solve the inequality. Write your answer using interval notation. x(x − 2)(x + 5) > 0

Question

solve the inequality. Write your answer using interval notation.
x(x − 2)(x + 5) > 0

Answer 1

Zeros are at -5, 0, and 2

If x < -5, x(x − 2)(x + 5) < 0

If -5 < x < 0, x(x − 2)(x + 5) > 0

 

Can you continue and state in interval notation.

Answer 2

x(x − 2)(x + 5) > 0

 

x>0

 

or

 

x-2 >0

 

x > 2

 

or

 

x+5 >0

x>-5

 

x>0,x > 2,x>-5

 

Answer:

-5<x<0    Or  x>2

 

Interval notation

(-5,0) υ (2,∞)

 

Answer 3

This inequality has three key points that make the left side = -5, 0 and 2.

These three points should be placed on the number-line, in ascending order.

That divides the domain into four intervals.

In the rightmost interval (2,infinity) the three different factors will all be positive.

Moving leftward from that interval, you encounter three different intervals where the function will be negative, positive and negative, in that order.

 

Therefore the solution is (-5,0) ∪ (2,infinity)

Answer 4

First, we find the zeros.  This is when the inequality is zero.

 

x = -5    ,        x = 0        ,      x = 2

 

 

We use these zeros as our reference points.  If we expand the inequality, we get a 3rd degree polynomial with a positive coefficient.  This means that the value of the inequality decreases as x decreases, and the value of the inequality increases as x increases.

 

Knowing that, you solution in interval notation is

 

(-5, 0)∪(2, infinity)