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Study Guides > Prealgebra

Simplifying and Evaluating Expressions With Integers

Learning Outcomes

  • Simplify integer expressions involving subtraction
  • Substitute and simplify integer expressions involving subtraction
Now that you have seen subtraction modeled with color counters, we can move on to performing subtraction of integers without the models.
  • Subtract −23−7-23 - 7. Think: We start with 2323 negative counters. We have to subtract 77 positives, but there are no positives to take away. So we add 77 neutral pairs to get the 77 positives. Now we take away the 77 positives. So what’s left? We have the original 2323 negatives plus 77 more negatives from the neutral pair. The result is 3030 negatives. −23−7=−30-23 - 7=-30 Notice, that to subtract 7,\text{7,} we added 77 negatives.
  • Subtract 30−(−12)30-\left(-12\right). Think: We start with 3030 positives. We have to subtract 1212 negatives, but there are no negatives to take away. So we add 1212 neutral pairs to the 3030 positives. Now we take away the 1212 negatives. What’s left? We have the original 3030 positives plus 1212 more positives from the neutral pairs. The result is 4242 positives. 30−(−12)=4230-\left(-12\right)=42 Notice that to subtract −12-12, we added 1212.
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

Subtracting a number is the same as adding it's opposite.

a−b=a+(−b)a-b=a+(-b)

  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 6−46 - 4 gives the same answer as 6+(−4)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 6−46 - 4 long ago. But knowing that 6−46 - 4 gives the same answer as 6+(−4)6+\left(-4\right) helps when we are subtracting negative numbers.  

example

Simplify:
  1. 13−8 and 13+(−8)13 - 8\text{ and }13+\left(-8\right)
  2. −17−9 and −17+(−9)-17 - 9\text{ and }-17+\left(-9\right)
Solution:
1.
13−813 - 8 and 13+(−8)13+\left(-8\right)
Subtract to simplify. 13−8=513 - 8=5
Add to simplify. 13+(−8)=513+\left(-8\right)=5
Subtracting 88 from 1313 is the same as adding −8−8 to 1313.
2.
−17−9-17 - 9 and −17+(−9)-17+\left(-9\right)
Subtract to simplify. −17−9=−26-17 - 9=-26
Add to simplify. −17+(−9)=−26-17+\left(-9\right)=-26
Subtracting 99 from −17−17 is the same as adding −9−9 to −17−17.
Now you can try a similar problem.   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−(−5)8-\left(-5\right) gives the same result as 8+58+5. Subtracting a negative number is like adding a positive. In the next example, we will see more examples of this concept.

example

Simplify:
  1. 9−(−15) and 9+159-\left(-15\right)\text{ and }9+15
  2. −7−(−4) and −7+4-7-\left(-4\right)\text{ and }-7+4

Answer: Solution:

1.
9−(−15)9-\left(-15\right) and 9+159+15
Subtract to simplify. 9−(−15)=−249-\left(-15\right)=-24
Add to simplify. 9+15=249+15=24
Subtracting −15−15 from 99 is the same as adding 1515 to 99.
2.
−7−(−4)-7-\left(-4\right) and −7+4-7+4
Subtract to simplify. −7−(−4)=−3-7-\left(-4\right)=-3
Add to simplify. −7+4=−3-7+4=-3
Subtracting −4−4 from −7−7 is the same as adding 44 to −7−7.

Now you can try a similar problem. The table below summarizes the four different scenarios we encountered in the previous examples, and how you would use counters to simplify.
Subtraction of Integers
5−35 - 3 −5−(−3)-5-\left(-3\right)
22 −2-2
22 positives 22 negatives
When there would be enough counters of the color to take away, subtract.
−5−3-5 - 3 5−(−3)5-\left(-3\right)
−8-8 88
55 negatives, want to subtract 33 positives 55 positives, want to subtract 33 negatives
need neutral pairs need neutral pairs
When there would not be enough of the counters to take away, add neutral pairs.
In our next example we show how to subtract a negative with two digit numbers.

example

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

Answer: Solution:

We are taking 5858 negatives away from 7474 negatives. −74−(−58)-74-\left(-58\right)
Subtract. −16-16

Now you can try a similar problem. In the following video we show another example of subtracting two digit integers. https://youtu.be/IfiN-mJZu2E Now let's increase the complexity of the examples a little bit. We will use the order of operations to simplify terms in parentheses before we subtract from left to right.

example

Simplify: 7−(−4−3)−97-\left(-4 - 3\right)-9.

Answer: 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. 7−(−4−3)−97-\left(-4 - 3\right)-9
Subtract from left to right. 7−(−7)−97-\left(-7\right)-9
Subtract. 14−−914--9
55

Now you try it. Watch the following video to see more examples of simplifying integer expressions that involve subtraction. https://youtu.be/mDkSpz0BPPc Now we will add another operation to an expression. Because multiplication and division come before addition and subtraction, we will multiply, then subtract.

example

Simplify: 3⋅7−4⋅7−5⋅83\cdot 7 - 4\cdot 7 - 5\cdot 8.

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

Multiply first. 3⋅7−4⋅7−5⋅83\cdot 7 - 4\cdot 7 - 5\cdot 8
Subtract from left to right. 21−−28−−4021--28--40
Subtract. −−7−−40--7--40
−−47--47

Now you try. Watch the following video to see another example of simplifying an integer expression involving multiplication and subtraction. https://youtu.be/42Su4r5UmoE

Evaluate Variable Expressions with Integers

Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.  

example

Evaluate x−4 whenx - 4\text{ when}
  1. x=3x=3
  2. x=−6x=-6.

Answer: Solution: 1. To evaluate x−4x - 4 when x=3x=3 , substitute 33 for xx in the expression.

x−−4x--4
Substitute 3 for x\text{Substitute }\color{red}{3}\text{ for }x 3−−4\color{red}{3}--4
Subtract. −−1--1
2. To evaluate x−4x - 4 when x=−6x=-6, substitute −6-6 for xx in the expression.
x−−4x--4
Substitute −−6 for x\text{Substitute }\color{red}{--6}\text{ for }x −−6−−4\color{red}{--6}--4
Subtract. −−10--10

Now you try. In the next example, we will subtract a positive and a negative.

example

Evaluate 20−z when20-z\text{ when}
  1. z=12z=12
  2. z=−12z=-12

Answer: Solution: 1. To evaluate 20−z when z=1220-z\text{ when }z=12, substitute 1212 for zz in the expression.

20−−z20--z
Substitute 12 for z\text{Substitute }\color{red}{12}\text{ for }z 20−−1220--\color{red}{12}
Subtract. 88
2. To evaluate 20−z when z=−12, substitute −12 for zin the expression.20-z\text{ when }z=-12,\text{ substitute }-12\text{ for }z\text{in the expression.}
20−−z20--z
Substitute −−12 for z\text{Substitute }\color{red}{--12}\text{ for }z 20−−(−−12)20--(\color{red}{--12})
Subtract. 3232

Now you try.  

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