Question

Number of positive integral values of x satisfying the inequality
(x-4)^2013. (X + 8)^2014 (x + 1) /x^2016(x - 2)^3 - (x + 3)^5 . (x-6) (x + 9)^2012 ≤ 0

(A)0
(B) 1
(C)2
(D)3​

Answer 1

The correct answer is (A) 0.

Explanation:

To determine the number of positive integral values of x satisfying the given inequality, let's analyze the factors involved.

The inequality can be rewritten as:

$\frac{(x-4)^{2013 } * (x + 8)^{2014} * (x + 1)}{x^{2016} * (x - 2)^3 - (x + 3)^5 * (x-6) * (x+9)^{2012} } \leq 0$

For this inequality to hold true, either the numerator should be negative and the denominator positive, or the numerator should be zero.

1. Numerator = 0:

 $(x-4)^{(2013) } *(x + 8)^{(2014) } * (x + 1)$ = 0

2. Numerator < 0 and Denominator > 0:

For this condition to hold, the signs of the factors in the numerator and denominator must alternate.

Analyzing the signs of the factors:

- (x-4) and (x + 8) are always positive.

- (x + 1) changes sign at x = -1.

- $x^{2016}$ and $(x - 2)^3$ are always positive.

- $(x + 3)^5$ changes sign at x = -3.

- (x-6) changes sign at x = 6.

- (x + 9) changes sign at x = -9.

Based on the analysis, we can conclude that there are no positive integral values of x satisfying the given inequality. Therefore, the correct answer is (A) 0.

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