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Simplifying Expressions with Integers

Simplifying 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.

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

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:

Subtraction Property

\(a-b=a+\left(\mathit{\text{−b}}\right)\)

Look at these two examples.

This figure has two columns. The first column has 6 minus 4. Underneath, there is a row of 6 blue circles, with the first 4 separated from the last 2. The first 4 are circled. Under this row there is 2. The second column has 6 plus negative 4. Underneath there is a row of 6 blue circles with the first 4 separated from the last 2. The first 4 are circled. Under the first four is a row of 4 red circles. Under this there is 2.

We see that \(6-4\) gives the same answer as \(6+\left(-4\right).\)

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

Optional Video: Subtacting Integers - The Basics

Example

Simplify:

  1. \(\phantom{\rule{0.2em}{0ex}}13-8\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}13+\left(-8\right)\phantom{\rule{1em}{0ex}}\)
  2. \(\phantom{\rule{0.2em}{0ex}}-17-9\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-17+\left(-9\right)\)

Solution

\(13-8\) and \(13+\left(-8\right)\)
Subtract to simplify. \(13-8=5\)
Add to simplify. \(13+\left(-8\right)=5\)
Subtracting 8 from 13 is the same as adding −8 to 13.
\(-17-9\) and \(-17+\left(-9\right)\)
Subtract to simplify. \(-17-9=-26\)
Add to simplify. \(-17+\left(-9\right)=-26\)
Subtracting 9 from −17 is the same as adding −9 to −17.

Now look what happens when we subtract a negative.

This figure has two columns. The first column has 8 minus negative 5. Underneath, there is a row of 13 blue circles. The first 8 are separated from the next 5. Under the last 5 blue circles there is a row of 5 red circles. They are circled. Under this there is 13. The second column has 8 plus 5. Underneath there is a row of 13 blue circles. The first 8 are separated from the last 5. Under this there is 13.

We see that \(8-\left(-5\right)\) gives the same result as \(8+5.\) Subtracting a negative number is like adding a positive.

Example

Simplify:

  1. \(\phantom{\rule{0.2em}{0ex}}9-\left(-15\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}9+15\phantom{\rule{1em}{0ex}}\)
  2. \(\phantom{\rule{0.2em}{0ex}}-7-\left(-4\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-7+4\)

Solution

\(9-\left(-15\right)\) and \(9+15\)
Subtract to simplify. \(9-\left(-15\right)=-24\)
Add to simplify. \(9+15=24\)
Subtracting −15 from 9 is the same as adding 15 to 9.
\(-7-\left(-4\right)\) and \(-7+4\)
Subtract to simplify. \(-7-\left(-4\right)=-3\)
Add to simplify. \(-7+4=-3\)
Subtracting −4 from −7 is the same as adding 4 to −7

Recall the results of the examples in the previous lesson.

Subtraction of Integers

\(5–3\) \(–5–\left(–3\right)\)
\(2\) \(–2\)
2 positives 2 negatives
When there would be enough counters of the color to take away, subtract.
\(–5–3\) \(5–\left(–3\right)\)
\(–8\) \(8\)
5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives
need neutral pairs need neutral pairs
When there would not be enough of the counters to take away, add neutral pairs.

Example

Simplify: \(-74-\left(-58\right).\)

Solution

We are taking 58 negatives away from 74 negatives. \(-74-\left(-58\right)\)
Subtract. \(-16\)

Example

Simplify: \(7-\left(-4-3\right)-9.\)

Solution

We use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right.

Simplify inside the parentheses first. .
Subtract from left to right. .
Subtract. .
.

Example

Simplify: \(3·7-4·7-5·8.\)

Solution

We use the order of operations to simplify this expression. First we multiply, and then subtract from left to right.

Multiply first. .
Subtract from left to right. .
Subtract. .
.

This lesson is part of:

Introducing Integers

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