Problem 78 - Algebra and Trigonometry

Question

In Exercises \(75-78,\) list the quadrant or quadrants satisfying each condition. \(x^{3}<0\) and \(y^{3}>0\)

Step-by-Step Solution

Answer

The quadrant satisfying the conditions \(x^{3}<0\) and \(y^{3}>0\) is the second quadrant.

Step 1: Determine sign of x and y

Since cubing preserves the sign of a number: \(x^3 < 0\) implies \(x < 0\), and \(y^3 > 0\) implies \(y > 0\).

Step 2: Identify the quadrant

Points where \(x < 0\) and \(y > 0\) lie in the second quadrant (Quadrant II).

Key Concepts

Quadrant Analysis

Quadrant analysis is a fundamental part of graphing on the Cartesian coordinate system. The system is divided into four sections, called quadrants, which are numbered counterclockwise starting from the upper right quadrant. Each quadrant has distinct properties based on the signs of the x (horizontal) and y (vertical) coordinates.

In the first quadrant, both x and y are positive. In the second quadrant, x is negative and y is positive, which is the scenario we see in our exercise with conditions stating that the cube of x is less than zero and the cube of y is greater than zero. This specific condition implies that the x-coordinate is negative and the y-coordinate is positive, making the second quadrant the focus of our analysis. Understanding which signs correspond to each quadrant helps in sketching and identifying the position of points and the solution to inequalities involving coordinates.

Coordinate Plane

The coordinate plane, also known as the xy-plane, is a two-dimensional surface where each point is defined by a pair of numerical coordinates: the x-coordinate (horizontal axis) and the y-coordinate (vertical axis). The plane is divided by two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical). Where these two axes intersect at the point (0,0) is known as the origin.

When plotting or analyzing points on the coordinate plane, it's crucial to understand the implications of the x and y values. A positive x moves the point right from the origin, while a negative x moves it left. A positive y moves the point up, and a negative y moves it down. For quadratic inequalities like in our exercise, analyzing the cube of x and y helps determine the quadrant in which the solution set lies as these powers can change the signs based on their odd or even characteristic.

Inequalities in Algebra

In algebra, inequalities are equations that use symbols such as <, >, ≤, or ≥ to show the relationship between two expressions. They are crucial in defining ranges of numbers rather than specific values. When working with inequalities, especially with powers, it is important to consider the behavior of negative and positive values.

For example, in the given problem statement, the inequality involves cubes of x and y. Since any negative value raised to an odd power will remain negative, and any positive value raised to any power will stay positive, it's easy to determine the signs of the coordinates based on the inequalities. This information can then be applied to the coordinate plane to locate the correct quadrant—demonstrating how algebraic inequalities can provide insights into geometric positioning on the Cartesian coordinate system.