If Pn(x)=(x-1)(x-2)...(x-n) solve the inequality P5(x)/P6(x)>=2

If Pn(x)=(x-1)(x-2)...(x-n), then P5(x) =
(x-1)(x-2)...(x-5) and P6(x) =
(x-1)(x-2)...(x-6)

P5(x)/P6(x) =
(x-1)(x-2)...(x-5)/(x-1)(x-2)...(x-5)(x-6)

We'll simplify
and we'll get:

P5(x)/P6(x) =
1/(x-6)

But, from enunciation, we know that P5(x)/P6(x)
> =2

1/(x-6) >=
2

We'll subtract 2 both
sides:

1/(x-6) - 2 >=
0

We'll multiply by (x-6):

(1
- 2x + 12)/(x - 6) > = 0

We'll combine like
terms:

(13 - 2x)/(x - 6)>=
0

A ratio is positive if and only if both numerator and
denominator, are positive or negative.

Case
1)

13 - 2x >= 0

-2x
>= -13

2x =<
13

x = < 13/2 = 6.5

x -
6 >= 0

x >=
6

The interval of admissible values for x, that makes
positive the ratio, is [6 ; 6.5].

Case
2)

13 - 2x = <  0

-2x =
< -13

2x >= 13

x
>= 13/2 = 6.5

x - 6 =<
0

x =< 6

There is no
common interval for admissible values for x, in this
case.

The only admissible interval of valid
solutions for x is [6 ; 6.5].