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

Question

Vinod · Accepted Answer

Let's approach this step-by-step to determine which statement must be true when (x^3 + y^6) is positive.
Understand the given conditionWe know that (x^3 + y^6) > 0Step 2: Analyze the components
x^3 can be positive, negative, or zeroy^6 is always non-negative for any real y (because even powers of real numbers are always non-negative)
Consider possible scenariosa) 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 positivey 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 positivey cannot be zero in this case
Draw conclusions about x and yx can be positive, negative, or zeroy can be any real number except zero
Determine the statement that must always be trueBased on our analysis, the only statement that must always be true is:y &ne; 0 (y is not equal to zero)This is because:
If y were zero, y^6 would be zeroWe need y^6 to be positive in case x^3 is zero or negativeEven 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