Simplifying Expressions with Integers
Simplifying Expressions with Integers
Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.
For example, if you want to add \(37+\left(-53\right),\) you don’t have to count out \(37\) blue counters and \(53\) red counters.
Picture \(37\) blue counters with \(53\) red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because \(53-37=16,\) there are \(16\) more negative counters.
Let’s try another one. We’ll add \(-74+\left(-27\right).\) Imagine \(74\) red counters and \(27\) more red counters, so we have \(101\) red counters all together. This means the sum is \(\text{−101.}\)
Recall the results of the previous lesson.
Table: Addition of Positive and Negative Integers
| \(5+3\) | \(-5+\left(-3\right)\) |
| both positive, sum positive | both negative, sum negative |
| When the signs are the same, the counters would be all the same color, so add them. | |
| \(-5+3\) | \(5+\left(-3\right)\) |
| different signs, more negatives | different signs, more positives |
| Sum negative | sum positive |
| When the signs are different, some counters would make neutral pairs; subtract to see how many are left. | |
Example
Simplify:
- \(\phantom{\rule{0.2em}{0ex}}19+\left(-47\right)\phantom{\rule{1em}{0ex}}\)
- \(\phantom{\rule{0.2em}{0ex}}-32+40\)
Solution
Since the signs are different, we subtract \(19\) from \(47.\) The answer will be negative because there are more negatives than positives.
The signs are different so we subtract \(32\) from \(40.\) The answer will be positive because there are more positives than negatives
\(\begin{array}{c}-32+40\\ 8\end{array}\)
Example
Simplify: \(-14+\left(-36\right).\)
Solution
Since the signs are the same, we add. The answer will be negative because there are only negatives.
\(\begin{array}{c}-14+\left(-36\right)\\ -50\end{array}\)The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.
Example
Simplify: \(-5+3\left(-2+7\right).\)
Solution
| \(-5+3\left(-2+7\right)\) | |
| Simplify inside the parentheses. | \(-5+3\left(5\right)\) |
| Multiply. | \(-5+15\) |
| Add left to right. | \(10\) |
Optional Video: Example on Adding Integers by Mathispower4u
This lesson is part of:
Introducing Integers