Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-2: If \(x<0\) and \(y>0\), determine the sign of the real number. (a) \(\frac{x}{y}\) (b) \(x y^{2}\) (c) \(\frac{x-y}{x y}\) (d) \(y(y-x)\)

Step-by-Step Solution

Answer

(a) Negative, (b) Negative, (c) Positive, (d) Positive.

Step 1: Analyzing Sign of Fraction

Given that \(x < 0\) and \(y > 0\), we begin by examining the expression \(\frac{x}{y}\). Since \(x\) is negative and \(y\) is positive, dividing a negative number by a positive number yields a negative result. Therefore, the sign of \(\frac{x}{y}\) is negative.

Step 2: Analyzing Sign of Product

Next, let's look at the expression \(x y^2\). Since \(x < 0\) is negative and \(y > 0\), raising \(y\) to any power (like \(y^2\)) keeps it positive. The product of a negative number (\(x\)) and a positive number (\(y^2\)) is negative. Thus, \(x y^2\) is negative.

Step 3: Complex Fraction Analysis

Now consider \(\frac{x-y}{xy}\). For \(x-y\), \(x < y\) because \(x < 0\) and \(y > 0\), so \(x-y\) is negative. For \(xy\), since \(x\) is negative and \(y\) is positive, the product \(xy\) is negative as well. Dividing a negative number by a negative number yields a positive result. Therefore, \(\frac{x-y}{xy}\) is positive.

Step 4: Product of Sum and Variable

Finally, we analyze \(y(y-x)\). Note that \(y-x\) is positive because \(y > x\). As \(y\) is also positive, the product of two positive numbers is positive. Therefore, \(y(y-x)\) is positive.

Key Concepts

Sign of Numbers

Understanding the sign of numbers is crucial when working with algebraic expressions. A number's sign can be either positive or negative. In algebra, the context of a problem often dictates the sign of variables like \( x \) and \( y \). For example, when we say \( x < 0 \), \( x \) is negative, whereas \( y > 0 \) indicates \( y \) is positive.

The combination of positive and negative signs in an expression can affect the overall result:

• Negative divided by positive equals negative.
• Positive times positive equals positive.
• Negative times positive equals negative.
• Negative divided by negative equals positive.

Remembering these rules helps greatly when analyzing the sign of algebraic expressions, as it's essential to first determine each component's sign before solving.

Inequalities

Inequalities are expressions that describe the relative size or order of two values. If an expression states \( x < 0 \) and \( y > 0 \), it is asserting that \( x \) is less than zero, and \( y \) is greater than zero.

Inequalities allow us to understand the relationships between numbers and thus determine the sign of more complex algebraic expressions. They are used to:

• Establish the direction of the sign (e.g., \(-5 < 0\), \(7 > 0\)).
• Guide how operations like multiplication or division affect overall outcomes.

It’s crucial to interpret inequalities correctly in order to predict the sign of an outcome. For instance, knowing \( x < y \) directly implies \( x-y \) is negative, or knowing \( y > x \) suggests \( y-x \) results in a positive value.

Fraction Analysis

Analyzing fractions involves more than just dividing two numbers; it also requires paying careful attention to the signs. Fractions like \( \frac{x}{y} \) represent division, and therefore the rules of sign division apply.

Key points in fraction analysis include:

• If the numerator (top) is negative and the denominator (bottom) is positive, the entire fraction is negative.
• If both the numerator and the denominator are negative, the fraction becomes positive.

Let's consider something more complex like \( \frac{x-y}{xy} \). For the numerator \( x-y \), the terms \( x \) (negative) and \( y \) (positive) result in a negative numerator. Since both \( x \) and \( y \) maintain the same signs, the product \( xy \) is negative, and hence the fraction \( \frac{x-y}{xy} \) becomes positive because a negative divided by a negative gives a positive result.

Product Analysis

Product analysis involves examining the factors that constitute a multiplication expression. When determining the sign of a product, we consider each factor's sign. In algebraic expressions like \( x y^2 \), we multiply \( x \) and \( y^2 \).

Factors in product analysis can be:

• Positive, if all factors have the same sign.
• Negative, if the factors have different signs.

Let's review \( y(y-x) \):

• \( y \) is positive as per our condition \( y > 0 \).
• \( y-x \), since \( y > x \) implies it's positive.

Thus, both factors in \( y(y-x) \) are positive, making the entire expression positive. On the other hand, in \( x y^2 \), since \( x \) is negative and \( y^2 \) is positive, the result is negative. Understanding the interaction between factors is key.