Problem 35 - A Graphical Approach to College Algebra

Question

Name the possible quadrants in which the point ( \(x, y\) ) can lie if the given condition is true. $$x y>0$$

Step-by-Step Solution

Answer

The point (x, y) can lie in Quadrants I and III.

Step 1: Understand the Condition

The condition given is \(xy > 0\). This means that the product of \(x\) and \(y\) is positive. A product is positive when both numbers have the same sign, either both are positive or both are negative.

Step 2: Analyze Quadrants for Positive Values

Quadrant I is where both \(x\) and \(y\) are positive. Therefore, if \(x > 0\) and \(y > 0\), their product \(xy > 0\), which satisfies the condition. Hence, Quadrant I is a possible solution.

Step 3: Analyze Quadrants for Negative Values

Quadrant III is where both \(x\) and \(y\) are negative. In this case, \(xy > 0\) because the product of two negative numbers is positive. Thus, Quadrant III also satisfies the condition.

Step 4: Exclude Other Quadrants

Quadrant II has \(x < 0\) and \(y > 0\), which gives \(xy < 0\) (product of a negative and positive is negative). Quadrant IV has \(x > 0\) and \(y < 0\), which also gives \(xy < 0\). Therefore, these quadrants do not meet the condition.

Key Concepts

Positive and Negative Numbers

In the coordinate plane, numbers can be either positive or negative, affecting their placement within various quadrants. Let's break it down:- **Positive Numbers**: These are numbers greater than zero. On a coordinate plane, positive values of \(x\) or \(y\) mean the point is located to the right of the vertical axis for \(x\) and above the horizontal axis for \(y\).- **Negative Numbers**: These are numbers less than zero. Negative values of \(x\) put a point to the left of the vertical axis, while negative values of \(y\) place a point below the horizontal axis.The sign of the numbers determines not only their position on the axes but also their interaction with other numbers. For example:- When two positive numbers are multiplied, the result is a positive number.- Similarly, when multiplying two negative numbers, the result is also positive.Understanding how positive and negative numbers interact is crucial in solving problems related to quadrants in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures using the coordinate plane. The plane is divided into four quadrants with specific rules about the signs of numbers:- **Quadrant I**: Here, \(x > 0\) and \(y > 0\). Points in this quadrant have both coordinates positive.- **Quadrant II**: In this region, \(x < 0\) and \(y > 0\) signify one positive and one negative coordinate.- **Quadrant III**: Both coordinates, \(x < 0\) and \(y < 0\), are negative in this quadrant.- **Quadrant IV**: This quadrant features \(x > 0\) and \(y < 0\), showing one positive and one negative coordinate.In coordinate geometry, we often use these rules to perform operations like locating points, determining distances, or solving equations. By analyzing the signs of \(x\) and \(y\), we can identify the quadrant in which a point lies, enabling us to determine various mathematical properties related to that point.

Quadrant Analysis

Quadrant analysis is a method we use to identify which quadrant a point with coordinates \( (x, y) \) belongs to, based on the signs of \(x\) and \(y\). When given a condition like \(xy > 0\), we know we are looking for quadrants where the multiplication of these coordinates yields a positive result.- **Condition**: The requirement \(xy > 0\) implies: - Both \(x\) and \(y\) are positive, or - Both \(x\) and \(y\) are negative.Through this analysis, we find:- **Quadrant I**: Since both coordinates are positive here, their product is also positive. Thus, a point (\(x, y\)) where \(x > 0\) and \(y > 0\) fits this condition.- **Quadrant III**: Here, both coordinates are negative, so the product \(xy\) remains positive. A point with \(x < 0\) and \(y < 0\) also satisfies \(xy > 0\).Other quadrants, such as Quadrant II and Quadrant IV, do not satisfy \(xy > 0\) because multiplying a positive by a negative produces a negative result. Thus, quadrant analysis helps us quickly ascertain where certain mathematical conditions hold true in the coordinate plane.