The number of possible integral values of the parameter $k$ $k$ for which the inequality $k^{2} x^{2} < (8 k - 3) (x + 6)$ $k^2x^2 < (8k -3)(x+6)$ holds true for all values of $x$ $x$ satisfying $x^{2} < x + 2$ $x^2 < x+2$ is?

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The number of possible integral values of the parameter $k$ $k$ for which the inequality $k^{2} x^{2} < (8 k - 3) (x + 6)$ $k^2x^2 < (8k -3)(x+6)$ holds true for all values of $x$ $x$ satisfying $x^{2} < x + 2$ $x^2 < x+2$ is?

A) 0
B) 1
C) 2
D) 3

1 Answer

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Answer 1 · 2017-11-27T18:24:07

$0$ $0$

Explanation:

$x^{2} < x + 2$ $x^2 < x + 2$ is true for $x \in (- 1, 2)$ $x in (-1,2)$

now solving for $k$ $k$

$k^{2} x^{2} - (8 k - 3) (x + 6) < 0$ $k^2 x^2 - (8 k - 3) (x + 6) < 0$ we have

$k \in (\frac{24 + 4 x - \sqrt{24^{2} + 192 x - 2 x^{2} - 3 x^{3}}}{x^{2}}, \frac{24 + 4 x + \sqrt{24^{2} + 192 x - 2 x^{2} - 3 x^{3}}}{x^{2}})$ $k in ((24 + 4 x - sqrt[24^2 + 192 x - 2 x^2 - 3 x^3])/x^2, (24 + 4 x + sqrt[24^2 + 192 x - 2 x^2 - 3 x^3])/x^2)$

but

$\frac{24 + 4 x + \sqrt{24^{2} + 192 x - 2 x^{2} - 3 x^{3}}}{x^{2}}$ $(24 + 4 x + sqrt[24^2 + 192 x - 2 x^2 - 3 x^3])/x^2$ is unbounded as $x$ $x$ approaches $0$ $0$ so the answer is $0$ $0$ integer values for $k$ $k$ obeying the two conditions.

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The number of possible integral values of the parameter $k$ $k$ for which the inequality $k^{2} x^{2} < (8 k - 3) (x + 6)$ $k^2x^2 < (8k -3)(x+6)$ holds true for all values of $x$ $x$ satisfying $x^{2} < x + 2$ $x^2 < x+2$ is?

A) 0
B) 1
C) 2
D) 3

1 Answer

Explanation:

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Science

Math

Humanities

... and beyond

The number of possible integral values of the parameter kk for which the inequality k2x2<(8k−3)(x+6)k^2x^2 < (8k -3)(x+6) holds true for all values of xx satisfying x2<x+2x^2 < x+2 is?

A) 0 B) 1 C) 2 D) 3

1 Answer

Explanation:

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Impact of this question

The number of possible integral values of the parameter $k$ $k$ for which the inequality $k^{2} x^{2} < (8 k - 3) (x + 6)$ $k^2x^2 < (8k -3)(x+6)$ holds true for all values of $x$ $x$ satisfying $x^{2} < x + 2$ $x^2 < x+2$ is?

A) 0
B) 1
C) 2
D) 3