Problem 77 - College Algebra Essentials

Question

List the quadrant or quadrants satisfying each condition. $$ x^{3}>0 \text { and } y^{3}<0 $$

Step-by-Step Solution

Answer

Quadrant IV

Step 1: Identify the conditions

The exercise gives us two conditions that must be satisfied simultaneously: \(x^{3} > 0\) and \(y^{3} < 0\). This implies that the x-values are positive (since \(x^{3} > 0)\) and the y-values are negative (since \(y^{3} < 0)\).

Step 2: Apply the Cartesian coordinate system concepts

We know that the Cartesian system is split into four quadrants. Quadrant I is where both x and y are positive, Quadrant II is where x is negative and y is positive, Quadrant III is where both x and y are negative, and Quadrant IV is where x is positive and y is negative. Given the conditions, we are looking for a quadrant where x is positive and y is negative.

Step 3: Identify the correct quadrant

Comparing our conditions with the properties of each quadrant, we see that Quadrant IV, where x is positive and y is negative, satisfies these conditions. Therefore, the solution is Quadrant IV.

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system, named after the mathematician René Descartes, is a two-dimensional system used to locate points in a plane. This system is essential for working with graphs and geometric figures.
It consists of two perpendicular lines called axes: the x-axis, which is horizontal, and the y-axis, which is vertical. The point where these axes intersect is known as the origin, labelled as (0,0).
The system allows each point on the plane to be represented as an ordered pair of numbers \(x, y\), where the first number is the x-coordinate and the second is the y-coordinate. Understanding this system is crucial for working with various mathematical concepts, including geometry and algebra.

Coordinate plane

The coordinate plane, also known as the Cartesian plane, is formed by the intersection of the x-axis and y-axis. This plane allows us to visualize and plot points, lines, and curves based on their coordinates.
By dividing the plane into four parts, called quadrants, it becomes easier to identify the position of points relative to the axes. Each point is expressed with coordinates that highlight their specific location on the plane.
This visual representation helps not only in solving mathematical problems but also in understanding spatial relationships, making the coordinate plane a powerful tool in mathematics.

Positive and negative values

In the context of the coordinate plane, positive and negative values play a critical role in determining a point's position. The x-axis and y-axis divide the plane into sections where these values vary.

• Positive x-values are found to the right of the origin.
• Negative x-values are found to the left of the origin.
• Positive y-values are above the origin.
• Negative y-values are below the origin.

Understanding which side of the axis each value lies on helps in plotting points and deducing the target areas or quadrants in various scenarios where these values need to be evaluated.

Quadrant identification

Quadrants are four distinct areas created by the intersection of the x-axis and y-axis in the coordinate plane. Each quadrant is denoted by Roman numerals I, II, III, and IV, following a counter-clockwise direction starting from the top right.

• Quadrant I: x is positive, y is positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: x is negative, y is negative.
• Quadrant IV: x is positive, y is negative.

In the given exercise, we look for a quadrant where x is positive and y is negative—a characteristic of Quadrant IV. Recognizing these quadrants helps understand how positive and negative values interact in the coordinate grid, which is pivotal for solving related mathematical problems.