Answer
Statement: $\forall x \in \mathbb{R}$, if x(x + 1) > 0 then x > 0 or x < −1.
Contrapositive: $\forall x \in \mathbb{R}$, if x ≤ 0 and x ≥ −1, then x(x + 1) ≤ 0.
Converse: $\forall x \in \mathbb{R}$, if x > 0 or x < −1 then x(x + 1) > 0.
Inverse: $\forall x \in \mathbb{R}$, if x(x + 1) ≤ 0 then x ≤ 0 and x ≥ −1.
The statement, its contrapositive, its converse, and its inverse are all true.
Work Step by Step
A statement of the form: $\forall x \in D$, if P(x) then Q(x),
has as its contrapositive statement: $\forall x \in D$, if ~Q(x) then ~P(x),
as its converse statement: $\forall x \in D$, if Q(x) then P(x),
and as its inverse statement: $\forall x \in D$, if ~P(x) then ~Q(x).
