Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-2: 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: Determine the sign of xy

Since \( x < 0 \) and \( y > 0 \), multiplying a negative number (\( x \)) by a positive number (\( y \)) results in a negative number. Thus, the sign of \( xy \) is negative.

Step 2: Determine the sign of x^2 y

\( x^2 \) is the square of \( x \), and squaring any real number, regardless of its sign, results in a non-negative number. Therefore, \( x^2 \geq0 \). Since \( y > 0 \), the product \( x^2 y \) will be the product of a non-negative number and a positive number, which is positive. Therefore, the sign of \( x^2 y \) is positive.

Step 3: Determine the sign of \(\frac{x}{y}+x\)

First, consider the term \(\frac{x}{y}\). Since \( x < 0 \) and \( y > 0 \), \(\frac{x}{y}\) is negative, similar to \( xy \). Adding \( x \) (also negative) to this negative fraction results in a further negative value. Therefore, \( \frac{x}{y} + x \) is negative.

Step 4: Determine the sign of y-x

The expression \( y - x \) involves subtracting a negative \( x \) from positive \( y \), which can be considered as \( y + (-x) \). Simplifying this, it becomes \( y + |x| \), which is clearly positive since you are adding two positive numbers. Therefore, the sign of \( y - x \) is positive.

Key Concepts

Real Numbers

Real numbers include all the numbers you can think of, from positive, negative, zero, and including fractions and decimals. They form a continuous line on the number line without any gaps. Here are some important characteristics to remember about real numbers:

• Positive numbers: Numbers greater than zero, found to the right of zero on the number line.
• Negative numbers: Numbers less than zero, found to the left of zero on the number line.
• Zero: Neither positive nor negative, centered on the number line.
• Fractions and decimals: Also considered real numbers, these represent numbers between whole numbers.

When dealing with real numbers in inequalities, such as "if \( x < 0\)", it means that \( x\) is a negative number. Understanding that \( y > 0\) signifies \( y\) is a positive number is key in interacting with inequalities.

Multiplication of Signed Numbers

Multiplying signed numbers can change the sign of the outcome based on the rules of multiplication. If you multiply two numbers with different signs, the result will always be negative. Here's why:

• Positive \( \times \) Positive = Positive
• Negative \( \times \) Negative = Positive (because two negatives cancel each other out)
• Positive \( \times \) Negative = Negative (and vice versa, Negative \( \times \) Positive = Negative)

For example, if we have \( x < 0\) and \( y > 0\), when these numbers are multiplied \( xy\), the result will be a negative number as one is negative and the other is positive.

Addition and Subtraction of Signed Numbers

Adding and subtracting signed numbers requires careful attention to their signs and how these influence the result:

• Adding two positive numbers: Always gives a positive result.
• Adding two negative numbers: Results in a negative number; for example, \(-3 + (-2) = -5\).
• Adding a positive and a negative number: The result depends on which has the larger absolute value. Subtract the smaller from the larger and keep the sign of the larger number.
• Subtracting a negative number: Equivalent to adding its positive opposite, e.g., \( y - x\) becomes \( y + |x|\) if \( x < 0\), enhancing the value positively.

For instance, \( \frac{x}{y} + x\) yields a negative sum because both terms are negative if \( x < 0\). Likewise, in \( y - x\), the subtraction of a negative is the same as adding a positive, resulting in a positive sum.

Squaring Numbers

Squaring a number means multiplying it by itself. This operation has a unique property: the square of any real number is always non-negative. Let's explore:

• Squaring Positive Numbers: The result is positive because multiplying two positive numbers yields a positive product.
• Squaring Negative Numbers: The result is also positive because multiplying two negative numbers results in a positive.

Therefore, if \( x < 0\), then \( x^2\) will still be a positive number since \((-x) \times (-x)\) or \(x \times x\) results in a positive value. This property is critical when assessing expressions like \( x^2 y\), which, given \( y > 0\), remains positive because the product of a non-negative and a positive number is always positive. Hence, \( x^2 y\) is positive.