solve the folowing inequality (x-9)^2(x+3)>0

Question

put the answer in interval notation

Answer 1

(x-9)²(x+3)>0

 

Since the first (squared) factor is always positive except if x = 9, we see that x ≠ 9 is a requirement.

 

That being the case, we merely must assurethat x+3 > 0, i.e. x> -3.

 

So the solution is x > -3 except excluding 9. Notice that this answer is different from that given by David W.

For example, his solution excludes x = 0, but mine accepts that and substitution shows the statement to be satisfied.

Answer 2

If we completely factor the inequality and replace the greater thansign with an equal sign, we get

 

(x - 9)(x - 9)(x + 3) = 0

 

 

This setup allows us to find the x-intercepts (points on the x-axis).  Thus the x-intercepts are

 

x = -3           and          x = 9

 

 

Since we have a positive cubic function, the graph will start off increasing from negative infinity, then increase to positive infinity at the end.  This is the basic shape of a cubic function.

 

Also, the graph needs to above the x-axis because the function is greater than zero.

 

 

x=9  has a multiplicity of 2, so at x=9, the graph is tangential to the x-axis.  x=-3 has a multiplicity of 1, so it crosses the x-axis.

 

The only time the graph will be above the x-axis, due to its shape and behavior is in the interval (-3, 9)∪(9, ∞).

 

 

You can use a graphing calculator to verify this.

 

 

Answer 3

(x-9)²(x+3)>0

 

(x-9)(x-9)(x+3)  is greater than zero (that is, positive) when:

  +     +      +       case A

  +     -       -        case B

  -      +      -        case C

  -      -       +       case D

 

Case A is true when x>9 and x>-3.

Case B cannot be true.

Case C cannot be true.

Case D is true when x<9 and x>-3.

 

Putting cases A and D together, x>-3.  In interval notation:

    (-3,∞)  and  x≠9