Problem 101 - 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. $$ p-n \text { is always a positive number. } $$

Step-by-Step Solution

Answer

True. The expression \( p-n \) simplifies to \( p + |n| \) which is positive.

Step 1: Understanding Terms

First, identify the properties of the variables given. Here, \( p \) is a positive number, meaning that \( p > 0 \). Also, \( n \) is a negative number, so \( n < 0 \).

Step 2: Write the Expression

The exercise asks about the expression \( p - n \). Substituting the known properties of \( n \), rewrite it as \( p - (-n) \). This becomes \( p + n \) since subtracting a negative number is the same as adding its positive counterpart.

Step 3: Analyzing the Expression

Since \( p > 0 \) and \( n < 0 \), then the absolute value of \( n \), which is \( -n \), is a positive number. Hence, \( p + |n| \) is always positive because it is the sum of two positive numbers.

Step 4: Conclusion

Having determined that \( p + |n| \) is the result of a positive number added to another positive number, the expression \( p-n \) is always positive. Thus, the statement is true.

Key Concepts

Positive and Negative Numbers

Positive and negative numbers form the basic building blocks of algebraic expressions. Positive numbers are always greater than zero. Negative numbers are always less than zero. To write a positive number, we use it as-is without any symbol, like this: 2, 5, or 8. To write a negative number, we place a minus sign before the number, such as -3, -7, or -9.

Understanding how to perform operations with these numbers is crucial. For instance, when subtracting a negative number, the two negatives combine to make a positive. This changes the operation into addition. When you have a situation like the expression stated in the exercise with $p - n$, where $p$ is positive and $n$ is negative, it turns into $p + (-n)$, which is the same as $p + |n|$.

So remember:

• Positive minus Negative = Positive + Positive
• Always consider the sign change when dealing with negative numbers in subtraction.

Properties of Numbers

Numbers have several properties that help us manipulate and simplify expressions. These properties include the commutative, associative, and distributive properties. While all these properties can help in different scenarios, let’s focus on a few specific ideas relevant to the problem:

• **Commutative Property:** For addition and multiplication, the order of the numbers does not matter. For example, $a + b = b + a$.
• **Addition of Opposites:** Subtracting a negative number is the same as adding its positive equivalent. Hence $a + (-b)$ equals $a - b$.
• **Positive & Negative Relationship:** When a positive number is added to another positive number, the result is always positive.

When working with expressions like $p - n$, understanding these properties is vital as they assure that certain operations yield consistent results. For example, if you were dealing with a numerical value problem, applying these properties ensures that the outcome behaves predictably.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form without changing their value. This makes understanding and solving them easier. It requires keen attention to the signs and properties involved.

Take the expression $p - n$ from the exercise. We've learned that subtracting a negative number, $n$, is the same as adding its positive counterpart. Thus, the expression simplifies to $p + |n|$. Simplifying involves recognizing such situations and applying algebraic rules to make expressions straightforward.

Some tips for simplifying expressions include:

• Look for terms that can cancel out.
• Combine like terms to reduce complexity.
• Use already known properties effectively to streamline the process.

Simplifying helps not only in manual calculations but also helps in developing a deeper understanding of how expressions and numbers work together.