Simplifying and Evaluating Expressions With Integers

Learning Outcomes

  • Add and subtract integers
  • Simplify variable expressions for a given value
  • Evaluate variable 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 [latex]37+\left(-53\right)[/latex], you don’t have to count out [latex]37[/latex] blue counters and [latex]53[/latex] red counters.

Picture [latex]37[/latex] blue counters with [latex]53[/latex] red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because [latex]53 - 37=16[/latex], there are [latex]16[/latex] more negative counters.

[latex]37+\left(-53\right)=-16[/latex]

Let’s try another one. We’ll add [latex]-74+\left(-27\right)[/latex]. Imagine [latex]74[/latex] red counters and [latex]27[/latex] more red counters, so we have [latex]101[/latex] red counters all together. This means the sum is [latex]\text{-101.}[/latex]

[latex]-74+\left(-27\right)=-101[/latex]

Look again at the results of [latex]-74-\left(27\right)[/latex].

Addition of Positive and Negative Integers
[latex]5+3[/latex] [latex]-5+\left(-3\right)[/latex]
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.
[latex]-5+3[/latex] [latex]5+\left(-3\right)[/latex]
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.

Exercises

Simplify:

  1. [latex]19+\left(-47\right)[/latex]
  2. [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]\begin{array}{c}19+\left(-47\right)\\ -28\end{array}[/latex]

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]\begin{array}{c}-32+40\\ 8\end{array}[/latex]

Tip: Think of positive numbers as money you have and negative numbers as money you owe.  This will help you determine if your answer is positive or negative.  (-4) + 7 would be owing $4 and having $7, once you settle up, you still have $3.  So the answer would be positive 3.

Another example is (-3) + (-5).  This means you owe $3 and you owe $5, so you owe $8, which would be represented by -8.

If you have 6 + (-10) and we think in terms of money, you have $6 but you owe $10.  Once you settle up, you still owe $4.  This gives you an answer of -4.

 

try it

 

example

Simplify: [latex]-14+\left(-36\right)[/latex]

 

try it

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]

 

try it

Watch the following video to see another example of how to simplify an expression that contains integer addition and multiplication.

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions. In our first example we will evaluate a simple variable expression for a negative value.

example

Evaluate [latex]x+7\text{ when}[/latex]

  1. [latex]x=-2[/latex]
  2. [latex]x=-11[/latex]

Now you can try a similar problem.

try it

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

  1. [latex]n+1[/latex]
  2. [latex]-n+1[/latex]

Now you can try a similar problem.

try it

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].

Now you can try a a similar problem.

try it

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].

Now you can try a similar problem.

try it