Exercises
Simplify:
- [latex]19+\left(-47\right)[/latex]
- [latex]-32+40[/latex]
Solution:
1. Since the signs are different, we subtract [latex]19[/latex] from [latex]47[/latex]. The answer will be negative because there are more negatives than positives.
[latex-display]\begin{array}{c}19+\left(-47\right)\\ -28\end{array}[/latex-display]
2. The signs are different so we subtract [latex]32[/latex] from [latex]40[/latex]. The answer will be positive because there are more positives than negatives
[latex-display]\begin{array}{c}-32+40\\ 8\end{array}[/latex-display]
example
Simplify: [latex]-14+\left(-36\right)[/latex].
Answer:
Solution:
Since the signs are the same, we add. The answer will be negative because there are only negatives.
[latex-display]\begin{array}{c}-14+\left(-36\right)\\ -50\end{array}[/latex-display]
The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.
example
Simplify: [latex]-5+3\left(-2+7\right)[/latex].
Answer:
Solution:
|
[latex]-5+3\left(-2+7\right)[/latex] |
| Simplify inside the parentheses. |
[latex]-5+3\left(5\right)[/latex] |
| Multiply. |
[latex]-5+15[/latex] |
| Add left to right. |
[latex]10[/latex] |
Watch the following video to see another example of how to simplify an expression that contains integer addition and multiplication.
https://youtu.be/RJ7uU9HbdqA
example
Evaluate [latex]x+7\text{ when}[/latex]
- [latex]x=-2[/latex]
- [latex]x=-11[/latex].
Answer:
Solution:
| 1. Evaluate [latex]x+7[/latex] when [latex]x=-2[/latex] |
|
|
|
| Substitute [latex]\color{red}{-2}[/latex] for x. |
[latex]\color{red}{-2}+7[/latex]
|
| Simplify. |
[latex]5[/latex] |
| 2. Evaluate [latex]x+7[/latex] when [latex]x=-11[/latex] |
|
|
[latex]x+7[/latex]
|
| Substitute [latex]\color{red}{-11}[/latex] for x. |
[latex]\color{red}{-11}+7[/latex] |
| Simplify. |
[latex]-4[/latex] |
Now you can try a similar problem.
In the next example, we are give two expressions,[latex]n+1[/latex], and [latex]-n+1[/latex]. We will evaluate both for a negative number. This practice will help you learn how to keep track of multiple negative signs in one expression.
example
When [latex]n=-5[/latex], evaluate
- [latex]n+1[/latex]
- [latex]-n+1[/latex].
Answer:
Solution:
| 1. Evaluate [latex]n+1[/latex] when [latex]n=-5[/latex] |
|
|
[latex]n+1[/latex] |
| Substitute [latex]\color{red}{-5}[/latex] for n. |
[latex]\color{red}{-5}+1[/latex] |
| Simplify. |
[latex]-4[/latex] |
| 2. Evaluate [latex]-n+1[/latex] when [latex]n=-5[/latex] |
|
|
[latex]-n+1[/latex] |
| Substitute [latex]\color{red}{-5}[/latex] for n. |
[latex]-(\color{red}{-5})+1[/latex] |
| Simplify. |
[latex]5+1[/latex] |
| Add. |
[latex]6[/latex] |
Now you can try a similar problem.
Next we'll evaluate an expression with two variables, where one of the variables is assigned a negative value.
example
Evaluate [latex]3a+b[/latex] when [latex]a=12[/latex] and [latex]b=-30[/latex].
Answer:
Solution:
|
[latex]3a+b[/latex] |
| Substitute [latex]\color{red}{12}[/latex] for a and [latex]\color{blue}{-30}[/latex] for b. |
[latex]3(\color{red}{12})+(\color{blue}{-30})[/latex] |
| Multiply. |
[latex]36+(-30)[/latex] |
| Add. |
|
Now you can try a a similar problem.
In the next example, the expression has an exponent as well as parentheses. It is important to remember the order of operations, you will need to simplify inside the parentheses first, then apply the exponent to the result.
example
Evaluate [latex]{\left(x+y\right)}^{2}[/latex] when [latex]x=-18[/latex] and [latex]y=24[/latex].
Answer:
Solution:
This expression has two variables. Substitute [latex]-18[/latex] for [latex]x[/latex] and [latex]24[/latex] for [latex]y[/latex].
|
[latex]{\left(x+y\right)}^{2}[/latex] |
| Substitute [latex]\color{red}{-18}[/latex] for x and [latex]\color{blue}{24}[/latex] for y. |
[latex]{\left(-18+24\right)}^{2}[/latex] |
| Add inside the parentheses. |
[latex]{\left(6\right)}^{2}[/latex] |
| Simplify |
[latex]36[/latex] |
Now you can try a similar problem.
