Problem 77 - 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 fourth quadrant is the quadrant that satisfies the conditions \(x^{3}>0\) and \(y^{3}<0\).

Step 1: Identification of Quadrants

In the Cartesian coordinate system, there are four quadrants: the first quadrant (upper right) where both x and y are positive, the second quadrant (upper left) where x is negative and y is positive, the third quadrant (lower left) where both x and y are negative, and the fourth quadrant (lower right) where x is positive and y is negative.

Step 2: Applying Given Conditions

The exercise asks for conditions where \(x^{3} > 0\) and \(y^{3} < 0\). This means x is positive and y is negative, as the cube of any positive number is positive and the cube of any negative number is negative.

Step 3: Conclusion

By comparing this to the properties of each quadrant, the quadrant satisfying these conditions is the fourth quadrant where x is positive and y is negative.

Key Concepts

Cartesian Coordinate System

The Cartesian coordinate system is a fundamental framework for navigation and analysis in various fields of mathematics, physics, engineering, and more. Imagine the system as a flat grid, drawn on a piece of paper, where every point on the grid represents a unique location in two-dimensional space. This system is built on two perpendicular lines, known as axes. The horizontal line is called the x-axis, and the vertical line is the y-axis.

These axes intersect at a point called the origin, denoted as (0,0). Moving away from the origin in any direction involves either positive or negative values along these axes. Positive values are typically drawn to the right for the x-axis and upward for the y-axis, while negative values go to the left and downward, respectively.

Quadrant Analysis

Breaking down the Cartesian plane, you'll find it divided into four regions, known as quadrants. They are labeled counterclockwise starting from the upper right quarter as the first quadrant (I), upper left as the second quadrant (II), lower left as the third quadrant (III), and lower right as the fourth quadrant (IV).

Each quadrant is characterized by the signs of the x and y coordinates of the points within them. For instance, in the first quadrant, both x and y coordinates of any point will be positive, while in the third quadrant, both will be negative. Researchers and students use quadrant analysis to quickly determine the relation between two variables on a graph, or, as in our original exercise, to deduce potential locations based on algebraic conditions.

Signs of Coordinates

Understanding the signs of coordinates within each quadrant is crucial for analyzing equations and inequalities. Positive x values are associated with the first and fourth quadrants, while positive y values occur in the first and second quadrants. Conversely, negative x values belong to the second and third quadrants, and negative y values to the third and fourth quadrants.

The signs of coordinates give us quick insights into the behavior of algebraic expressions. For example, the condition \(x^{3} > 0\) implies that x is positive because a positive number remains positive when raised to any odd power. Similarly, \(y^{3} < 0\) implies y is negative as a negative number remains negative when raised to any odd power. These sign rules lead us to the exercise's conclusion: the fourth quadrant, where x is positive and y is negative, is the solution.