If (x^3 + y^6) is positive, which of the following must be true?

Let's approach this step-by-step to determine which statement must be true when (x^3 + y^6) is positive.

Understand the given condition
We know that (x^3 + y^6) > 0
Step 2: Analyze the components

x^3 can be positive, negative, or zero
y^6 is always non-negative for any real y (because even powers of real numbers are always non-negative)

Consider possible scenarios
a) If x^3 > 0:
y^6 can be any non-negative number (including zero)
The sum will be positive

b) If x^3 = 0 (when x = 0):
y^6 must be positive for the sum to be positive
y cannot be zero in this case

c) If x^3 < 0:
y^6 must be positive
|y^6| must be greater than |x^3| for the sum to be positive
y cannot be zero in this case

Draw conclusions about x and y
x can be positive, negative, or zero
y can be any real number except zero

Determine the statement that must always be true
Based on our analysis, the only statement that must always be true is:
y ≠ 0 (y is not equal to zero)
This is because:

If y were zero, y^6 would be zero
We need y^6 to be positive in case x^3 is zero or negative
Even when x^3 is positive, y can be zero, but it doesn't have to be

Therefore, the only condition we can be certain of is that y cannot be zero. Copy