Add and subtract positive and negative numbers

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Suppose you took a job as a peer tutor and the person you were tutoring asked you to give examples of the Commutative Property of Addition, Associative Property of Addition, and Identity Property of Addition. Could you do it? Would you be able to distinguish between the different properties? After completing this Concept, you'll have a firm understanding of these properties so that you can easily add integers.

Guidance

A football team gains 11 yards on one play, then loses 5 yards on the next play, and then loses 2 yards on the third play. What is the total loss or gain of yardage?

A loss can be expressed as a negative integer. A gain can be expressed as a positive integer. To find the net gain or loss, the individual values must be added together. Therefore, the sum is 11+(5)+(2)=4\begin{align*}11 + (-5) + (-2) = 4\end{align*}. The team has a net gain of 4 yards.

Addition can also be shown using a number line. If you need to add 2+3\begin{align*}2 + 3\end{align*}, start by making a point at the value of 2 and move three integers to the right. The ending value represents the sum of the values.

Example A

Find the sum of 2+3\begin{align*}-2 + 3\end{align*} using a number line.

Solution: Begin by making a point at –2 and moving three units to the right. The final value is 1, so 2+3=1.\begin{align*}-2 + 3 = 1.\end{align*}

When the value that is being added is positive, we jump to the right. If the value is negative, we jump to the left (in a negative direction).

Example B

Find the sum of 23\begin{align*}2 - 3\end{align*} using a number line.

Solution: Begin by making a point at 2. The expression represents subtraction, so we will count three jumps to the left.

The solution is: 23=1\begin{align*}2 - 3 = -1\end{align*}.

In the previous Concept, you learned the Additive Inverse Property. This property states that the sum of a number and its opposite is zero. Algebra has many other properties that help you manipulate and organize information.

The Commutative Property of Addition: For all real numbers a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*}, a+b=b+a\begin{align*}a + b = b + a\end{align*}.

To commute means to change locations, so the Commutative Property of Addition allows you to rearrange the objects in an addition problem.

The Associative Property of Addition: For all real numbers a, b,\begin{align*}a, \ b,\end{align*} and c\begin{align*}c\end{align*}, (a+b)+c=a+(b+c).\begin{align*}(a + b)+ c = a + (b + c).\end{align*}

To associate means to group together, so the Associative Property of Addition allows you to regroup the objects in an addition problem.

The Identity Property of Addition: For any real number a, a+0=a.\begin{align*}a, \ a + 0 = a.\end{align*}

This property allows you to use the fact that the sum of any number and zero is the original value.

These properties apply to all real numbers, but in this lesson we are applying them to integers, which are just a special kind of real number.

Example C

Nadia and Peter are building sand castles on the beach. Nadia built a castle two feet tall, stopped for ice-cream, and then added one more foot to her castle. Peter built a castle one foot tall before stopping for a sandwich. After his sandwich, he built up his castle by two more feet. Whose castle is taller?

Solution:

Nadia’s castle is (2+1)\begin{align*}(2 + 1)\end{align*} feet tall. Peter’s castle is (1+2)\begin{align*}(1 + 2)\end{align*} feet tall. According to the Commutative Property of Addition, the two castles are the same height.

Guided Practice

Simplify the following using the properties of addition:

a) 9+(1+22)\begin{align*}9 + ( 1 + 22)\end{align*}

b) 4,211+0\begin{align*}4,211 + 0\end{align*}

Solution:

a) It is easier to regroup 9+1\begin{align*}9 + 1\end{align*}, so by applying the Associative Property of Addition, (9+1)+22=10+22=32.\begin{align*}(9 + 1) + 22 = 10 + 22 = 32.\end{align*}

b) The Additive Identity Property states the sum of a number and zero is itself; therefore, 4,211+0=4,211.\begin{align*}4,211 + 0 = 4,211.\end{align*}

Practice

In exercises 1 and 2, write the sum represented by the moves on the number line.

In 3 – 6, which property of addition does each situation involve?

1. Shari’s age minus the negative of Jerry’s age equals the sum of the two ages.
2. Kerri has 16 apples and has added zero additional apples. Her current total is 16 apples.
3. A blue whale dives 160 feet below the surface and then rises 8 feet. Write the addition problem and find the sum.
4. The temperature in Chicago, Illinois one morning was 8F\begin{align*}-8^\circ F\end{align*}. Over the next six hours the temperature rose 25 degrees Fahrenheit. What was the new temperature?

Mixed Review

1. Find the opposite of –72.
2. Evaluate |1629+78114|.\begin{align*}|16-29+78-114|.\end{align*}
3. What is the domain and range of the following: {(2,1),(3,0),(4,6),(1,3)}?\begin{align*}\left \{(2,-1),(3,0),(4,6),(1,-3) \right \}?\end{align*}
4. Write a rule for the following table:
Volume (in inches3\begin{align*}inches^3\end{align*}) Mass (in grams)
1 20.1
2 40.2
3 60.3
4 80.4

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