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Chapter 1: Problem 77

List the quadrant or quadrants satisfying each condition. $$x^{3}>0 \text { and } y^{3}<0$$

Short Answer

Expert verified
The given conditions are met in Quadrant IV.

Step by step solution

01

Interpret the First Condition

When \(x^3 > 0\), this implies that the cube of x is positive. Since the cube of a positive number is positive and the cube of negative a negative number is negative, we can deduce that 'x' must be positive. This places us in the regions of the Cartesian coordinate system where x is positive, which are Quadrant I and Quadrant IV.
02

Interpret the Second Condition

Similarly, for the condition \(y^3 < 0\), this indicates that the y coordinate must be negative since cubing a negative value results in a negative value; a positive number cubed remains positive. Therefore, 'y' is negative, which places us in the regions of the Cartesian coordinate system where 'y' is negative. This corresponds to Quadrant III and Quadrant IV.
03

Find the Common Quadrant

To meet both conditions simultaneously, we must find the quadrant(s) that satisfy both our conditions - x is positive and y is negative. Comparing Quadrants from Step 1 and Step 2, the only common Quadrant, where 'x' is positive and 'y' is negative, is Quadrant IV.

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