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The coordinate plane is a two-dimensional surface defined by a horizontal axis, called the x-axis, and a vertical axis, called the y-axis. These two axes intersect at a point known as the origin, which has coordinates (0, 0).

On the coordinate plane, each point is represented by a pair of numbers, \(x, y\), which show its position relative to the origin. The x-coordinate measures horizontal distance from the origin, while the y-coordinate measures vertical distance.

The coordinate plane is a valuable tool in mathematics for visually representing equations and relationships between numbers. It helps make sense of algebraic equations and inequalities by providing a visual format. Whether plotting simple points or more complex curves, the coordinate plane offers a straightforward way to study and understand analytical geometry.

In the coordinate plane, the x-axis and y-axis divide the plane into four sections called quadrants. These quadrants are like the four quarters of a pie, each with distinct sign combinations for coordinates.

Quadrants are numbered starting from the "top-right" and proceeding counterclockwise:

• First Quadrant: This quadrant is where both x and y coordinates are positive \(x > 0, y > 0\).
• Second Quadrant: Here, x is negative and y is positive \(x < 0, y > 0\). This is one area of interest for our inequality because the product \(xy\) is negative here.
• Third Quadrant: In this quadrant, both x and y are negative \(x < 0, y < 0\).
• Fourth Quadrant: This quadrant features a positive x and a negative y \(x > 0, y < 0\). It is also significant because \(xy\) results in a negative product here.

Understanding quadrants is crucial because it helps us determine the location and behavior of points described by inequalities such as \(xy < 0\).

Inequalities are mathematical expressions that relate two values or expressions that are not necessarily equal but instead use symbols like \( <, >, \leq, \geq\). They are used to compare numbers or expressions to find out if one is greater, less, or simply not equal to the other.

In the case of our problem, \(xy < 0\), the inequality tells us that we are looking for points where the product of x and y is less than zero. Essentially, this means the x and y coordinates must have opposite signs.

To find where \(xy < 0\) on the coordinate plane, we consider the signs of x and y in the four quadrants:

• In the first and third quadrants, \(xy > 0\) because both coordinates are either positive or negative, respectively, making the product positive.
• In the second quadrant, \(x < 0\) and \(y > 0\), resulting in a negative product.
• In the fourth quadrant, \(x > 0\) and \(y < 0\), also resulting in a negative product.

Once you grasp the basic concept of inequalities, you can use them to determine and sketch regions on the coordinate plane where certain conditions are met, providing a deeper understanding of spatial relationships and relationships between functions.