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أدلة الدراسة > Prealgebra

Simplifying and Evaluating Expressions With Integers

Learning Outcomes

  • Add and subtract integers
  • Simplify variables expressions for a given value
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+(−53)37+\left(-53\right), you don’t have to count out 3737 blue counters and 5353 red counters. Picture 3737 blue counters with 5353 red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because 53−37=1653 - 37=16, there are 1616 more negative counters. 37+(−53)=−1637+\left(-53\right)=-16 Let’s try another one. We’ll add −74+(−27)-74+\left(-27\right). Imagine 7474 red counters and 2727 more red counters, so we have 101101 red counters all together. This means the sum is -101.\text{-101.} −74+(−27)=−101-74+\left(-27\right)=-101 Look again at the results of −74−(27)-74-\left(27\right).
Addition of Positive and Negative Integers
5+35+3 −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+3 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.

Exercises

Simplify:
  1. 19+(−47)19+\left(-47\right)
  2. −32+40-32+40
Solution: 1. Since the signs are different, we subtract 1919 from 4747. The answer will be negative because there are more negatives than positives. 19+(−47)−28\begin{array}{c}19+\left(-47\right)\\ -28\end{array} 2. The signs are different so we subtract 3232 from 4040. The answer will be positive because there are more positives than negatives −32+408\begin{array}{c}-32+40\\ 8\end{array}
   

example

Simplify: −14+(−36)-14+\left(-36\right).

Answer: Solution: Since the signs are the same, we add. The answer will be negative because there are only negatives. −14+(−36)−50\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(−2+7)-5+3\left(-2+7\right).

Answer: Solution:

−5+3(−2+7)-5+3\left(-2+7\right)
Simplify inside the parentheses. −5+3(5)-5+3\left(5\right)
Multiply. −5+15-5+15
Add left to right. 1010

  Watch the following video to see another example of how to simplify an expression that contains integer addition and multiplication. https://youtu.be/RJ7uU9HbdqA

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 x+7 whenx+7\text{ when}
  1. x=−2x=-2
  2. x=−11x=-11.

Answer: Solution:

1. Evaluate x+7x+7 when x=−2x=-2
Substitute −2\color{red}{-2} for x. −2+7\color{red}{-2}+7
Simplify. 55
2. Evaluate x+7x+7 when x=−11x=-11
x+7x+7
Substitute −11\color{red}{-11} for x. −11+7\color{red}{-11}+7
Simplify. −4-4

Now you can try a similar problem.   In the next example, we are give two expressions,n+1n+1, and −n+1-n+1. 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 n=−5n=-5, evaluate
  1. n+1n+1
  2. −n+1-n+1.

Answer: Solution:

1. Evaluate n+1n+1 when n=−5n=-5
n+1n+1
Substitute −5\color{red}{-5} for n. −5+1\color{red}{-5}+1
Simplify. −4-4
2. Evaluate −n+1-n+1 when n=−5n=-5
−n+1-n+1
Substitute −5\color{red}{-5} for n. −(−5)+1-(\color{red}{-5})+1
Simplify. 5+15+1
Add. 66

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 3a+b3a+b when a=12a=12 and b=−30b=-30.

Answer: Solution:

3a+b3a+b
Substitute 12\color{red}{12} for a and −30\color{blue}{-30} for b. 3(12)+(−30)3(\color{red}{12})+(\color{blue}{-30})
Multiply. 36+(−30)36+(-30)
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 (x+y)2{\left(x+y\right)}^{2} when x=−18x=-18 and y=24y=24.

Answer: Solution: This expression has two variables. Substitute −18-18 for xx and 2424 for yy.

(x+y)2{\left(x+y\right)}^{2}
Substitute −18\color{red}{-18} for x and 24\color{blue}{24} for y. (−18+24)2{\left(-18+24\right)}^{2}
Add inside the parentheses. (6)2{\left(6\right)}^{2}
Simplify 3636

Now you can try a similar problem.  

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