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07 Mar 2024, 07:49
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B
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If \(x = -\sqrt[200]{0.2}\), then \(x\) satisfies which of the following inequalities?
A. \(x < -1\)
B. \(-1 < x < -0.9\)
C. \(-0.9 < x < -0.5\)
D. \(-0.5 < x < -0.1\)
E. \(-0.1 < x < 0\)
A. \(x < -1\)
B. \(-1 < x < -0.9\)
C. \(-0.9 < x < -0.5\)
D. \(-0.5 < x < -0.1\)
E. \(-0.1 < x < 0\)
15 Mar 2024, 19:45
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Official Solution:
If \(x = -\sqrt[200]{0.2}\), then \(x\) satisfies which of the following inequalities?
A. \(x < -1\)
B. \(-1 < x < -0.9\)
C. \(-0.9 < x < -0.5\)
D. \(-0.5 < x < -0.1\)
E. \(-0.1 < x < 0\)
Such a large root from 5 would still be more than 1, however very close to it. Therefore:
Finally, \(-\frac{1}{1.001}\) will be more than -1, however very close to it.
Answer: B
If \(x = -\sqrt[200]{0.2}\), then \(x\) satisfies which of the following inequalities?
A. \(x < -1\)
B. \(-1 < x < -0.9\)
C. \(-0.9 < x < -0.5\)
D. \(-0.5 < x < -0.1\)
E. \(-0.1 < x < 0\)
\(x = -\sqrt[200]{0.2} = \)
\(= -\sqrt[200]{\frac{1}{5}} = \)
\(= -\frac{1}{\sqrt[200]{5}}\)
\(= -\sqrt[200]{\frac{1}{5}} = \)
\(= -\frac{1}{\sqrt[200]{5}}\)
Such a large root from 5 would still be more than 1, however very close to it. Therefore:
\(-\frac{1}{\sqrt[200]{5}}\approx \)
\(\approx -\frac{1}{1.001} \)
\(\approx -\frac{1}{1.001} \)
Finally, \(-\frac{1}{1.001}\) will be more than -1, however very close to it.
Answer: B
General Discussion
07 Mar 2024, 08:05
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Bunuel
Assume
\(x^1 = 0.2 ⇒ x = 0.2\)
Let's see the effect of increasing the power on x. Assume the power on \(x\) was 2
\(x^2 = 0.2 ⇒ x = \sqrt[2]{0.2} = \pm \sqrt[2]{\frac{2}{10}}\)
\(=\pm \frac{\sqrt[2]{2}}{\sqrt[2]{10}} \)\(\approx \pm \frac{1.5}{3} \approx \pm {0.5}\)
So we see that the value of power on x increases, and the value of x increases.
i.e
- when the power = 1 ⇒ \(x = 0.2\)
- when the power = 2 ⇒ \(x \approx 0.5\)
\(x = -\sqrt[200]{0.2}\)
Raising to the power of 200 on both sides
\(x^{200} = 0.2\)
From our analysis above we can conclude that x is either greater than 0.5 or less than -0.5. Hence, we can eliminate D and E. The value will not exceed 1 or be less than -1 because the root value will not be in decimals in that case. Hence, eliminate A.
Between B and C, we can infer that the value of x would be a value very close to 1 or -1
As we have a negative sign in the question, the value will be very close to -1
Option B
