inequalityIf Pn(x) = (x-1)(x-2)...(x-n) solve the inequality P5(x)/P6(x) >= 6

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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) > =6

1/(x-6) >= 6

We'll subtract 6 both sides:

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

We'll multiply by (x-6):

(1 - 6x + 36)/(x - 6) > = 0

We'll combine like terms:

(37 - 6x)/(x - 6)>= 0

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

Case 1)

37 - 6x >= 0

-6x >= -37

6x =< 37

x = < 37/6 = 6.166

x - 6 >= 0

x >= 6

The interval of admissible values for x, that makes positive the ratio, is [6 ; 6.166].

Case 2)

37 - 6x = <  0

-6x = < -37

6x >= 37

x >= 37/6 = 6.166

x - 6 =< 0

x =< 6

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

The only admissible range of possible solutions for x is [6 ; 6.166].

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