Expert: Abe Mantell - 12/26/2007

Question
I am studying absolute value inequalities and am experiencing an embarassingly fundamental confusion.

About the inequality |x|<4, the text's author says "The question is,'what real numbers are closer to 0 than 4 is?'

My confusion is: does "less than" mean closer to zero, or does it mean farther negative on a number line?

For a positive number they both mean the same. but for a negative number they don't mean the same.  E.g. is -5 "less than" -4? It is not closer to zero.

When an author says one number is "smaller" than another, does she mean more negative or closer to zero.

Answer
I think the phrase:
'what real numbers are closer to 0 than 4 is?'
is quite confusing!  To explain what values of x
satisfy |x|<4, I'd ask "what values of x are
within 4 units of zero?"...thus, -4<x<4

Yes, -5<-4...

Hmm, "smaller than" may be ambiguous...I'd say "less than"
Or, one can compare magnitudes (i.e. absolute values) and
say "smaller than"...

What book are you reading?  Seems the author may not be
much of a mathematician!

Good luck...