BTGmoderatorDC wrote:If f(x) = x³ + 9, is f(x) positive?
(1) x < −1
(2) x > −3
Given: f(x) = x³ + 9
Target question: Is f(x) positive?
Statement 1: x < −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)³ + 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)³ + 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 > −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)³ + 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)³ + 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)³ + 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)³ + 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
