Is x^2>9 true if and only if x is not equal to 2?

[Trail_Builder]

Homework Statement

Insert either "if", "only if", or "if and only if"..

x>2 ... x^2>9

Homework Equations

The Attempt at a Solution

I don't think any fit :S coonffuuuuuused

 

[kehler]

x can be any real number? In that case I *think* it should be "if".
But I'm no expert at this :P

 

[dirk_mec1]

if equals =>
only if equals <=
if and only if equals <=>

 

[_Andreas]

Homework Statement

Insert either "if", "only if", or "if and only if"..

x>2 ... x^2>9

Homework Equations

The Attempt at a Solution

I don't think any fit :S coonffuuuuuused

One fits perfectly. Think about what dirk said.

 

[kehler]

Oops sorry, guess I was wrong.

 

[Trail_Builder]

still don't know :S

I know the x^2>9 is equivalent to x<-3 or x>3.

so I am guessing that rules out the <=>

and if it was the x>3, then I would stick the <= one in. ("only if").

"if" obviously won't work.

but don't you have to consider the x<-3 case too? or am I imposing somekind of weird necessary/sufficient crap in where i shouldn't?

 

[_Andreas]

still don't know :S

I know the x^2>9 is equivalent to x<-3 or x>3.

so I am guessing that rules out the <=>

and if it was the x>3, then I would stick the <= one in. ("only if").

"if" obviously won't work.

but don't you have to consider the x<-3 case too? or am I imposing somekind of weird necessary/sufficient crap in where i shouldn't?

Ponder the following. First, assume that x>2 is true. Does this imply that x^2>9 is true? If yes, then "if x>2, then x^2>9". If no, then not "if x>2, then x^2>9". (This is the "sufficient" condition.) Second, assume that x^2>9 is true. Does this imply that x>2 is true? If yes, then "only if x>2, then x^2>9". If no, then not "only if x>2, then x^2>9". (This is the "necessary" condition.) If yes to both, then "if and only if".

 

[epenguin]

Don't you also need to insert 'then' somewhere?

 

[_Andreas]

but don't you have to consider the x<-3 case too? or am I imposing somekind of weird necessary/sufficient crap in where i shouldn't?

I think you need to know if there are any constraints on x. Can it be only a positive integer, perhaps? What does it say in your book?

 

[Trail_Builder]

Ponder the following. First, assume that x>2 is true. Does this imply that x^2>9 is true? If yes, then "if x>2, then x^2>9". If no, then not "if x>2, then x^2>9". (This is the "sufficient" condition.) Second, assume that x^2>9 is true. Does this imply that x>2 is true? If yes, then "only if x>2, then x^2>9". If no, then not "only if x>2, then x^2>9". (This is the "necessary" condition.) If yes to both, then "if and only if".

thanks for the clarification. I kind just used my intuition before but glad to know how to properly do it, lol :).

the answer is no and no then, because there isn't a constaint on x... :S

so was I right in saying none?

 

[Trail_Builder]

I think you need to know if there are any constraints on x. Can it be only a positive integer, perhaps? What does it say in your book?

no specified constaints

 

[_Andreas]

thanks for the clarification. I kind just used my intuition before but glad to know how to properly do it, lol :).

the answer is no and no then, because there isn't a constaint on x... :S

so was I right in saying none?

Yeah, if there are no constraints on x, then it seems to me that you were correct in that none fit. It is not necessary that x>2 for x^2>9. What's necessary is that x is not equal to 2. Neither is x>2 sufficient for x^2>9. (In other words I was wrong when I said one fits perfectly.)