Rules of Absolute Values

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Lesson Summary: 
We present here some basic rules of absolute values needed for many proofs.

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Lesson: 

The absolute value of a real number can take on three equivalent forms:

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where the $\max$ function is defined here.

The absolute value is always a positive number (<Sign in to see all the formulas>).

In the table below we assume that $x$, $a$ and $b$ are real numbers that can be either positive or negative. We also assume that <Sign in to see all the formulas> is a real number that can only be positive (<Sign in to see all the formulas>).

Rule of Absolute Values
  1. <Sign in to see all the formulas>
  2. <Sign in to see all the formulas>
  3. <Sign in to see all the formulas>
  4. <Sign in to see all the formulas>
  5. <Sign in to see all the formulas>
  6. <Sign in to see all the formulas>
  7. <Sign in to see all the formulas>
  8. <Sign in to see all the formulas>
  9. The triangle inequality: <Sign in to see all the formulas>
  10. <Sign in to see all the formulas>