1.4 Additive Inverses and Absolute Values
Learning Objectives
- Understand that any number plus plus its opposite (its additive inverse) is always zero.
- Express what absolute values are and use them in expressions and equations.
Introduction
Suppose that you are creating a budget and that you have expenses of $2,500 per month. How much money would you have to bring in each month in order to break even? Would the expenses be thought of as a positive or negative number? What would be the additive inverse of the expenses? What would be the absolute value? After finishing this Concept, you'll be able to answer questions such as these so that you don't break your budget!
Guided Learning
Graph and Compare Integers
More specific than the rational numbers are the integers. Integers are whole numbers and their negatives. When comparing integers, you will use the math verbs such as less than, greater than, approximately equal to, and equal to. To graph an integer on a number line, place a dot above the number you want to represent.
Example A
Compare the numbers 2 and –5.
Solution: First, we will plot the two numbers on a number line.
We can compare integers by noting which is the greatest and which is the least. The greatest number is farthest to the right, and the least is farthest to the left.
In the diagram above, we can see that 2 is farther to the right on the number line than –5, so we say that 2 is greater than –5. We use the symbol > to mean “greater than.”
Therefore, .
Numbers and Their Opposites
Every real number, including integers, has an opposite, which represents the same distance from zero but in the other direction.
A special situation arises when adding a number to its opposite. The sum is zero. This is summarized in the following property.
The Additive Inverse Property: For any real number .
We see that is the additive inverse, or opposite, of .
Example B
Find the opposite number of the following numbers. Use the Additive Inverse Property to show that they are opposites.
a.) -5
b.) 1/2
c.) 5.1
Solutions:
a.) The opposite number of -5 is 5. Using the Additive Inverse Property: .
b.) The opposite number of 1/2 is -1/2. The Additive Inverse Property shows us that they are opposites: .
c.) The opposite number of 5.1 is -5.1. The Additive Inverse Property gives: 5.1 + (-5.1)=0.
Absolute Value
Absolute value represents the distance from zero when graphed on a number line. For example, the number 7 is 7 units away from zero. The number –7 is also 7 units away from zero. The absolute value of a number is the distance it is from zero, so the absolute value of 7 and the absolute value of –7 are both 7. A number and its additive inverse are always the same distance from zero, and so they have the same absolute value.
We write the absolute value of –7 like this: |-7|.
We read the expression |x| like this: “the absolute value of x.”
- Treat absolute value expressions like parentheses. If there is an operation inside the absolute value symbols, evaluate that operation first.
- The absolute value of a number or an expression is always positive or zero. It cannot be negative. With absolute value, we are only interested in how far a number is from zero, not the direction.
Example C
Evaluate the following absolute value expressions.
a) |5 + 4|
b) 3 - |4 - 9|
c) |-5 - 11|
d) |-5 + 10|
Solution:
a) 9
b) -2
c) 16
d) 5
Review
Opposite
For all real numbers there is a number equal distance from zero on the other side of the number line.
Integers
The set of whole numbers and their negatives.
Absolute value
Represents the distance from zero when graphed on a number line.
Additive Inverse Property:
Any number plus it’s opposite is always zero
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