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Chapter 1: Problem 56

Find the opposite, or additive inverse. $$ -11 $$

Short Answer

Expert verified
The additive inverse of -11 is 11.

Step by step solution

01

Understanding the opposite or additive inverse

The opposite, or additive inverse, of a number is that number which, when added to the original number, results in zero. Essentially, you need to find the number that when combined with -11 gives zero.
02

Identify the additive inverse

Consider the number -11. To find its opposite or additive inverse, we need a number that satisfies the equation: -11 + x = 0.
03

Solve for the additive inverse

To solve for x, isolate x on one side of the equation: -11 + x = 0. This can be rewritten as x = 11. Therefore, the additive inverse of -11 is 11.

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Key Concepts

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

Opposite Numbers
Opposite numbers are numbers that are the same distance from zero on a number line but in opposite directions. For any number 'a,' its opposite is '-a.' For example, the opposite of 5 is -5.
Opposite numbers are crucial when discussing additive inverses. The additive inverse refers to what you need to add to a number to get zero. For -11, the opposite number is 11 because -11 + 11 = 0.
Understanding opposite numbers helps in solving problems involving additive inverses and equations.
  • Opposite number of 7 is -7
  • Opposite number of -12 is 12
Solving Equations
Solving equations involves finding the value of a variable that makes the equation true. Let's look specifically at additive inverses in equations.
We have the equation:
\(-11 + x = 0\)
To isolate x, we must add 11 to -11, so we get \(x = 11\).
Solving equations often starts by isolating the variable, using inverse operations to cancel out terms on either side. For instance, in
\(a + b = c\), subtracting a from both sides gives \(b = c - a\).
  • Identify the variable.
  • Isolate the variable.
  • Perform inverse operations as needed.
Mastery of solving equations simplifies a broad range of math problems.
Integer Properties
Integers are whole numbers, including negative numbers, zero, and positive numbers. Key properties of integers include:
  • Integers are closed under addition, subtraction, and multiplication. This means adding, subtracting, or multiplying any two integers results in another integer.
  • The additive inverse of any integer 'a' is '-a.'
  • Zero is the only integer that is neither positive nor negative and serves as the identity element for addition (i.e., any number plus zero equals the number itself).
Knowing the properties of integers helps understand the behavior of numbers in various mathematical contexts.
For example, knowing that \(-11 + 11 = 0\) relies on the property that integers are closed under addition and the concept of additive inverses.

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