Problem 76 - Algebra and Trigonometry

Question

In Exercises \(75-78,\) 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 Quadrants II and IV.

Step 1: Determine when y/x < 0

The fraction \(\frac{y}{x} < 0\) when \(x\) and \(y\) have opposite signs.

Step 2: Identify the quadrants

• Quadrant II: \(x < 0, y > 0\), so \(\frac{y}{x} < 0\) \(\checkmark\)
• Quadrant IV: \(x > 0, y < 0\), so \(\frac{y}{x} < 0\) \(\checkmark\)

The condition is satisfied in \(\textbf{Quadrants II and IV}\).

Key Concepts

Inequalities in Quadrants

Understanding inequalities within the context of a coordinate plane is crucial for students who are delving into algebraic concepts. An inequality like \(\frac{y}{x}<0\) specifies a relationship between two variables and asks us where, in the coordinate system, this condition holds true.

Visualizing a coordinate plane divided into four quadrants, each defined by positive and negative values of x (the horizontal component) and y (the vertical component), enables us to analyze this inequality. In the first quadrant, both x and y are positive, thus dividing y by x yields a positive value, not satisfying the condition that it must be less than zero. However, in the second quadrant, x is negative and y is positive, which means \(\frac{y}{x}\) would indeed be negative, meeting our inequality's requirement.

At the same time, the third quadrant has both x and y as negative, which, like the first quadrant, gives a positive result when dividing y by x. Contrarily, in the fourth quadrant, where y is negative and x is positive, the inequality is once again fulfilled. It's this type of logical quadrant by quadrant assessment that leads us to the precise and clear conclusion of our inequality challenge.

Coordinate Plane Analysis

Delving deeper into coordinate plane analysis can uncover a myriad of information about the relationships between algebraic expressions and their graphical representations. A coordinate plane is a two-dimensional surface where each point is determined by an x and y coordinate. These axes divide the plane into four quadrants, and it's essential to understand that each quadrant has unique characteristics based on the signs of x and y.

When exploring inequalities like \(\frac{y}{x}<0\), it's important to not only consider the numerical values but also to recognize the impact of the sign of these numbers. The rules are consistent: in Quadrants I and III, the signs of x and y are the same, and in Quadrants II and IV, they are different. This knowledge allows students to predict the sign of any division or multiplication of coordinates without performing the actual calculation, streamlining their problem-solving process and enhancing their understanding of coordinate geometry.

Signs of Coordinates

A core aspect of breaking down problems within the coordinate plane is acknowledging the signs of the coordinates, which inherently define each quadrant's nature. The first quadrant is where both x and y are positive; the second quadrant has a positive y and a negative x; the third quadrant contains negative values for both x and y; and the fourth quadrant has a positive x and a negative y.

For an inequality like \(\frac{y}{x}<0\), the signs of x and y become paramount. An easy mnemonic is 'same signs divide positive, different signs divide negative,' meaning if the signs of x and y are the same (as in Quadrants I and III), the division \(\frac{y}{x}\) results in a positive number. Conversely, if the signs are different (as in Quadrants II and IV), the result is a negative number. Understanding the signs of the coordinates in a problem helps students quickly establish the results of operations and the truth of inequalities across the four quadrants.