List the quadrant or quadrants satisfying each condition. x³ > 0 and y³ <0

Verified step by step guidance

1. Understand the problem: We are tasked with identifying the quadrant(s) where the conditions x^3 > 0 and y^3 < 0 are satisfied. Recall that the Cartesian plane is divided into four quadrants, and the signs of x and y in each quadrant determine the conditions.
2. Analyze the condition x^3 > 0: The cube of x is positive when x itself is positive. This means x > 0. On the Cartesian plane, x > 0 corresponds to Quadrants I and IV.
3. Analyze the condition y^3 < 0: The cube of y is negative when y itself is negative. This means y < 0. On the Cartesian plane, y < 0 corresponds to Quadrants III and IV.
4. Combine the conditions: For both conditions to be true simultaneously, we need a quadrant where x > 0 and y < 0. From the analysis, this occurs in Quadrant IV.
5. Conclude the solution: The quadrant that satisfies the conditions x^3 > 0 and y^3 < 0 is Quadrant IV.

Key Concepts

Quadrants of the Cartesian Plane

The Cartesian plane is divided into four quadrants based on the signs of the x and y coordinates. Quadrant I has both x and y positive, Quadrant II has x negative and y positive, Quadrant III has both x and y negative, and Quadrant IV has x positive and y negative. Understanding these quadrants is essential for determining where specific conditions hold true.

Inequalities and Their Solutions

Inequalities express a relationship where one side is not equal to the other, often involving greater than (>) or less than (<) symbols. In this case, x^3 > 0 indicates that x must be positive, while y^3 < 0 indicates that y must be negative. Solving these inequalities helps identify the values of x and y that satisfy the given conditions.

Cubic Functions and Their Behavior

Cubic functions, such as f(x) = x^3, exhibit specific behaviors based on the sign of x. For positive x, the output is positive, and for negative x, the output is negative. This characteristic is crucial for understanding the conditions x^3 > 0 and y^3 < 0, as it directly influences the signs of x and y in the context of the quadrants.