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

Verified step by step guidance

1. insert step 1> Identify the conditions given in the problem: $x^3>0$ and $y^3<0$.
2. insert step 2> Analyze the condition $x^3>0$. Since the cube of a number is positive if the number itself is positive, this implies $x>0$.
3. insert step 3> Analyze the condition $y^3<0$. Since the cube of a number is negative if the number itself is negative, this implies $y<0$.
4. insert step 4> Determine the quadrant(s) where $x>0$ and $y<0$. In the Cartesian coordinate system, the quadrant where $x$ is positive and $y$ is negative is the fourth quadrant.
5. insert step 5> Conclude that the conditions $x^3>0$ and $y^3<0$ are satisfied in the fourth quadrant.

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.