \leq |a| + |b|\Big\} $$

How did it jump to that conclusion?

Answer 1

The definition of absolute value is: $$|x| = \begin{cases} x & \text{if $x\geq 0$} \\ -x & \text{if $x<0$.} \end{cases}$$ So assume $a+b\geq 0$. Then $|a+b| = a+b\leq |a|+|b|$ by the inequality you've shown. If $a+b<0$, then $a+b = -|a+b|$, so $-(|a|+|b|)\leq -|a+b| \Longleftrightarrow |a+b|\leq |a|+|b|$ (the inequality flips since we divide by $-1$).

Answer 2

so you have $-|a| - |b| \le a + b \le |a| + |b|$ that is $(a+b)$ in magnitude is less or equal to the nonnegative quantity $|a| +|b|$ writing this using absolute value notation is $$ |a +b | \le |a| +|b| $$ called the triangle inequality.