Transcript Adding and Subtracting Integers

-7
-10
-5
+7
0
5
10
A number line has many functions. Previously, we learned
that numbers to the right of zero are positive and numbers to
the left of zero are negative. By putting points on the
number line, we can graph values.
If one were to start at zero and move seven places to the right,
this would represent a value of positive seven. If one were to
start at zero and move seven places to the left, this would
represent a value of negative seven.
-7
-10
-5
+7
0
5
10
Both of these numbers, positive seven (+7) and negative seven
(-7), represent a point that is seven units away from the origin.
The absolute value of a number is the distance between that
number and zero on a number line. Absolute value is shown by
placing two vertical bars around the number as follows:
5  The absolute value of five is five.
-3  The absolute value of negative three is three.
What is 5 + 7?
We can show how to do this by using algebra tiles.
1
1
1
1
1
+
1
1
1
1
5
+
What is –5 + -7?
-1
-1
-1
-1
-5
-1
+
-1
1
=
1
7
-1
+
1
-1
-1
-7
=
-1
-1
=
-1
=
1
1
1
1
1
1
1
1
1
1
1
1
12
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-12
We can show this same idea using a number line.
9
-10
-5
0
5
What is 5 + 4?
Move five (5) units to the right from zero.
Now move four more units to the right.
The final point is at 9 on the number line.
Therefore, 5 + 4 = 9.
10
-9
-10
-5
0
5
What is -5 + (-4)?
Move five (5) units to the left from zero.
Now move four more units to the left.
The final point is at -9 on the number line.
Therefore, -5 + (-4) = -9.
10
To add integers with the same sign, add the absolute
values of the integers. Give the answer the same sign as
the integers.
Examples
Solution
1.
15  12
2.
 14  (33)
 47
3.
 4x  (12x)
 16x
27
What is (-7) + 7?
-5
0
5
-10
10
To show this, we can begin at zero and move seven units to
the left.
Now, move seven units to the right.
Notice, we are back at zero (0).
For every positive integer on the number line, there is a
corresponding negative integer. These integer pairs are
opposites or additive inverses.
Additive Inverse Property – For every number a, a + (-a) = 0
When using algebra tiles, the additive inverses make what
is called a zero pair.
For example, the following is a zero pair.
-1
1
1 + (-1) = 0.
This also represents a zero pair.
x
-x
x + (-x) = 0
Add the following integers:
-10
-5
(-4) + 7.
0
Start at zero and move four units to the left.
Now move seven units to the right.
The final position is at 3.
Therefore, (-4) + 7 = 3.
5
10
Add the following integers:
-10
-5
(-4) + 7.
0
5
Notice that seven minus four also equals three.
In our example, the number with the larger absolute
value was positive and our solution was positive.
Let’s try another one.
10
Add (-9) + 3
-10
-5
0
Start at zero and move nine places to the left.
Now move three places to the right.
The final position is at negative six, (-6).
Therefore, (-9) + 3 = -6.
5
10
Add (-9) + 3
-10
-5
0
5
10
In this example, the number with the larger absolute
value is negative. The number with the smaller absolute
value is positive.
We know that 9-3 = 6. However, (-9) + 3 = -6.
6 and –6 are opposites. Comparing these two examples
shows us that the answer will have the same sign as the
number with the larger absolute value.
To add integers with different signs determine the absolute
value of the two numbers. Subtract the smaller absolute value
from the larger absolute value. The solution will have the same
sign as the number with the larger absolute value.
Example
Subtract
Solution
12  15
15  12
3
17  (25)
25  17
8
We can show the addition of numbers with opposite signs by
using algebra tiles. For example, 3 + (-5) would look like this:
1
1
1
+
-1
1
-1
-1
1
-1
-1
1
-1
-1
-1
Group the zero pairs.
Remove the zero pairs
-1
-1
We can show the addition of numbers with opposite signs by
using algebra tiles. For example, 3 + (-5) would look like this:
-1
1
1
1
+
-1
-1
=
-1
-1
-1
Group the zero pairs.
Remove the zero pairs
The remainder is the solution.
Therefore, 3 + (-5) = -2
-1
-1
-1
Let’s see how subtraction works using algebra tiles for 3 - 5.
Begin with three algebra tiles.
1
1
-1
1
1
-1
1
Now, take away or remove five tiles from these three. It
can’t be done. However, we can add zero pairs until we
have five ones because adding zero doesn’t change the value
of the number.
Let’s see how subtraction works using algebra tiles for 3 - 5.
Begin with three algebra tiles.
1
1
-1
1
1
-1
1
Now remove the five ones.
Let’s see how subtraction works using algebra tiles for 3 - 5.
Begin with three algebra tiles.
-1
-1
Now remove the five ones.
The remainder is negative two (-2).
Therefore, 3 – 5 = -2. This is the same as 3 + (-5).
To subtract a number, add its additive inverse.
For any numbers a and b, a – b = a + (-b).
Find each sum or difference.
1.
-24 – 11
2.
18 + (-40)
3.
-9 + 9
4.
-16 – (-14)
5.
13 – 35
6.
-29 + 65
9c – (-12c)
9.
-7x + 45x
Simplify each expression.
7.
18r – 27r
8.
Evaluate each expression.
10. 89 
12.
- -7 
11. -14 
13. c + 5  if c = -19
1.
 24  11  24  ( 11)
  35
2.
18  ( 40)   ( 40  18)
  22
3.
4.
99 0
 16  ( 14)  16  14
  (16  14)
 2
5.
13  35  13  ( 35)
  ( 35  13)
  22
6.
 29  65  65  29
 36
7.
18r  27r  18r  ( 27r )
  ( 27r  18r )
  9r
8.
9c  ( 12c )  9c  12c
 21c
9.
10.
11.
 7x  45x  45x  7x
 38x
89  89
13. c + 5  if c = 19
c  5  19  5
14  14
 14
 14
12.
 7   ( 7)

7