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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.

Adding a Series of Integers

You probably have no trouble solving this example:

2 + 3 + 1 = 6

Adding from left to right:

  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

Adding from left to right

  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:

  1. For terms in parentheses, change the subtraction signs (–) to addition (+).
  2. 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.
  3. 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)

Finally, complete the addition:

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
  1. Change each subtraction to addition and switch the sign of the number that immediately follows
12 + ( – 2) + ( – 4) +  ( +5)
  1. 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:

Combining Addition and Subtraction

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

David L. Heiserman, Editor

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All Rights Reserved

Revised: June 06, 2015