91Ó°ÊÓ

The coordinate plane is a two-dimensional plane that helps us visualize and solve mathematical problems involving variables. It consists of two number lines that intersect at a right angle. The horizontal line is called the "x-axis," and the vertical line is known as the "y-axis." These two axes split the plane into four sections, called "quadrants." Each point on this plane can be defined by a unique pair of numbers which are known as coordinates.

In mathematical terms, coordinates are written as ordered pairs \((x, y)\), where "x" represents the position on the horizontal axis and "y" on the vertical axis. The point where the two axes intersect is called the origin, designated as (0, 0). Understanding the coordinate plane is crucial when working with inequalities because it provides a clear representation of where different solutions can exist.

Negative fractions are fractions where the numerator and the denominator have opposite signs. This results in a value that is less than zero. In the context of the inequality \(\frac{y}{x} < 0\), we need to find scenarios where this holds true.

For a fraction to be negative:

• The numerator \(y\) should be positive when the denominator \(x\) is negative.
• Alternatively, the numerator \(y\) should be negative when the denominator \(x\) is positive.

While dealing with fractions on a coordinate plane, knowing how to determine the sign of the fraction is essential because it influences the regions where the solutions to inequalities will be found. In our inequality case, \(\frac{y}{x}<0\) suggests that one value is positive and the other is negative, guiding us to specific quadrants.

The coordinate plane is divided into four quadrants. Each quadrant represents a distinct combination of positive and negative values for \(x\) and \(y\). Understanding these quadrants allows us to determine where specific inequality expressions are true.

The quadrants are:

• Quadrant I: \((x > 0, y > 0)\)
• Quadrant II: \((x < 0, y > 0)\)
• Quadrant III: \((x < 0, y < 0)\)
• Quadrant IV: \((x > 0, y < 0)\)

For the inequality \(\frac{y}{x} < 0\), we need points where the coordinates have opposite signs. These are located in:

• Quadrant II, where \(x\) is negative and \(y\) is positive.
• Quadrant IV, where \(x\) is positive and \(y\) is negative.

Identifying these quadrants helps to solve inequalities by pinpointing the areas on the plane that meet specific conditions.