In Exercises 75–78, list the quadrant or quadrants satisfying each condition. x^3 > 0 and y^3 <0

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 positive x and y values, Quadrant II has negative x and positive y values, Quadrant III has negative x and y values, and Quadrant IV has positive x and negative y values. Understanding these quadrants is essential for determining where specific inequalities 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 regions in the Cartesian plane that satisfy the given conditions.

Cubic Functions and Their Behavior

Cubic functions, such as f(x) = x^3, have specific characteristics based on the sign of x. For positive x, the output is positive, and for negative x, the output is negative. This behavior is crucial for interpreting the inequalities x^3 > 0 and y^3 < 0, as it directly informs us about the possible values of x and y in the context of the quadrants.