Write the inverse, converse, and contrapositive of the following statement:

upside down A x E R, if (x + 2)(x - 3) > 0, then x < -2 or x >3

Indicate which among the statement, its converse, ints inverse, and its contrapositive are true and which are false. Give a counterexample for each that is false.

I don't quite understand what your trying to say here. Is this suppose to be an Inequality or Algebraic Equation or what is it that I am actually suppose to solve?

Here's a summary drawn from:

http://www.jimloy.com/logic/converse.htm

statement: if p then q
converse: if q then p
inverse: if not p then not q
contrapositive: if not q then not p

For the given example:
Statement:
∀x∈R, if(x+2)(x-3) > 0 then x<-2 or x>3

So
p: if (x+2)(x-3)
q: x<-2 ∨ x>3

Substitute into the above to find the inverse, converse and contrapositive. Post your answers for checking if you wish.

correction:

p: (x+2)(x-3) >0
q: x<-2 ∨ x>3

The example is to apply mathematical logic to the results of the solution of an algebraic inequality.

There is nothing to do algebraically, but to modify the original statement to make the converse, inverse and contrapositive.

If the original statement were:
"If it rains, then I go to the park".
Compare with the symbolic logic statement:
if p then q ( p -> q )
we conclude that
p = it rains (condition)
q = I stay home (consequence).

The converse is then
if q then p (q -> p)
which translated in words:
if I stay home then it rains.

The inverse is:
if ~p then ~q (~p -> ~q)
which translates to:
If it does not rain, then I do not stay home.

The contrapositive is:
if ~q then ~p (~q -> ~p)
which translates to:
If I do not stay home, then it does not rain.

So for the given question, you only need to substitute
p = (x+2)(x-3) > 0
q = x<-2 ∨ x>3
and repeat the above exercise.

Let me see if I have this right and what your saying:

(x + 2) (x - 3) > 0

x squared - 3x + 2x -6 > 0

x squared - 1x - 6 > 0

Is this how you solve the problem?

I don't see any numerical value of what p and q are equal to so I am only understanding that the problem in front of me and what I am looking at shows me an inequality. What is it that I am missing here? I don't quite understand exactly what it is that I am suppose to insert or substitute.

There is no algebraic manipulation to be done, just like the raining and staying home.

Let
p = (x+2)(x-3) > 0
q = x<-2 ∨ x>3

Then
p->q, or "if p then q", would translate to the original statement of

If (x+2)(x-3)>0 then x<-2 ∨ x>3.

The converse is then
q->p, or "if q then p" translates to
If x<-2 ∨ x>3 then (x+2)(x-3)>0

There is no algebraic calculation to be done. You would substitute
(x+2)(x-3)>0 for p, and
x<-2 ∨ x>3 for q.

I will let you do the inverse (~p->~q) and the contrapositive (~q->~p).

I am not sure what this one is but I will see whether or not I get this right.

If x < -2 V x < 3 then (x + 2) (x - 3) < 0

Also, what does the wavy line mean when you mentioned the Inverse:

If ~ p then ~ q (~ p - > ~ q)

Another question what does the dash mean after the p in the parentheses? Is that supposed to be a dash, hyphen, minus, less than or equal to, or greater than or equal to?

Another question also is the example where you did the converse is that true or false?