Question
Given real numbers $a$ and $b$, establish the following formulas: $|a+b| \leq$ $|a|+|b|,|| a|-| b|| \leq|a-b|, \max \{a, b\}=\frac{1}{2}(a+b+|a-b|)$, and $\min \{a, b\}=$ $\frac{1}{2}(a+b-|a-b|)$.
Step 1
To prove this, we can consider the definitions of absolute values. By the properties of absolute values, we have: \[ |a| = a \quad \text{if } a \geq 0 \quad \text{and} \quad |a| = -a \quad \text{if } a < 0 \] \[ |b| = b \quad \text{if } b \geq 0 \quad \text{and} Show more…
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