inequality help please

Question

Solve the inequality x(x - 3)(x +5) < 0 and express the solution set in interval notation. Show Work.

Answer 1

This problem is best solved by the following trick:

 

1) Consider zeroes of the expression x(x-3)(x+5). They are x=-5, x=0, and x=3. Break the number line into intervals (-∞;-5), (-5;0), (0;3), and (3;∞). For x from each interval construct the table of signs of all factors, x, (x-3) , and (x+5). It will look like follows:

 

           (-∞;-5)  (-5;0)   (0;3)   (3;∞)

(x+5)      -           +          +          +

x             -            -          +          +

(x-3)       -           -            -           +

 

Since the product of three numbers is less than zero when either all three are negative or just one is negative, the solution of the original equation is as follows:

 

(-∞;-5)∪(0;3)