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Chapter 2—Integers

2-6 Combining Integer Addition and Subtraction

 When you complete the work for this section, you should be able to: Add a series of three or more integers Subtract a series of three or more integers Perform combinations of addition and subtraction operations on a series of three or more integers

When you are working in one of the modern trades, technologies, and business, it isn't unusual to find problems that require you to add and subtract a series of signed integers. This isn't always a simple 2 + 2 = 4 kind of world. Often, it is more like a  2 + 4 – 8 + ( –1) = –3 sort of world.   This lesson will help you make sure you understand how to handle problems of this kind.

You probably have no trouble solving this example:

2 + 3 + 1 = 6

1. You combine the first two terms: 2 + 3 + 1 = 5 + 1
2. Then you combine the result to the final term: 5 + 1 = 6
 Procedure When performing a series of addition and subtraction operations, make a habit of working the problem from left to right.

What if there is a negative integer in the series of addends?  For example:

2 + ( –3 ) + 1 = 0

1. Combine the first two terms:  2 + ( –3 ) + 1 = –1 + 1
2. Combine the result to the final term: –1 + 1 = 0

Examples and Exercises 1

 Adding a Series of Integers Use these interactive examples and exercises to strengthen your understanding and build your skills:

Subtracting a Series of Integers

Recall that subtracting integers is a matter of changing the subtraction to addition, and then completing the addition.

 Procedure When a series of three or more terms include subtraction operations: For terms in parentheses, change the subtraction signs (–) to addition (+). Switch the sign attached to the term that follows the operation you just changed. If it's positive, change to negative. If it's negative, change to positive. Complete the addition of all terms.

Suppose you see this:

8 – ( +10) = ?

Change the subtraction to addition and the sign of the number that follows:

8 – ( +10) = 8 + ( –10)

8 + ( –10) = – 2

So 8 – 10 = – 2

But you should already know that from the previous lesson. Now let's make it a little more complicated.

Example

 Problem 12 – 2 – 4 – ( –5) = _____ Procedure Change each subtraction to addition and switch the sign of the number that immediately follows 12 + ( – 2) + ( – 4) +  ( +5) Complete the addition, from left to right 12 + ( – 2) + ( – 4) +  ( +5) = 10 + ( – 4) +  ( +5) 10 + ( – 4) +   ( +5) = 6 +  ( +5) 6 + ( +5) = 11 Solution 12 – 2 – 4 – ( –5) = 11

Examples and Exercises 2

 Subtracting a Series of Integers Use these interactive examples and exercises to strengthen your understanding and build your skills:

Always perform a series of addition and subtraction operations from left to right.

Example

 Problem 8 + 2 – 3 = _____ Procedure Add the first two terms Change the subtraction to addition Complete the operation 8 + 2 – 3 = 10 – 3 10 – 3 = 10 + ( –3) 10 + ( –3) = 7 Solution 8 + 2 – 3 = 7

Example

 Problem 2 + 4 – 8 + ( –1) =  _____ Procedure Add the first two terms Change the subtraction to addition Do the resulting addition Add the remaining terms 2 + 4 – 8 + ( –1) = 6 – 8 + ( –1) 6 – 8 + ( –1) = 6 + ( – 8) + ( –1) 6 + ( – 8) + ( –1) = –2 + ( –1) –2 + ( –1) = –3 Solution 2 + 4 – 8 + ( –1) =  –3

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills:

Simplifying Signed-Integer Expressions for Addition and Subtraction

 Remember Much of the confusion about subtracting signed integers is the result of having to use the same symbols for two entirely different purposes: The + is used for indicating both the addition operation and a positive integer value. The – is used for indicating both the subtraction operation and a negative integer value. What can be done about this source of confusion? Nothing. You have to consider the signs very carefully until the differences become second nature.

You have been seeing a lot of parentheses in this lesson. Pre-algebra teachers and textbooks tend to "overuse" the parentheses in order to clarify the different ways that  plus and minus signs are used. An expression such as ( + 8) – ( + 6) + (– 2) is really very cumbersome, but it clearly say + 6 is to be subtracted from + 8, and the result is added to – 2. Once you have mastered the concepts of adding and subtracting positive and negative numbers, you can simplify these expressions—and without changing their meaning.

Here are some simple examples of removing unnecessary parentheses:

( +2 ) is the same as 2

( –2 ) is the same as – 2

( +2 ) + ( +3 ) is the same as  2 + 3

( +2 )( +3 ) is the same as  2 – 3

( –2 )( +3 ) is the same as  –23

( +2 )( –3 ) is the same as  2 + 3

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