Question

Two numbers A and B are graphed on a number line. It is always,sometimes, or never true that A- |B| < A+B and A > |B| Explain

Answer 1

Consider the cases when A is not > |B|:

In these cases "(A- |B| < A+B) and (A > |B|)" is never true, because the second part of the proposition (sentence in " ") is not true.


So consider the cases when A>|B|

A is clearly positive, since |B| is positive.

case 1:

both A and B are positive, so |A|=A and |B|=B, and A>|B|,

in this case

A-|B|=A-B and clearly A-B<A+B as this is equivalent to -B<B, which is true.


case 2:

B is negative, so |B|=-B,

thus A-|B|=A-(-B)=A+B

so A-|B|=A+B, thus A-|B| is not < A+B

case 3:

B=0,

A-|B|=A and A+B=A,

so A-|B|=A+B, thus A-|B| is not < A+B.


Answer: sometimes:

Precisely when A>B, and both A and B are positive.