Mathematics 90 (3122)
CA (Class Addendum) 4: Identity Properties
Mt. San Jacinto College
Menifee Valley Campus
Fall 2009
____Solutions________
Name
This class handout is worth a maximum of five (5) points. It is due no later than the end
of class on Friday, 9 October.
NOTE: You may need to study this entire handout carefully several times before you
begin the exercises it contains. You may need to study example exercises carefully
several times before you attempt the exercise sets that follow them. Also, the order in
which the exercises occur may not necessarily be the order in which you complete them.
If you find the solutions to a particular exercise set elude you, skip to another one. You
are being given two weeks to complete this handout because you’ll probably need to
study it, attempt some of the exercises and then take a break, continuing with it a day or
two later.
The sum of a number and zero is the number itself. In other words, when you add zero to
a number, you do not change the value of that number. For example, 3 + 0 = 3; when you
add zero to three, you still have three! Similarly, ( x 2  4 x  1 ) + 0 = x 2  4 x  1 .
At first, this information seems not only obvious but useless. However, you will see that
it not only useful but necessary. The following property makes precise the notion of
addition by zero:
The Addition Property of Zero If x is a real number, then
x+0 = x
(1).
The following three equations are expressions of The Addition Property of Zero:
-2 + [-8 + 8] = -2
( x 4  x 2  1) + ( x 2  x 2 ) = x 4  x 2  1
( x 2  6x  3 ) + ( 9  9 ) = x 2  6x  3
The product of a number and one is the number itself. In other words, if you multiply a
number by one, you don’t change the value of the number. For example, 6 · 1 = 6.
Similarly, 7 · 1 = 7 .
Once again, this information appears obvious and perhaps useless, initially. However, to
create equivalent fractions when adding fractions, to rationalize denominators and to
1
simplify complex fractions, multiplying by one is the way to go! The following property
makes precise the notion of multiplication by one:
The Multiplication Property of One If x is a real number
x 1 
x
(2).
The following three equations are expressions of the Multiplication Property of One:
13 
23
= 13
23
-32t 
1
2 5
5x
= -32t
5x

5
5
=
(provided x ≠ 0)
1
2 5
Example 1. Which identity property is expressed by the following equation?
1
2 5

5
5
=
5
10
Solution: We are asked to determine whether the equation above is an expression of the
Addition Property of Zero or the Multiplication Property of One. In other words, does
the equation take the form of formula (1) or formula (2)? Notice that we have
multiplication (between the first two fractions, reading the equation from left to right) and
the second fraction is equivalent to one (the square root of 5 divided by itself is one).
This information strongly suggests that the given equation takes the form of formula (2)
x 1 
where x corresponds to
1
x
and 1 corresponds to
5
(If you vertically align formula
2 5
5
(2) with the given equation, you will see this correspondence.) While this much
information is enough (in this particular case) to conclude that the answer is the
Multiplication Property of One, the right side of the equation also corresponds to x. That
5
is, x also corresponds to
. The answer is: The Multiplication Property of One.
10
2
NOTES: Given that the equation in Example 1 above is either an expression of the
Addition Property of Zero or the Multiplication Property of One, the reasoning applied in
Example 1 is sufficient. However, in general, when comparing an equation to a property
formula, you should ensure that every occurrence of a variable in the property formula
corresponds to equivalent expressions in the equation (on both sides of the equal signs).
Also, if the variable in a property formula corresponds to different looking expressions
5
1
(as x does to both
and
in Example 1), make sure that these expressions are in
10
2 5
fact equivalent (equal).
In fact, the identity properties are so useful precisely because they allow “plastic surgery”
to be performed! That is, they allow changing the way an expression appears without
changing its value. Therefore, you should expect to find different looking expressions
corresponding to x in formulas (1) and (2) whenever an identity property is employed.
While this is not the case for the examples that preceded Example 1, it will be the case in
general.
Example 2. Which identity property is expressed by the following equation?
( x 2  6x  1 ) + (8 – 8) = ( x 2  6x  9 ) - 8
Solution: There is quite a bit of addition (including subtraction, which is really addition
of the opposite) here. However, this alone is not sufficient to conclude that it is the
Addition Property of Zero that is being expressed. Since formula (1) (for the Addition
Property of Zero) contains a zero and formula (2) (for the Multiplication Property of One)
contains a one, we can begin by searching for a zero or a one in the equation. There is a
zero on the left-hand side: 8 – 8. There is a plus sign just to its left. This suggests we
ought to compare this equation to formula (1):
x+0 = x
Comparing left-hand sides, we see that the x in formula (1) corresponds to x 2  6x  1 in
the equation and 0 in the formula corresponds to 8 – 8. Comparing the right-hand sides,
we see x in formula (1) also corresponds to ( x 2  6x  9 ) - 8. But, simplifying
( x 2  6x  9 ) - 8 yields x 2  6x  1 . That is, every occurrence of x in formula (1)
corresponds to equivalent (equal) expressions in the equation. Therefore, the equation is
an expression of the Addition Property of Zero.
The answer is: The Addition Property of Zero
3
Exercise 1. (To receive full credit (one point), you must complete at least three of the
following four parts correctly).
Which identity property is expressed by the following equation?
a.
( x 2  10 x  19 ) + (25 – 25) = ( x 2  10 x  25 ) - 6
Solution: The Addition Property of Zero
b.
21v
7v 3
=

15
5 3
Solution: The Multiplication Property of One
c.
( x 2  16 x  64 ) - 41 = (64 – 64) + ( x 2  16 x  23 )
Solution: The Addition Property of Zero
d.
7 4w
28w

=
6r
24rw
4w
Solution: The Multiplication Property of One
In Exercise 2, you will be asked to “finish” applications of one of the two identity laws.
In other words, as written, each equation will be missing a symbol (or more) that you
must insert to create an equation that expresses one of the identity properties. Here are
two examples.
Example 3. Insert the missing symbol(s) (e.g. a constant or a variable expression) to
create an equation that expresses an identity property.
( x 2  4x  4 ) + 1 = (4 –
) + ( x 2  4x  5 )
Solution: Notice that the entire left-hand side of the equation above is equivalent
to x 2  4 x  5 (add the like terms 4 and 1). This same expression, x 2  4 x  5 , is
present on the right-side side. As previously discussed, this is what we’d expect for any
equation that expresses an identity property. The question is: Does this equation take the
form of formula (1) or formula (2)? Since we see addition to the left of x 2  4 x  5 on
4
the right-hand side, we suspect the answer is formula (1), x + 0 = x (in this case,
rewritten as: x = 0 + x). However, for this to be the case, we must have a zero to the left
of the plus symbol. Notice that if we place a 4 inside the parentheses to the right of the
minus sign, we’ll have zero inside the parentheses. That is, we’ll have 4 – 4 inside the
parentheses. Doing so completes the exercise, since the given equation would then take
the form
x = 0+x
where the letter x corresponds to ( x 2  4x  4 ) + 1 (the left –hand side of the equation)
and x 2  4 x  5 (located on the right-hand side of the equation) in the equation and zero
in the formula corresponds the expression 4 – 4 in the equation. That is, the modified
equation now expresses the Addition Property of Zero.
Example 4. Insert the missing symbol(s) (e.g. a constant or a variable expression) to
create an equation that expresses an identity property.
21xy
7 x 3y

=
24 y
8
Solution: Since we see multiplication (on the left-hand side of the equation), we suspect
this equation could be modified to express the Multiplication Property of One. If so, the
equation must either already contain a one or we must be able to construct a one
somewhere in the equation. Notice that unless x = 8/7, the fraction 7x/8 does not equal
one. Therefore, since x can be any real number, the fraction 7x/8 is not equivalent to one.
However, to the right of the multiplication symbol, we see a fraction that is missing its
denominator. We could easily create a one here by placing a copy of the numerator in the
denominator. If we do so, we get an equation of the form
x 1 
x
where x corresponds to the fraction 7x/8, on the left-hand side, and the fraction
21xy/(24y), on the right-hand side. Moreover, we have constructed a one, of the form
3y/(3y), in the equation that corresponds to the constant 1 in the formula. .But the
equation x · 1 = x is formula (2)! Therefore, inserting 3y as we did modifies the equation
to express the Multiplication Property of One and thus completes the exercise.
Exercise 2. Insert the missing symbol(s) (e.g. a constant or a variable expression) to
create an equation that expresses an identity property. (To receive full credit (two
points), you must complete at least six of the following eight parts correctly. If you
complete between three and five parts correctly, you will receive one point). PLEASE
USE A PENCIL OR INK OTHER THAN BLACK!
5
a.
b.
c.
d.
3x 5z
15 xz

=
4
5z
20 z
( y 2  12 y  36 ) - 27 = ( 36 – 36 ) + ( y 2  12 y  9 )
21p
35
=
3p 7

5
7
25 
25
 25 25 

2
( t 2  5t  3 ) + 

) + 3 

 = ( t  5t 
4 
4
4 
 4

e.
5t
2r
10rt

=
9az 2r
18arz
f.
( z 2  7z 
g.
h.
49
25
) 4
4
49 
 49
2
= 

 + ( z  7 z  6)
4 
 4
7 9c
63c

=
6 jx 9c
54cjx
( p 2  11 p  4 ) + (
121
121
121
105
–
) = ( p 2  11p 
) 4
4
4
4
In order to recognize an application of a real number property, it is necessary to
determine how a mathematical expression, say in an exercise, corresponds to a variable in
the formula for the property. The following examples utilize the identity properties to
illustrate this correspondence.
Example 5. The following equation expresses the Addition Property of Zero.
6
( u 2  4u  4 ) + 1 = (4 – 4) + ( u 2  4u  5 )
Comparing it to formula (1), which expressions in the equation correspond to the variable
x in formula (1)?
Solution: The equation above and formula (1), rewritten as x = 0 + x, are both
expressions of the Addition Property of Zero. One way to determine the correspondence
as requested is to write down both equations in a vertical format as follows:
( u 2  4u  4 ) + 1 = (4 – 4) + ( u 2  4u  5 )
x
=
0
+
x
Notice that as we read both equations simultaneously as we would read a book (from left
to right), we see that the expressions ( u 2  4u  4 ) + 1 and u 2  4u  5 in the upper
equation correspond to the variable expression x in the lower equation. Therefore, the
answer is: ( u 2  4u  4 ) + 1 , u 2  4u  5 .
Example 6. The following equation expresses The Multiplication Property of One.
7 2
10
=
7
5 2

2
2
Comparing it to formula (2), which expressions in the equation correspond to the variable
x in formula (2)?
Solution: The equation above and formula (2), rewritten as x = x ∙ 1, are both expressions
of the Multiplication Property of One. One way to determine the correspondence
requested is to write down both equations in a vertical format as follows:
7 2
10
=
x
=
7
5 2
x

2
2
· 1
Notice that as we read both equations simultaneously as we would read a book (from left
7 2
7
to right), we see that the expressions
and
in the upper equation correspond
10
5 2
to the variable x in the lower equation (notice also that the fraction
to the number one in the formula). Therefore, the answer is:
7
2 / 2 corresponds
7 2
7
,
10 5 2
Exercise 3. To receive full credit (two points), you must complete at least three of the
following four parts correctly. If you complete two parts correctly, you’ll earn one point.
If you complete less than two parts correctly, you’ll receive zero points.
a. The following equation expresses the Addition Property of Zero.
( x 2  10 x  8 ) + (25 – 25) = ( x 2  10 x  25 ) - 17
Comparing it to formula (1), which expressions in the equation correspond to the variable
x in formula (1)?
Solution: x corresponds to both
x 2  10 x  8
and
x 2  10 x  25 - 17.
b. The following equation expresses The Multiplication Property of One.
21s
15
=
7s 3

5 3
Comparing it to formula (2), which expressions in the equation correspond to the variable
x in formula (2)?
Solution: x corresponds to both
21s
15
and
7s
.
5
8
c. The following equation expresses the Addition Property of Zero.
( x 2  16 x  64 ) - 51 = (64 – 64) + ( x 2  16 x  13 )
Comparing it to formula (1), which expressions in the equation correspond to the variable
x in formula (1)?
Solution: x corresponds to
x 2  16 x  64 - 51
and
x 2  16 x  13 .
d. The following equation expresses The Multiplication Property of One.
1
3
2 a ( a  2)

7a  2
2a a  2 
1
=
3
1
2a ( a  2)
a  a2

2a  4 2
1
Comparing it to formula (2), which expressions in the equation correspond to the variable
x in formula (2)?
Solution: x corresponds to
1
3

2a
a2
3
1

2a  4
2
and
7a  2
.
a  a2
9