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Chapter 1: Problem 76

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

Short Answer

Expert verified
The ratio \( \frac{y}{x} \) is less than zero in the second and fourth quadrants.

Step by step solution

01

Analyze the condition

The condition states that the ratio \( \frac{y}{x} \) should be negative. This means that y and x should have opposite signs, as the division of two numbers with opposite signs results in a negative number.
02

Determine the quadrants

In the coordinate plane, the signs of x and y change depending on the quadrant. This leads to the conclusion that the condition is met in the second and the fourth quadrant. In the second quadrant, x is negative and y is positive and in the fourth quadrant, x is positive and y is negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadrants
In the coordinate plane, we divide the plane into four distinct sections, known as quadrants. These quadrants help to determine the positions of points based on their x and y coordinates. Understanding quadrants is essential because it helps us visualize how various equations and inequalities behave.

Each quadrant in the coordinate plane is defined by a unique combination of positive or negative values of x and y:
  • First Quadrant: Both x and y coordinates are positive \((+, +)\).
  • Second Quadrant: x is negative and y is positive \((- , +)\).
  • Third Quadrant: Both x and y coordinates are negative \((- , -)\).
  • Fourth Quadrant: x is positive and y is negative \((+ , -)\).
By learning the distinctiveness of each quadrant, we can quickly determine the behavior of points or equations in the coordinate plane.
Signs of x and y
The signs of x and y coordinates help to identify which quadrant a point is located in, without needing a graph. This provides an efficient way to understand various mathematical concepts related to the coordinate system.

When we talk about the sign of x and y:
  • Positive x-coordinate: Positioned to the right of the y-axis.
  • Negative x-coordinate: Positioned to the left of the y-axis.
  • Positive y-coordinate: Positioned above the x-axis.
  • Negative y-coordinate: Positioned below the x-axis.
These descriptions help predict where the points lie without physical plotting, simplifying the process of solving inequalities or determining quadrants based on conditions like signs.
Negative Ratio
The concept of a negative ratio, such as \( \frac{y}{x} < 0 \), is crucial in determining which quadrants satisfy certain conditions. To have \( \frac{y}{x} < 0 \), y and x must have opposite signs. This condition helps us understand how expressions behave in different quadrants.

Let's break it down:
  • Second Quadrant: Here, x is negative, and y is positive, resulting in a negative ratio \((- \text{ and } +)\).
  • Fourth Quadrant: In this quadrant, x is positive, and y is negative, which also results in a negative ratio \((+ \text{ and } -)\).
By recognizing this property, one can easily solve related mathematical problems by determining the quadrant where opposite signs occur. This aids in visualizing the solution and verifying results effectively.

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