Necessary and sufficient condition for inequality

[songoku]

Homework Statement

consider two conditions x²-3x-10 < 0 and |x-2| < A on a real number x, where A is positive real number

(i) find the range of values of A such that |x-2| < A is a necessary condition for x²-3x-10 < 0
(ii) find the range of values of A such that |x-2| < A is a sufficient condition for x²-3x-10 < 0

Homework Equations

The Attempt at a Solution

what is necessary and sufficient condition? I tried googling but found nothing about it...

thanks

 

[jamesmo]

Homework Statement

consider two conditions x²-3x-10 < 0 and |x-2| < A on a real number x, where A is positive real number

(i) find the range of values of A such that |x-2| < A is a necessary condition for x²-3x-10 < 0
(ii) find the range of values of A such that |x-2| < A is a sufficient condition for x²-3x-10 < 0

Homework Equations

The Attempt at a Solution

what is necessary and sufficient condition? I tried googling but found nothing about it...

thanks

Well, the first step is to solve the equality x²-3x-10 = 0. You will get two solutions. Then you will have to think about what makes the inequality true. The necessary condition is the one that is requried to make the statement true: "For hot dogs to taste good, they must have mustard". The sufficient condition is the one that says if the condition is met, the statement is true, "As long as hot dogs have mustard, hot dogs are good." The necessary and sufficient condition that if that condition is met, by necessity, the statement is true "only hot dogs that have mustard are good (necessity) and if they have mustard they need nothing else to be good (sufficency)".

 

[songoku]

Well, the first step is to solve the equality x²-3x-10 = 0. You will get two solutions. Then you will have to think about what makes the inequality true. The necessary condition is the one that is requried to make the statement true: "For hot dogs to taste good, they must have mustard". The sufficient condition is the one that says if the condition is met, the statement is true, "As long as hot dogs have mustard, hot dogs are good." The necessary and sufficient condition that if that condition is met, by necessity, the statement is true "only hot dogs that have mustard are good (necessity) and if they have mustard they need nothing else to be good (sufficency)".

the answer for the inequality : -2 < x < 5

then,
|x-2| < A
-A < x-2 < A
2-A < x < 2+A

I still don't understand how to obtain the necessary and sufficient condition.

thanks

 

[jamesmo]

the answer for the inequality : -2 < x < 5

then,
|x-2| < A
-A < x-2 < A
2-A < x < 2+A

I still don't understand how to obtain the necessary and sufficient condition.

thanks

Now think about the other inequality. At x= -2 => |-2-2| = 4 and at x = 5 => |5 - 2| = 3

The necessary condition is the one that has to be met, though there may be other conditions required to make it true. I will let you work out sufficient.

 

[The Chaz]

...

The Attempt at a Solution

what is necessary and sufficient condition? I tried googling but found nothing about it...

thanks

Really??! Not to be contrary (though I usually AM), but you must not have looked too closely. The wikipedia page for "necessary and sufficient conditions" is pretty informative. It requires a bit of logic.

There are statements P and Q.
P <=> |x-2| < A
Q <=> x2-3x-10 < 0

"Formally, a statement P is a necessary condition of a statement Q if Q implies P."
Q is equivalent to (x-5)(x+2) < 0, which is equivalent to -2 < x < 5. If you subtract two from each part of the compound inequality, you get
-4 < x - 2 < 3.
I guess from here you could say (and it requires a bit of creative insight) that
-4 < -3 < x - 2 < 3.
Drop the "-4"
-3 < x - 2 < 3, which is equivalent to |x-2| < 3.

 

[songoku]

Now think about the other inequality. At x= -2 => |-2-2| = 4 and at x = 5 => |5 - 2| = 3

The necessary condition is the one that has to be met, though there may be other conditions required to make it true. I will let you work out sufficient.

I don't get your hint..don't understand how to relate the examples about x = -2 and x = 5 to find the answer.

Really??! Not to be contrary (though I usually AM), but you must not have looked too closely. The wikipedia page for "necessary and sufficient conditions" is pretty informative. It requires a bit of logic.

There are statements P and Q.
P <=> |x-2| < A
Q <=> x2-3x-10 < 0

"Formally, a statement P is a necessary condition of a statement Q if Q implies P."
Q is equivalent to (x-5)(x+2) < 0, which is equivalent to -2 < x < 5. If you subtract two from each part of the compound inequality, you get
-4 < x - 2 < 3.
I guess from here you could say (and it requires a bit of creative insight) that
-4 < -3 < x - 2 < 3.
Drop the "-4"
-3 < x - 2 < 3, which is equivalent to |x-2| < 3.

My bad, I meant that I didn't find an example necessary and sufficient condition used directly to inequality. I read the wiki page but I am facing difficulty to apply it to my question.

I understand your work, but still not be able to translate your work to find the answer...

Here's my other attempt :
range of values of A such that |x-2| < A is a necessary condition for x²-3x-10 < 0 is A≥4. Is this right?

thanks

 

[The Chaz]

No. Look at the last line of my post and compare it with what you're looking for...
"|x - 2| < 3" compared with
"|x - 2| < A".

What is "A"?

 

[songoku]

No. Look at the last line of my post and compare it with what you're looking for...
"|x - 2| < 3" compared with
"|x - 2| < A".

What is "A"?

A = 3 but the question asks about the range of A so I think the answer won't be just a number.

My thought :
|x-2| < A
-A < x-2 < A
2-A < x < 2+A

For : A ≤ 3, the min. value of (2-A) is -1 (not fit to -2 < x < 5)
For : A ≥ 3, the value of 2-A < x < 2+A can be -1 < x < 5 (not fit to -2 < x < 5)

So, for -2 < x < 5 fits in 2-A < x < 2+A, A should be at least 4, then A≥4

Where is my mistake?

Thanks

 

[The Chaz]

The absolute value can't be negative, so A has to be >= 0

 

[songoku]

The absolute value can't be negative, so A has to be >= 0

hm...I don't really get what you mean. My answer is A≥4 so it is obvious that A ≥ 0.

Can you tell me where my mistake is in my work?

Thanks

 

[jamesmo]

hm...I don't really get what you mean. My answer is A≥4 so it is obvious that A ≥ 0.

Can you tell me where my mistake is in my work?

Thanks

Think about it if A were say, 5 what would that imply for what x is? (-3 or 7) Is the original inequality true? (it isn't) Once you have done that thought experiment, would the inequality be right for 3.5? or 2?

I have found messing with inequalities can be tricky, especially when trying to get signs right. They are often times easier to understand with logic rather than algebra.

 

[songoku]

Think about it if A were say, 5 what would that imply for what x is? (-3 or 7) Is the original inequality true? (it isn't) Once you have done that thought experiment, would the inequality be right for 3.5? or 2?

I have found messing with inequalities can be tricky, especially when trying to get signs right. They are often times easier to understand with logic rather than algebra.

For the original inequality to be true :
2-A = -2
A = 4

2+A = 5
A = 3

there can't be single value for A...

My logic doesn't work here. I'm very confused..

Thanks

 

[jamesmo]

For the original inequality to be true :
2-A = -2
A = 4

2+A = 5
A = 3

there can't be single value for A...

My logic doesn't work here. I'm very confused..

Thanks

There isn't a singular value, one is the necessary condition the other is the sufficient.

 

[songoku]

There isn't a singular value, one is the necessary condition the other is the sufficient.

Let me try again. So, necessary and sufficient condition are related to 3 or 4.

necessary condition is the one that is required to make the statement true, then maybe it is the same as my answer before, A ≥ 4.

sufficient condition is the one that says if the condition is met, the statement is true, then maybe 0 < a ≤ 3.

Are those right answers?

Thanks

 

[jamesmo]

Let me try again. So, necessary and sufficient condition are related to 3 or 4.

necessary condition is the one that is required to make the statement true, then maybe it is the same as my answer before, A ≥ 4.

sufficient condition is the one that says if the condition is met, the statement is true, then maybe 0 < a ≤ 3.

Are those right answers?

Thanks

You are almost there, but the inequality for the necessary condition is backwards. Does the concept make sense now?

 

[songoku]

You are almost there, but the inequality for the necessary condition is backwards. Does the concept make sense now?

For this one I think I get it. Hope it will be easier for the other questions.

thanks a lot for your help jamesmo and the chaz