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

BTGmoderatorDC wrote:If f(x) = xÂ³ + 9, is f(x) positive?

(1) x < âˆ’1
(2) x > âˆ’3

Given: f(x) = xÂ³ + 9

Target question: Is f(x) positive?

Statement 1: x < âˆ’1
Let's TEST some values.
There are several values of x that satisfy statement 1. Here are two:
Case a: x = -2. In this case, f(-2) = (-2)Â³ + 9 = (-8) + 9 = 1. So, the answer to the target question is YES, f(x) IS positive
Case b: x = -5/2. In this case, f(-5/2) = (-5/2)Â³ + 9 = (-125/8) + 9 = -15 5/8 + 9 = some NEGATIVE number. So, the answer to the target question is NO, f(x) is NOT positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x > âˆ’3
Let's TEST some values.
NOTE: Always see if you can reuse any test cases from statement 1. In this case, we can reuse both of them.
There are several values of x that satisfy statement 2. Here are two:
Case a: x = -2. In this case, f(-2) = (-2)Â³ + 9 = (-8) + 9 = 1. So, the answer to the target question is YES, f(x) IS positive
Case b: x = -5/2. In this case, f(-5/2) = (-5/2)Â³ + 9 = (-125/8) + 9 = -15 5/8 + 9 = some NEGATIVE number. So, the answer to the target question is NO, f(x) is NOT positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: x = -2. In this case, f(-2) = (-2)Â³ + 9 = (-8) + 9 = 1. So, the answer to the target question is YES, f(x) IS positive
Case b: x = -5/2. In this case, f(-5/2) = (-5/2)Â³ + 9 = (-125/8) + 9 = -15 5/8 + 9 = some NEGATIVE number. So, the answer to the target question is NO, f(x) is NOT positive
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

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