Now that you have seen subtraction modeled with color counters, we can move on to performing subtraction of integers without the models.

• Subtract [latex]-23 - 7[/latex]. Think: We start with [latex]23[/latex] negative counters. We have to subtract [latex]7[/latex] positives, but there are no positives to take away. So we add [latex]7[/latex] neutral pairs to get the [latex]7[/latex] positives. Now we take away the [latex]7[/latex] positives. So what’s left? We have the original [latex]23[/latex] negatives plus [latex]7[/latex] more negatives from the neutral pair. The result is [latex]30[/latex] negatives. [latex]-23 - 7=-30[/latex] Notice, that to subtract [latex]\text{7,}[/latex] we added [latex]7[/latex] negatives.
• Subtract [latex]30-\left(-12\right)[/latex]. Think: We start with [latex]30[/latex] positives. We have to subtract [latex]12[/latex] negatives, but there are no negatives to take away. So we add [latex]12[/latex] neutral pairs to the [latex]30[/latex] positives. Now we take away the [latex]12[/latex] negatives. What’s left? We have the original [latex]30[/latex] positives plus [latex]12[/latex] more positives from the neutral pairs. The result is [latex]42[/latex] positives. [latex]30-\left(-12\right)=42[/latex] Notice that to subtract [latex]-12[/latex], we added [latex]12[/latex].

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 [latex]6 - 4[/latex] gives the same answer as [latex]6+\left(-4\right)[/latex].

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 [latex]6 - 4[/latex] long ago. But knowing that [latex]6 - 4[/latex] gives the same answer as [latex]6+\left(-4\right)[/latex] helps when we are subtracting negative numbers.

Now you can try a similar problem.

Now look what happens when we subtract a negative.

We see that [latex]8-\left(-5\right)[/latex] gives the same result as [latex]8+5[/latex]. Subtracting a negative number is like adding a positive. In the next example, we will see more examples of this concept.

Now you can try a similar problem.

The table below summarizes the four different scenarios we encountered in the previous examples, and how you would use counters to simplify.

In our next example we show how to subtract a negative with two digit numbers.

Now you can try a similar problem.

In the following video we show another example of subtracting two digit integers.

Now let’s increase the complexity of the examples a little bit. We will use the order of operations to simplify terms in parentheses before we subtract from left to right.

Now you try it.

Watch the following video to see more examples of simplifying integer expressions that involve subtraction.

Now we will add another operation to an expression. Because multiplication and division come before addition and subtraction, we will multiply, then subtract.

Now you try.

Watch the following video to see another example of simplifying an integer expression involving multiplication and subtraction.

Evaluate Variable Expressions with Integers

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

Now you try.

In the next example, we will subtract a positive and a negative.

Now you try.