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Chapter 1: Problem 142

If \(n\) is positive and \(m\) is negative, what is the sign of \(n+(-m) ?\) Why?

Short Answer

Expert verified
The sign is positive because adding two positive numbers results in a positive number.

Step by step solution

01

Understand the Given Conditions

The problem states that n is a positive number (greater than 0) and m is a negative number (less than 0). We need to find the sign of the expression n + (-m).
02

Simplify the Expression

Rewrite the expression n + (-m). Here, -m means the opposite of m. Since m is negative, -m will be positive. For example, if m = -5, then -m = 5. Thus, n + (-m) simplifies to n + a positive number.
03

Determine the Sign

Since n is a positive number and -m is also a positive number, adding these two positive numbers will result in a positive number. Therefore, the sign of n + (-m) is positive.
04

Explain the Result

The intuition behind this is that when a positive number is added to another positive number, the result will always be positive. In this case, n is positive, and -m, being the opposite of a negative number m, is also positive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

positive and negative numbers
In elementary algebra, dealing with positive and negative numbers is a fundamental concept. A positive number is any number greater than zero, while a negative number is any number less than zero. These numbers behave differently when performing various operations.
When you see a negative sign in front of a number, it signifies that the number is on the opposite side of zero on a number line compared to positive numbers. For example:
  • A positive number: 7
  • A negative number: -3
  • The opposite of -3 is 3
Understanding the behavior of positive and negative numbers helps in solving and simplifying expressions.
simplification of expressions
The simplification of expressions is a key skill in algebra that involves combining like terms to make an equation easier to work with. For instance, the expression given in the problem is n + (-m).
First, recall that if m is a negative number, then -m will be positive. This is because the negative sign in front of m reverses its sign. For example, if m = -8, then -m = 8.
  • Original expression: n + (-m)
  • If m = -8, then -m = 8
  • So, n + (-m) becomes n + 8
In this case, the expression simplifies to adding two positive numbers together.
addition of integers
Adding integers, both positive and negative, follows straightforward rules. When you add two positive numbers together, their sum is always positive. For example, if you add 2 and 3, the result is 5.
Let's look at the exercise where we have n (a positive integer) and -m (m is negative, so -m is positive):
  • If n = 7 and m = -4
  • Then -m becomes 4
  • So n + (-m) turns into 7 + 4 = 11
Thus, the sum of two positive integers will always yield a positive result. This aligns with our initial conclusion in the problem where the sign of n + (-m) is positive.

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