Prove that $a\,b\leq|a|\,|b|$ for all real numbers $a$ and $b$.

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Lesson Parent: 
Rules of Absolute Values
Problem: 

Prove that $a\,b\leq|a|\,|b|$ for all real numbers $a$ and $b$.

Answer: 

It is true that $a\,b\leq|a|\,|b|$ 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 $a\,b$ 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>.