Problem 8 - Beginning and Intermediate Algebra

Question

Suppose that \(x\) represents a positive number and \(y\) represents a negative \mathrm{number. Determine whether the given expression represents a positive or } a \text { negative number. } $$ y-x $$

Step-by-Step Solution

Answer

The expression y - x represents a negative number.

Step 1: Identify the signs of the variables

Recognize that in the expression, variable x represents a positive number and variable y represents a negative number.

Step 2: Substitute variables with their respective signs

Substitute x with a positive number and y with a negative number. For simplicity, let’s use x = a (where a > 0) and y = -b (where b > 0).

Step 3: Simplify the expression

Substitute the values: y - x becomes -b - a. Both b and a are positive, so -b - a is negative.

Step 4: Draw a conclusion

Since the expression -b - a is a sum of two negative numbers, the result is negative.

Key Concepts

positive and negative numbers

Positive and negative numbers play a fundamental role in algebraic expressions. Understanding this concept is crucial when simplifying expressions and solving equations.

Positive numbers are greater than zero and lie to the right of zero on the number line. Examples include 1, 2, and 100. Negative numbers are less than zero and lie to the left of zero on the number line. Examples include -1, -2, and -100.

When combining positive and negative numbers, recall these important rules:

• Adding a positive number to a negative number can be seen as moving right and then left on the number line. This can result in either a positive or negative number.
• Subtracting a positive number from a negative number intensifies the negative value, pushing it further left on the number line.
• Adding two negative numbers results in a more negative number, as their combined value goes further left on the number line.

Understanding these rules will help you determine whether an expression involving both positive and negative numbers results in a positive or negative outcome.

substitution

Substitution is the process of replacing variables with actual numbers or values. It is a common step in solving algebraic expressions.

In the original exercise, we are given that x is a positive number and y is a negative number. By substituting these with specific positive and negative values, we clarify the expression's outcome.

For example:

• Let x = a, where a > 0 (positive)
• Let y = -b, where b > 0 (negative)

By substituting these into the expression y - x, it becomes -b - a. This clearly shows both terms are negative.

Substitution helps simplify the understanding of how positive and negative numbers interact within an expression. Through correctly applying substitution, we can proceed to simplify the expression more easily.

simplifying expressions

Simplifying expressions involves combining and reducing terms to their simplest form, which makes it easier to understand and solve.

In the context of the exercise, after substitution, we have the expression -b - a. Here's how to simplify it:

• First, notice that both terms are negative.
• Combine the negative terms by simply adding their absolute values and keeping the negative sign.

Therefore, -b - a becomes a single negative value, which is interpreted as the sum of both absolute values but with a negative sign.

To conclude, simplifying expressions allows us to reduce complex equations into simpler forms, making it easier to understand and solve them. Applying these simplification techniques consistently can lead to quicker and more accurate solutions in algebra.