# Thread: Some Summer Calculus Homework Help...

1. ## Some Summer Calculus Homework Help...

Okay I have two questions that involve absolute value and inequalities that I am having some difficulty figuring/solving out....

Here they are,

Problem #1: Show that if |a-b|<s and |c-b|<t, then |a-c|<s+t

Next problem is a similar one to #1.

Problem #2: Show that if |x-1|<(e/3) , then |(3x+2)-5|<e

Any and all help is appreciated, thanks!

2. Originally Posted by forkball42
Problem #2: Show that if |x-1|<(e/3) , then |(3x+2)-5|<e
$|x - 1| < \frac{e}{3}$

$3|x - 1| < e$

$|3x - 3| < e$

$|3x + (2 - 5)| < e$

$|(3x + 2) - 5| < e$

-Dan

3. Originally Posted by forkball42
Problem #1: Show that if |a-b|<s and |c-b|<t, then |a-c|<s+t
$|a - b| < s$
and
$|c - b| < t$

$|a - b| + |c - b| < s + t$

Now, the "triangle inequality" (which is equivalent to the "Cauchy-Schwartz inequality" in this case) says that:
$|a - c| \leq |a - b| + |c - b|$ <-- It is slightly easier to see that $|a - c| \leq |a - b| + |b - c|$, but as $|c - b| = |b - c|$ these are the same.

Thus
$|a - c| < s + t$

-Dan