Problem 76 - College Algebra Essentials

Question

List the quadrant or quadrants satisfying each condition. $$ \frac{y}{x}<0 $$

Step-by-Step Solution

Answer

The condition \(\frac{y}{x}<0\) is satisfied in Quadrant II and Quadrant IV.

Step 1: Analyze the condition

The expression \(\frac{y}{x}<0\) can either be negative when y is positive and x is negative (which happens in the II quadrant), or when y is negative and x is positive (which happens in the IV quadrant).

Step 2: Apply the condition to each quadrant

Given the equation \(\frac{y}{x}<0\), check each quadrant: \n\n- In Quadrant I, both x and y are positive, making the fraction positive.\n\n- In Quadrant II, x is negative and y is positive, making the fraction negative.\n\n- In Quadrant III, both x and y are negative, making the fraction positive.\n\n- In Quadrant IV, x is positive and y is negative, making the fraction negative.

Step 3: Identify the quadrants

Based on the above analysis, the condition \(\frac{y}{x}<0\) is satisfied in Quadrant II (where x is negative and y is positive) and in Quadrant IV (where x is positive and y is negative).

Key Concepts

Quadrants

In coordinate geometry, the coordinate plane is divided into four sections called quadrants. Each quadrant is formed by the intersection of two perpendicular lines, the x-axis, and the y-axis, which creates four right angles. Here's a simple breakdown of the quadrants:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y coordinates are negative.
• Quadrant IV: x is positive, y is negative.

Understanding which signs x and y have in each quadrant is key. It helps in identifying where in the plane certain inequalities hold true.

Inequalities

Inequalities are mathematical expressions that describe a relationship of one value being less than, greater than, or not equal to another. In coordinate geometry, they help in determining regions on a graph where certain conditions are met. For instance, the inequality \(\frac{y}{x} < 0\) specifies where the fraction of y over x is negative.This inequality suggests y and x have opposite signs:

• If y is positive and x is negative, the fraction is negative, which happens in Quadrant II.
• If y is negative and x is positive, the fraction is again negative, which happens in Quadrant IV.

Understanding inequalities can guide us in graphically representing solutions on the coordinate plane.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. Expressions can include arithmetic operations like addition, subtraction, multiplication, and division. They are the core building blocks for equations and inequalities like the one given in this exercise, \(\frac{y}{x}\).When interpreting algebraic expressions in the context of coordinate geometry:

• The expression \(\frac{y}{x}\) compares the ratio between y and x values.
• Algebraic expressions help translate geometric properties into solvable mathematical forms.

This transformation from verbal statements or geometric configurations into algebraic forms allows more systematic solving of problems.

Signs of Coordinates

Coordinates come in pairs of values, usually noted as (x, y). The sign of each coordinate indicates where the point lies on the coordinate plane.

• Positive and Positive (I Quadrant): Means both coordinates are positive.
• Negative and Positive (II Quadrant): x is negative, y is positive.
• Negative and Negative (III Quadrant): Both coordinates are negative.
• Positive and Negative (IV Quadrant): x is positive, y is negative.

Recognizing the signs of coordinates enables us to understand the relative position of a point on the graph. Without a clear sense of whether values are positive or negative, we could misinterpret the quadrant in which a condition like \(\frac{y}{x} < 0\) is true. Identifying signs is fundamental to placing coordinates within the appropriate section of the grid.