Problem 1 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

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

Step 1: Analyze the expression for (a)

For expression \(xy\), we know that \(x < 0\) implies \(x\) is a negative number, and \(y > 0\) implies \(y\) is a positive number. The product of a negative number and a positive number is negative. Thus, \(xy < 0\).

Step 2: Analyze the expression for (b)

For \(x^{2}y\), note that \(x^{2}\) is the square of a negative number, which results in a positive number (since any real number squared is positive). Given \(y > 0\), the product of two positive numbers \(x^{2}\) and \(y\) is positive. Therefore, \(x^{2}y > 0\).

Step 3: Analyze the expression for (c)

Consider the expression \(\frac{x}{y} + x\). First, observe that \(\frac{x}{y}\) results in a negative number because \(x < 0\) (negative) and \(y > 0\) (positive), hence \(\frac{x}{y} < 0\). Adding \(x\) (which is also negative) to another negative number, \(\frac{x}{y} + x\), results in a negative expression. Thus, \(\frac{x}{y} + x < 0\).

Step 4: Analyze the expression for (d)

For \(y - x\), we know \(y > 0\) and \(-x > 0\) because \(x < 0\). Subtracting a negative number is equivalent to adding its positive, so \(y - x = y + (-x) > 0\) because both \(y\) and \(-x\) are positive, making the entire expression positive.

Key Concepts

Product of Numbers

When we talk about the product of numbers, it involves multiplying two or more numbers together. Understanding the signs of numbers involved is key in determining if their product is positive or negative. A useful rule to remember is:

• The product of two positive numbers is positive.
• The product of two negative numbers is positive, as the negatives cancel each other out.
• However, the product of a positive number and a negative number is always negative, because the negativity does not cancel out.

For example, if we consider the expression \(x y\) where \(x < 0\) and \(y > 0\), since one number is negative and the other is positive, the product will be negative \(x y < 0\). This rule is consistent and helps in solving such problems efficiently.

Square of a Number

The square of a number refers to multiplying the number by itself. An important characteristic of squaring a number is the impact on its sign:

• The square of a positive number is positive.
• Interestingly, the square of a negative number is also positive, because multiplying two negative numbers results in a positive number.

For instance, in the expression \(x^2 y\) where \(x < 0\) and \(y > 0\), squaring \(x\) changes it from negative to positive \(x^2 > 0\). Thus, multiplying \(x^2\) by a positive \(y\) gives a positive result \(x^2 y > 0\). Remember, squaring a number always results in a non-negative value (either positive or zero). This is particularly vital in solving and understanding algebraic problems.

Negative and Positive Numbers

Understanding the interplay between negative and positive numbers underpins many concepts in algebra. The behavior of sums and differences of such numbers can greatly affect the result:

• Adding two positive numbers gives a positive result.
• Adding two negative numbers gives a negative result.
• Adding a positive and a negative number depends on their absolute values; the number with the larger absolute value determines the sign.
• Double negatives turn into a positive effect when subtracted; that is, subtracting a negative is the same as adding a positive.

Consider the expression \( y - x \). Since \( y > 0 \) and \( x < 0 \), subtracting \( x \) (a negative number) is like adding its positive equivalent, which leads to a positive result \( y + (-x) > 0 \). This approach helps simplify calculations involving negative and positive terms.

Fraction and Sum of Numbers

Fractions are expressions representing divisions of numbers and understanding how fractions behave with negative and positive numbers is crucial. A fraction is negative if only one of the numerator or denominator is negative, and positive if both are positive or negative.

• If the numerator is negative and the denominator is positive, the fraction is negative.
• The sum of two negative numbers is always negative.
• The sum of a negative and a positive number depends on their magnitudes.

In the expression \( \frac{x}{y} + x \), we have \( x < 0 \) and \( y > 0 \), making \( \frac{x}{y} \) negative. Adding \(x\) (also negative) to \( \frac{x}{y} \) results in an expression that remains negative \, \( \frac{x}{y} + x < 0 \). Understanding fractions in this way can greatly aid in solving complex algebraic equations.