Mathematics 96 (3581)
CA (Class Addendum) 3: Inverse Properties
Mt. San Jacinto College
Menifee Valley Campus
Spring 2013
__________________
Name
This class handout is worth a maximum of five (5) points. It is due no later than the end
of class on Friday, 8 March.
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 that the solutions to a particular exercise set elude you, skip to another one.
You are being given approximately 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.
Every real number has an additive inverse. That is, every number has an opposite in the
sense that whenever the number and its opposite are added, the result is zero. For
example, 3 and -3 are additive inverses of one another since 3 + (-3) = 0. Similarly, -7
and 7 are opposites of one another since -7 + 7 = 0. Zero is its own opposite: 0 + 0 = 0.
Each variable expression has its own opposite. For example, 8v and -8v are additive
inverses, as are  4z 2 and 4z 2 . The following property makes the notion of additive
inverse precise:
The Inverse Property of Addition If x is a real number, then
x + (-x) = 0
(1)
The following three equations are expressions of The Inverse Property of Addition:
-2 + 2 = 0
6a + (-6a) = 0
(5x – 4y) + [- (5x – 4y)] = 0.
Every nonzero real number has a multiplicative inverse. That is, every nonzero number
has a reciprocal in the sense that whenever the number and its reciprocal are multiplied,
the result is one. For example, 6 and the fraction 1/6 are multiplicative inverses since
6(1/6) = 1. Similarly, -1/9 and -9 are reciprocals since (-1/9)(-9) = 1. Zero does not have
a multiplicative inverse since the expression 1/0 is undefined.
1
Each nonzero variable expression has its own reciprocal. For example, provided y is not
zero, 5y and 1/(5y) are reciprocals. Similarly, provided x is not zero or z is not zero,
-3xz and 1/(-3xz) are multiplicative inverses. The following property makes precise the
notion of multiplicative inverse:
The Inverse Property of Multiplication If x is a real number and x is not zero, then
x
1
x
 1 , x0
(2)
The following three equations are expressions of the Inverse Property of Multiplication:
13 
1
= 1
13
-72t 
1
= 1
 72t
1
 [8 – 5h] = 1
8  5h
Example 1. Write an equation that expresses the Inverse Property of Addition utilizing
the real number expression 4d.
Solution: We are asked to display that the sum of 4d and its opposite is zero. First, we
must express the opposite of 4d. By placing a minus sign in front of 4d we obtain -4d, an
expression that will always represent the opposite of 4d (for more on this, see the
discussion in the NOTE below.). We can complete the exercise by following the pattern
displayed in formula (1). That is, we’ll substitute 4d for x in formula (1). Then, formula
(1)
x + (-x) = 0
becomes
4d + (-4d) = 0.
One answer is: 4d + (-4d) = 0. Another answer is: -4d + (4d) = 0.
(NOTE: While not a mathematical proof, the following table provides suitable evidence
that 4d and -4d are always opposites of one another. Since actual values of both 4d and
-4d ultimately depend on the specific value of d itself, let’s start by choosing some
arbitrary values for d. The left-hand column of the table lists five such values: 3,-4,7,-5
2
and 0. For each of these values for d, the corresponding values of 4d and -4d are given in
the center and right-hand columns, respectively.
d
3
-4
7
-5
0
4d
12
-16
28
-20
0
-4d
-12
16
-28
20
0
Note that for each real value of d, the resulting values for 4d and -4d are always opposites
of one another. For example, consider the third row of the table. In the first column, we
see the value of d is -4. Evaluating 4d when d = -4 (i.e. replacing the d with parentheses
and inserting -4) yields 4(-4) = -16. Note that this is the value found in the second
column, third row. Similarly, -4d yields the value -4(-4) = 16 (the value in the third
column, third row). Since 16 and -16 are opposites, so are 4d and -4d! That is, when d =
-4, the table shows that 4d and -4d are indeed opposite real numbers. The same is true for
the other four values of d.)
Example 2. Write an equation that expresses the Inverse Property of Multiplication
utilizing the real number expression -5t.
Solution: Since -5t must be nonzero to have a multiplicative inverse, we will assume that
-5t is not zero. Therefore, we may assume that t itself is not zero. Since the operation is
multiplication, we don’t want the opposite of -5t. Rather, we want the reciprocal of -5t.
That is, we’re looking for the expression that, when multiplied to -5t, yields the real
number one. Constructing a fraction with -5t as the denominator and the number one as
the numerator does the trick. That is, the reciprocal of -5t is: 1/(-5t) or -1/(5t). Note that
these fractions represent real numbers. Since t is not zero by assumption, neither is -5t or
5t. Therefore, we are not dividing by zero so both fractions represent real quantities. We
can complete the exercise by following the pattern displayed in formula (2). We’ll
replace the x in formula (2) with -5t. That is, formula (2)
x
1
x
 1 , x0
becomes
-5t 
One answer is: -5t 
1
= 1, t ≠ 0
 5t
1
1
= 1, t ≠ 0 . Another answer is:
 (-5t) = 1, t ≠ 0.
 5t
 5t
3
Exercise 1. (To receive full credit (one point), you must complete at least three of the
following four parts correctly).
a. Write an equation that expresses the Inverse Property of Addition utilizing the real
number expression 8k.
b. Write an equation that expresses the Inverse Property of Multiplication utilizing the
real number expression -3fg
c. Write an equation that expresses the Inverse Property of Addition utilizing the
following real number expression: 6 - t.
d. Write an equation that expresses the Inverse Property of Multiplication utilizing the
following real number expression: 7 (u  w6 ) .
There is nothing special about the letter x utilized in formulas (1) and (2) above. In
Exercise 2 you will be asked to express the two inverse properties utilizing a variety of
symbols. Here are two examples.
Example 3. Express the Inverse Property of Addition using the variable c.
Solution: One way to interpret this request is that we are being asked to display the
Inverse Property of Addition using c rather than x in formula (1). Therefore, replacing x
with c in x + (-x) = 0, we have
c + (-c) = 0
One solution is: c + (-c) = 0. Another solution is: 0 = c + (-c).
Example 4. Express the Inverse Property of Multiplication utilizing the symbol ◊.
Solution: As in Example 3, we can interpret this request as a restatement of formula (2)
using ◊ rather than x. That is, with ◊ substituted for x,
4
x
1
x
 1 , x0
becomes

1

 1 , 0
One solution is:

1

 1 , 0
Another solution is:
1
  1 , 0

Exercise 2. (To receive full credit (one point), you must complete at least three of the
following four parts correctly).
a. Express the Inverse Property of Addition using the variable d.
b. Express the Inverse Property of Multiplication utilizing the variable v.
c. Express the Inverse Property of Addition using the symbol $.
d. Express the Inverse Property of Multiplication utilizing the symbol Ω.
5
Notice that Exercise 2 provides four additional ways to express the inverse properties!
That is, (a) and (c) are simply restatements of formula (1) and (b) and (d) are just
restatements of formula (2).
In Exercise 3, you will be asked to “finish” applications of one of the two inverse laws.
In other words, as written, each equation will be missing a symbol (or two) that you must
insert to create an equation that expresses one of the inverse properties. Here are two
examples.
Example 5. Insert the missing symbol(s) (e.g. parentheses, a constant or a variable
expression) to create an equation that expresses an inverse property.
0 =
+ 8
Solution: Since we see addition, and the equal sign, it appears we are to complete an
application of the Inverse Property of Addition. The left side of the equation, 0, looks
complete, in that it could already be one side of an equation that expresses an inverse
property (of addition). The right side, + 8, appears to be missing something!! That is,
if we inserted the constant -8 between the equal sign the plus sign, we’d have the
expression -8 + 8. The entire equation would then become:
0 = -8 + 8.
This equation takes the form
0 = a + (-a)
where the letter a corresponds to the constant -8 (and -a corresponds to the constant 8).
But the equation 0 = a + (-a) is equivalent to formula (1), with a substituted for x).
Therefore, inserting a -8 between the equal sign and the plus sign completes an
application of the Inverse Property of Addition and thus completes the exercise.
Example 6. Insert the missing symbol(s) (e.g. parentheses, a constant or a variable
expression) to create an equation that expresses an inverse property.
1
= 1
 3z
Solution: Since we see division on one side of the equation, and a one alone on the other
side, it appears we might be able to insert symbols to complete an application of the
Inverse Property of Multiplication. Since we have a one, and only a one, on the right
hand side of the equation, it looks like the right hand side is complete, in that it could
already be one side of an equation that expresses an inverse property (of multiplication).
As written, the left hand side doesn’t even appear equivalent to the right hand side (unless
z = -1/3), much less an equation that expresses an inverse property. We need to multiply
the fraction 1/(-3z) by a number so that product equals one. What number is needed?:
6
The reciprocal of 1/(-3z). That is, if we insert the factor -3z on the left hand side, we’ll
be done. We’d then have
-3z 
1
= 1
 3z
Notice that this equation now has the form
x
1
x
 1 , x0
where x corresponds to -3z ( and 1/x corresponds to 1/(-3z)). But the equation x · 1/x =
1, x ≠ 0, is formula (2)! Therefore, inserting -3z as we did completes an application of
the Inverse Property of Multiplication and thus completes the exercise.
Exercise 3. Insert the missing symbol(s) (e.g. parentheses, a constant or a variable
expression) to create an equation that expresses an inverse property. (To receive full
credit (one point), you must complete at least three of the following four parts correctly).
PLEASE USE A PENCIL OR INK OTHER THAN BLACK!
a.
m+
= 0
b.
1 = ( 7x )
c.
(9 - r) +
d.
p(6 - j)
r
= 0
= 1
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 inverse properties to
illustrate this correspondence.
Example 7. The following equation expresses the Inverse Property of Addition.
5d - g + [- (5d – g)] = 0
Comparing it to formula (1), what expression in the equation corresponds to the variable
expression -x in formula (1)?
7
Solution: The equation above and formula (1), x + (-x) = 0, are both expressions of the
Inverse Property of Addition. One way to determine the correspondence as requested is to
write down both equations in a vertical format as follows:
5d - g + [- (5d – g)] = 0
x + (x ) = 0
Notice that as we read both equations simultaneously as we would read a book (from left
to right), we see that the expression –(5d – g) in the upper equation corresponds to the
variable expression -x in the lower equation. Therefore, the answer is: -(5d – g)
corresponds to -x (or -x corresponds to –(5d – g)).
Example 8. The following equation expresses The Inverse Property of Multiplication.
1
(3 – x) = 1
3x
Comparing it to formula (2), which expression in the equation corresponds to the variable
1
expression
in formula (2)?
x
1
 1 , are both expressions of the
x
inverse property of multiplication. One way to determine the correspondence requested in
Example 8 is write down both equations in a vertical format as follows:
Solution: The equation above and formula (2), x 
1
(3 – x) = 1
3x
x 
1
x
 1
Notice that as we read both equations simultaneously as we would read a book (from left
1
to right), we see that the expression
in the upper equation corresponds to the
3x
variable x in the lower equation. Reading further, we see that the expression 3 – x in the
1
equation corresponds to
in formula (2). Therefore, the answer is: 3 –x corresponds to
x
1
1
( or
corresponds to 3 –x ).
x
x
NOTE: To successfully complete the exercise that follows, it is important to use
formulas (1) and (2) as written. That is, even though multiplication and addition are
8
commutative, it is important not to change the order of x and –x in formula (1) and x and
1/x in formula (2) when answering the questions in the following exercise. At the same
time, it is important to understand that, in general, when it comes to representing
opposites with variables, either of the opposites can be represented by x. Then, the other
number can (and must always) be represented by –x (read: “the opposite of x” or
“negative x”). For example, 7 and -7 are opposites. If x represents 7, then –x must
represent -7. The correspondence between –x and -7 feels very natural in the sense that
the expressions –x and -7 are both negative quantities. But x could just as well represent
the -7. In this case, -x would represent (positive) 7. YES, A VARIABLE
EXPRESSION WITH A NEGATIVE SIGN CAN REPRESENT A POSITIVE
QUANTITY!!
Exercise 4. To receive full credit (two points), you must complete all of the following
four parts correctly. If you complete two or three parts correctly, you’ll earn one point.
If you complete less than two parts correctly, you’ll receive zero points. NOTE: Even
though addition and multiplication are commutative, you must use the equations in
formulas (1) and (2) exactly as they appear when making the comparisons below.
a. The following equation is an expression of the Inverse Property of Addition.
-(45+ 8q) + (45 + 8q) = 0
Comparing it to formula (1), which expression in the equation corresponds to the variable
x in formula (1)?
9
b. The following equation is an expression of the Inverse Property of Multiplication.
1
 (f–l) = 1
f l
Comparing it to formula (2), which expression in the equation corresponds to the variable
1
in formula (2)?
x
c. The following equation is an expression of the Inverse Property of Addition.
(2 - a) + (a - 2) = 0
Comparing it to formula (1), which expression in the equation corresponds to the variable
-x in formula (1)?
10
d. The following equation is an expression of the Inverse Property of Multiplication.
(-5p)
1
= 1
 5p
Comparing it to formula (2), which expression in the equation corresponds to the variable
x in formula (2)?
11