Solve (2-x)(3+x)<0

$$( 2 - x ) ( 3 + x ) < 0$$

Solve for x

$$x\in \left(-\infty,-3\right)\cup \left(2,\infty\right)$$

Solution Steps

For the product to be negative, x+3 and 2-x have to be of the opposite signs. Consider the case when x+3 is positive and 2-x is negative.

$$x+3>0 2-x<0$$

The solution satisfying both inequalities is x>2.

$$x>2$$

Consider the case when 2-x is positive and x+3 is negative.

$$2-x>0 x+3<0$$

The solution satisfying both inequalities is x<-3.

$$x<-3$$

The final solution is the union of the obtained solutions.

$$x<-3\text{; }x>2$$