# Integers

## Subtract Integers


### Model Subtraction of Integers

Remember the story in the last section about the toddler and the cookies? Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.
We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.
Perhaps when you were younger, you read  as five take away three. When we use counters, we can think of subtraction the same way.
We will model four subtraction facts using the numbers  and 
Notice that  and  are very much alike.
1. First, we subtracted  positives from  positives to get  positives.
2. Then we subtracted  negatives from  negatives to get  negatives.
Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.
Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.

### Simplify Expressions with Integers

Do you see a pattern?  Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters.
1. Subtract 
Think: We start with  negative counters.
We have to subtract  positives, but there are no positives to take away.
So we add  neutral pairs to get the  positives. Now we take away the  positives.
So what’s left? We have the original  negatives plus  more negatives from the neutral pair. The result is  negatives.

Notice, that to subtract  we added  negatives.
2. Subtract 
Think: We start with  positives.
We have to subtract  negatives, but there are no negatives to take away.
So we add  neutral pairs to the  positives. Now we take away the  negatives.
What’s left? We have the original  positives plus  more positives from the neutral pairs. The result is  positives.

Notice that to subtract  we added 
While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.
Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows:
Look at these two examples.
We see that  gives the same answer as 
Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract  long ago. But knowing that  gives the same answer as  helps when we are subtracting negative numbers.
Now look what happens when we subtract a negative.
We see that  gives the same result as  Subtracting a negative number is like adding a positive.
Look again at the results of  - .

### Evaluate Variable Expressions with Integers

Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.

### Translate Word Phrases to Algebraic Expressions

When we first introduced the operation symbols, we saw that the expression  may be read in several ways as shown below.
Be careful to get  and  in the right order!

### Subtract Integers in Applications

It’s hard to find something if we don’t know what we’re looking for or what to call it. So when we solve an application problem, we first need to determine what we are asked to find. Then we can write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.
Geography provides another application of negative numbers with the elevations of places below sea level.
Managing your money can involve both positive and negative numbers. You might have overdraft protection on your checking account. This means the bank lets you write checks for more money than you have in your account (as long as they know they can get it back from you!)

### Key Concepts

1. Subtraction of Integers
2. Subtraction Property


3. Solve Application Problems


### Practice Makes Perfect

Model Subtraction of Integers 
In the following exercises, model each expression and simplify.
Simplify Expressions with Integers 
In the following exercises, simplify each expression.
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers 
In the following exercises, evaluate each expression for the given values.
Translate Word Phrases to Algebraic Expressions 
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Subtract Integers in Applications 
In the following exercises, solve the following applications.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
