Prove that ab≤|a||b| for all real numbers a and b.

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Lesson Parent: 
Rules of Absolute Values
Problem: 

Prove that ab≤|a||b| for all real numbers a and b.

Answer: 

It is true that ab≤|a||b| for all real numbers a and b.

SAMPLE SOLUTION

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Solution: 

To prove that <Sign in to see all the formulas> for all real numbers a and b we will do it by two main cases: when a and b have the same sign and when they don't.

In the first case we assume that a and b have the same sign. They are either both positive or both negative. If they are both positive then a=|a| and b=|b| and <Sign in to see all the formulas>. If they are both negative then ab is positive and <Sign in to see all the formulas>.

In the second case a and b do not have the same sign. Without loss of generality, we will assume that <Sign in to see all the formulas> and <Sign in to see all the formulas>. In this case <Sign in to see all the formulas>. We also know that <Sign in to see all the formulas>. Therefore <Sign in to see all the formulas> and thus <Sign in to see all the formulas>.