Problem 75 - College Algebra Essentials

Question

List the quadrant or quadrants satisfying each condition. $$ x y>0 $$

Step-by-Step Solution

Answer

The condition \(x y > 0\) is satisfied in Quadrant I and Quadrant III.

Step 1: Identify the signs of \(x\) and \(y\) in each quadrant

In Quadrant I, both \(x\) and \(y\) are positive (+/+), so the product \(x y\) will be positive. In Quadrant II, \(x\) is negative and \(y\) is positive (-/+), so the product \(x y\) will be negative. In Quadrant III, \(x\) and \(y\) are both negative (-/-), meaning the product \(x y\) will be positive. In Quadrant IV, \(x\) is positive and \(y\) is negative (+/-), therefore the product \(x y\) will be negative.

Step 2: Determine which quadrants satisfy the condition

The product of \(x\) and \(y\) is positive in Quadrants I and III because in both of these quadrants \(x\) and \(y\) have the same sign. In the other two quadrants, Quadrant II where \(x\) is negative and \(y\) is positive, and Quadrant IV where \(x\) is positive and \(y\) is negative, the product would be negative. Thus, the condition \(x y > 0\) is satisfied in Quadrant I and Quadrant III.

Key Concepts

Inequalities in Algebra

When studying inequalities in algebra, it is crucial to recognize the relationship between variables. An inequality indicates a comparative relation between two expressions, showing that one is less than, greater than, less than or equal to, or greater than or equal to the other. It's a way to express the notion that two sides are not equal. For instance, the inequality \( xy > 0 \) states that the product of \( x \) and \( y \) is more than zero.

To solve such inequalities, one must consider the possible values of \( x \) and \( y \) that make the inequality true. In the context of a coordinate plane, this involves analyzing the signs of the variables in different quadrants. Notably, the inequality does not need to involve a strict greater than (>) or less than (<) sign; it could also include \'greater than or equal to\' (\( \geq \)) or \'less than or equal to\' (\( \leq \)). Understanding inequalities is essential in multiple areas of algebra, from solving simple equations to graphing linear inequalities on the coordinate plane.

Quadrant Analysis

The coordinate plane is divided into four areas, known as quadrants, which are numbered counterclockwise starting from the upper right quadrant.

Quadrant I (QI):

Both the \( x \)- and \( y \)-coordinates are positive.

Quadrant II (QII):

Here, \( x \)-coordinates are negative, while \( y \)-coordinates are positive.

Quadrant III (QIII):

Both coordinates are negative in this quadrant.

Quadrant IV (QIV):

\( x \)-coordinates are positive, but \( y \)-coordinates are negative.

Knowing the signs of coordinates in each quadrant, you can determine the sign of the product \( xy \) in any given quadrant. This is particularly useful when dealing with inequalities that involve products of variables, such as \( xy > 0 \). Such an inequality would only be true in quadrants where both coordinates have the same sign and thus their product is positive, which in this case refers to Quadrant I and Quadrant III.

Signs of Products in Algebra

In algebra, the sign of a product is determined by the signs of the factors being multiplied. When two numbers of the same sign are multiplied (both positive or both negative), the result is always positive. Conversely, when two numbers of different signs are multiplied, the result is negative.

This rule is reflected on the coordinate plane when multiplying the values of \( x \) and \( y \) within each quadrant. For instance:

• In Quadrants I and III, where the coordinates have the same sign, their product will yield a positive result.
• In Quadrants II and IV, where the coordinates have differing signs, their product will be negative.

Therefore, understanding the relationship between the signs of factors is critical when working with product-based inequalities like \( xy > 0 \) in algebra.