Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Name the possible quadrants in which the point ( \(x, y\) ) can lie if the given condition is true. (Hint: Consider the rules for determining the product and the quotient of signed numbers.) $$x y<0$$

Step-by-Step Solution

Answer

The point (x, y) can lie in Quadrants II and IV.

Step 1: Understanding Quadrant Rules

In the Cartesian coordinate system, there are four quadrants. Quadrant I has both coordinates positive ( both \(x > 0\) and \(y > 0\) ), Quadrant II has a negative x-coordinate and positive y-coordinate (\(x < 0, y > 0\) ), Quadrant III has both coordinates negative (\(x < 0, y < 0\) ), and Quadrant IV has a positive x-coordinate and negative y-coordinate (\(x > 0, y < 0\) ).

Step 2: Analyzing the Condition \(xy < 0\)

The condition \(xy < 0\) means that the product of \(x\) and \(y\) is negative. For a product to be negative, one number must be negative, and the other must be positive.

Step 3: Identifying Valid Quadrants

To satisfy \(xy < 0\), we need one positive and one negative coordinate. In Quadrant II, \(x < 0\) and \(y > 0\), the product > 0, so it's not valid. In Quadrant IV, \(x > 0\) and \(y < 0\), the product is < 0, so it's valid. Therefore, Quadrants II and IV are possible.

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system, named after the French mathematician René Descartes, is a two-dimensional plane used to uniquely determine points by numerical coordinates. This system divides the plane into four distinct regions, known as quadrants.

• Quadrant I: Both x and y are positive ( x > 0, y > 0 ).
• Quadrant II: x is negative, y is positive ( x < 0, y > 0 ).
• Quadrant III: Both x and y are negative ( x < 0, y < 0 ).
• Quadrant IV: x is positive, y is negative ( x > 0, y < 0 ).

Understanding this system helps us analyze and solve geometric problems by relating algebraic equations to a graphic representation, giving us a visual insight into the behavior of mathematical functions.

Signed numbers

In mathematics, signed numbers are numbers that can be either positive or negative, indicated by a plus or minus sign.

The sign of a number affects its behavior in calculations:

• Positive numbers are written without a sign or with a plus sign ( x > 0 or +x ).
• Negative numbers are preceded by a minus sign ( x < 0 or -x ).

Signed numbers allow us to express opposites, such as directions (up vs down) or financial statuses (profit vs loss). They are essential when working in a Cartesian coordinate system since the sign of the number helps determine the quadrant in which a point lies. For instance, a negative x-coordinate points to the left, while a positive y-coordinate points upwards.

Product of coordinates

The product of coordinates in a Cartesian plane involves multiplying the x and y values of a point (x, y) . This product provides useful information about the relationship between the coordinates and helps in solving many geometry and algebra problems.

In a general sense:

• If both coordinates are positive, the product xy will also be positive.
• If one coordinate is negative and the other is positive, the result will be a negative product.
• If both coordinates are negative, the product will be positive, since multiplying two negative numbers results in a positive number.

Understanding how products work between coordinates allows us to infer the nature of their signs and further predict in which quadrant the point can lie.

Negative product condition

In the context of coordinate geometry, when the condition is set as xy < 0 , we are looking for situations where the product of x and y results in a negative number.

For a product to be negative, it is essential that the two values involved - x and y - have opposite signs.

• If x is positive and y is negative, as in Quadrant IV ( x > 0, y < 0 ), the product will indeed be negative.
• Alternatively, if x is negative and y is positive, as in Quadrant II ( x < 0, y > 0 ), the result will similarly be negative.

Recognizing this condition is valuable when solving problems that require determining possible coordinate properties based on equations or inequalities. So, with the condition xy < 0 , the points must strictly lie in those quadrants where one value is positive and the other negative, namely, Quadrants II and IV.