Problem 102 - Algebra A Combined Function

Question

If \(p\) is a positive number and \(n\) is a negative number, determine whether each statement is true or false. Explain your answer. $$ n-p \text { is always a negative number. } $$

Step-by-Step Solution

Answer

True; \( n - p \) is always negative since \( n \) is negative and \( p \) is positive.

Step 1: Understand the Given Statement

We are given the statement \( n-p \) and need to determine whether it is always a negative number, under the condition that \( p \) is a positive number and \( n \) is a negative number.

Step 2: Analyze the Expression

The expression \( n-p \) represents a negative number \( n \) subtracted by a positive number \( p \). This operation means taking away a positive value from a negative value, which would further decrease the value.

Step 3: Consider the Mathematical Outcome

Since \( n \) is negative and \( p \) is positive, the expression \( n - p \) simplifies to moving \( n \) further into the negative territory. Formally, if \( n < 0 \) and \( p > 0 \), then \( n - p < n \); since \( n \) is already negative, \( n - p \) will also be negative.

Step 4: Verify with an Example

Consider \( n = -2 \) and \( p = 3 \). Subtract \( p \) from \( n \): \( n - p = -2 - 3 = -5 \), which is negative. No matter the values, as long as \( n \) and \( p \) maintain their signs, \( n - p \) will always be negative.

Step 5: Conclusion

Given the conditions that \( n < 0 \) and \( p > 0 \), the result of \( n-p \) is always negative, thus making the statement true.

Key Concepts

Subtraction

Subtraction is an essential mathematical operation that involves finding the difference between two numbers. In the context of negative numbers, subtraction can be a bit tricky because of the involvement of negative values. Primarily, subtraction can be understood as the act of taking away a certain value from another. This operation is crucial when dealing with both positive and negative numbers:

• Subtracting a positive number from another positive reduces the overall value. For example, \(7 - 3 = 4\).
• Subtracting a positive number from a negative number increases the negativity of the result. For instance, in the expression \(n - p\), where \(n = -2\) and \(p = 3\), the result is \(-2 - 3 = -5\), which is more negative than \(n\) alone.
• Subtracting a negative number is equivalent to adding its positive counterpart, which makes the result more positive.

Understanding subtraction in terms of negative and positive numbers helps us predict whether the result of an expression will be positive or negative.

Positive Numbers

Positive numbers are those greater than zero. They are found to the right of zero on the number line and are fundamental in arithmetic operations. Often represented without a sign, positive numbers impact subtraction significantly when used in combination with negative numbers:

• Subtracting a positive number from another positive one typically yields a smaller positive number, as it involves reducing the total value.
• When a positive number subtracts a negative value, it effectively increases the total value. For example, \(5 - (-3) = 5 + 3 = 8\).
• In the expression \(n - p\) like the problem focuses on, the positive \(p\) enhances the negativity of a negative \(n\) when subtracted, showing how powerful the role of positive numbers can be in various calculations.

Knowing how positive numbers interact in mathematical operations paves the way for correctly solving and understanding complex expressions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, operators, and sometimes variables that represent a mathematical situation or problem. Understanding how these components interact is crucial for solving algebraic problems effectively. When evaluating expressions involving subtraction, like \(n - p\) where \(n\) is negative and \(p\) is positive, you have to consider the nature of negative and positive numbers:

• In expressions, the sequence of operations and the signs of numbers play a crucial role in determining whether the overall result is positive, negative, or zero.
• Mathematical expressions often require simplification by operating in an orderly sequence dictated by the rules of arithmetic operations.
• Using real-life context, such as owing and spending money, can help simplify understanding these abstract expressions. For instance, if you have a \(-2\) (representing debt) and lose 3 more (\(-2 - 3\)), it amplifies the negative debt resulting in \(-5\).

By dissecting mathematical expressions, students can better appreciate the logic behind equations and develop the skills necessary to navigate numeric and algebraic problems efficiently.