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Bunuel
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Bunuel
If \(f(x) = x^3 + 9\), is \(f(x)\) positive?

(1) \(x < −1\)

(2) \(x > −3\)

Official solution from Veritas Prep.

If \(x<−1\), then x could be \(−2\) in which case \(f(x)\) is positive or \(x\) could be \(−3\) in which case \(f(x)\) is negative. Therefore, Statement 1 alone is not sufficient.

If \(x>−3\), the x could be \(−2\) in which case \(f(x)\) is positive. However, the problem never stated that \(x\) is an integer. Then, \(x\) could be \(-2.99999\), that is almost \(-3\). In this case, \(f(x)\) will be a negative. \(f(x)=(-2.99999)^3+9 = -17.99973\) Therefore, Statement 2 is not sufficient.

Combining Statement 1 and Statement 2, \(f(x)\) could still either be positive or negative. Therefore, the answer is \(E\).
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I think it should be B.

(-2.9999)^3 is less than 9 and therefor -899999+9 >0

Hence it should be B
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I think it should be B.

(-2.9999)^3 is less than 9 and therefor -899999+9 >0

Hence it should be B

(-2.9999)^3 = -26.997300 < 0. :)
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Quote:
If f(x) = x^3 + 9, is f(x) positive?
Step 1: Understanding the question
When f(x) = 0
x^3 + 9 = 0
x = \((-9)^{1/3}\)
x ~ -2.1
When x = -3; f(x) is negative
When x = -1; f(x) is positive
ie. on left side of -2.1, the function is negative, and on right side of -2.1 the function is positive

Step 2: Understanding statement 1 alone
(1) x < −1
When x = -1.5; f(x) is positive
When x = -2.5; f(x) is negative
Insufficient

Step 3: Understanding statement 2 alone
(2) x > −3
from above reasoning; Insufficient

Step 4: Combining
x lies between -3 and -1 ie -3<x<-1. Still sign of f(x) is unknown.
Insufficient

E is correct
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