Properties of Absolute Value with Two Abs Values

[askor]

Is it true that ##\frac{|a|}{|b|} = |\frac{a}{b}|## and ##|a| < |b| = a^2 < b^2##?

 

[etotheipi]

##|a| < |b| = a^2 < b^2##?

Do you mean ##|a| < |b| \iff a^2 < b^2##?

 

[mfb]

Your textbook should cover the first one at least.

The second one depends on what a,b can be (assuming you mean what @etotheipi wrote)

 

[askor]

Do you mean ##|a| < |b| \iff a^2 < b^2##?

Yes.

 

[romsofia]

Define ##|x| = \sqrt{x^2}##. Can you use certain properties of the square root to show that ##\frac{|a|}{|b|} = |\frac{a}{b}|##?

 

[askor]

Define ##|x| = \sqrt{x^2}##. Can you use certain properties of the square root to show that ##\frac{|a|}{|b|} = |\frac{a}{b}|##?

I don't understand. Please tell me the point.

 

[etotheipi]

I don't understand. Please tell me the point.

$$\frac{|a|}{|b|} = \frac{\sqrt{a^2}}{\sqrt{b^2}} = \sqrt{\frac{a^2}{b^2}} = \sqrt{\left(\frac{a}{b}\right)^2} = \dots$$