# Related range of values of f(x) necessary/sufficient/ with absolute values

• May 31st 2011, 07:48 PM
Zellator
Related range of values of f(x) necessary/sufficient/ with absolute values
Hi Forum.

But here I go again:

Consider two conditions
\$\displaystyle x^2-3x-10\$ and \$\displaystyle |x-2|<a\$

Where x is a real number, and a is a positive real number.

The range of a, so that \$\displaystyle |x-2|<a\$ is a necessary condition for \$\displaystyle x^2-3x-10<0\$

The range of a, so that \$\displaystyle |x-2|<a\$ is a sufficient condition for \$\displaystyle x^2-3x-10<0\$

This problem is mainly about logic, right?
Here is something that I thought, after calculating the range of \$\displaystyle x^2-3x-10<0\$, that is \$\displaystyle -2<x<5\$

If we have \$\displaystyle -2<x<5\$
-2________________________________5
|________________________________|
___________________________a=0 ______________________________ a=3
|_________________________________________________ _| <-|x-2|<a

Edit: This didn't work so well but, the a=0 is right below -2 and the a=3 is right below 5

This is our sufficient condition: 0<a<3
I'm not sure what the absolute value does here.
What does the absolute value mean?
Absolute values are only the distance to x=0, right?

After calculating -correctly- you get a nicer solution that is:
\$\displaystyle 0<a\leqslant 3\$

But how do we get such solution?

\$\displaystyle |x-2|<a \$ is
\$\displaystyle x-2\$ if \$\displaystyle x-2\geqslant a\$ and
\$\displaystyle -x+2\$ if \$\displaystyle x-2<a\$

This relates to two range of values so this got a little out of control.
Somehow confusing, I wish I knew somewhere to find exercises of this kind!

I'm confused about that necessary/sufficient part

Sufficient
If you mow the lawn, you receive 10 dollars.
____________________Necessary

The \$\displaystyle 0<a\leqslant 3\$ range is sufficient for f(x) to work, or something like that?
But where does necessity comes in this? I know that if something is necessary it is not sufficient.

A--->B
A implies B
A sufficient, B necessary

I appreciate any help!
Thanks!(Bow)
• Jun 1st 2011, 01:47 AM
Ackbeet
Interesting problem. Your problem is dealing with implication. Let's let \$\displaystyle P\$ be the proposition that

\$\displaystyle x^{2}-3x-10<0,\$

and let's let \$\displaystyle Q(a)\$ be the proposition that

\$\displaystyle |x-2|<a.\$

Now, if you want to find the range of \$\displaystyle a\$ such that \$\displaystyle Q(a)\$ is sufficient for \$\displaystyle P,\$ then that's the same thing as finding the range of \$\displaystyle a\$ such that \$\displaystyle Q(a)\$ implies \$\displaystyle P.\$

Conversely (literally!), if you want to find the range of \$\displaystyle a\$ such that \$\displaystyle Q(a)\$ is necessary for \$\displaystyle P,\$ then that's the same thing as finding the range of \$\displaystyle a\$ such that \$\displaystyle P\$ implies \$\displaystyle Q(a).\$ So here, you just reverse the direction of the implication.

I agree with your 'sufficient' answer. \$\displaystyle 0<a\le 3\$ is exactly correct.

But now, with the 'necessary' answer, you want \$\displaystyle P\$ to imply \$\displaystyle Q(a).\$ How could that happen?