At first glance, it is a little difficult to see what you are being asked to
However, a good way to start is to consider carefully the definitions of each of
three numbers in the equation.
a = given real number
-1 = the additive inverse of 1
-a = the additive inverse of a
By showing that (-1)a has the same properties as the
additive inverse of a, you
will be showing that (-1)a must be the additive inverse of a.
Because you have shown that
you can now use the fact that
-a + a = 0 to conclude that . From this, you
complete the proof as follows.
Shown in first part
Cancellation Property of Addition
The list of additional properties of real numbers forms a
very important part of algebra . Knowing the names of the properties is not especially important, but
knowing how to use each property is extremely important. The next two examples
show how several of the properties are used to solvecommon problems in algebra.
In Exercises 69 and 70, use the properties of real
numbers to prove the statement.
69. If ac = bc and c≠0, then a = b
70. (-1)(-a) = a
In Exercises 71–74, identify the property of real numbers
that justifies each step.
In Exercises 75–80, use the Distributive Property to
the arithmetic mentally . For example, you work in an industry
where the wage is $14 per hour with “time and a half ”
for overtime. So, your hourly wage for overtime is
Dividends In Exercises 81–84, the dividends paid
share of common stock by the Proctor & Gamble Company
for the years 1994 through 2001 are approximated by the
0.113t + 0.13.
In this expression, the dividend per share is measured in
dollars and t represents the year, with corresponding
to 1994 (see figure). (Source: Proctor & Gamble
81. Use the graph to approximate the dividend paid in
82. Use the expression to approximate the annual
increase in the dividend paid per share.
83. Use the expression to forecast the dividend per share
84. In 2000, the actual dividend paid per share of
common stock was $1.28. Compare this with the
approximation given by the expression.
85. Geometry The figure shows two adjoining rectangles.
Find the total area of the rectangles in two
86. Geometry The figure shows two adjoining rectangles.
Find the total area of the two rectangles in two
True or False? In Exercises 87–90, de termine whether
the statement is true or false. Justify your answer.
91. Does every real number have a multiplicative
92. What is the additive inverse of a real number? Give
an example of the Additive Inverse Property.
93. What is the multiplicative inverse of a real number?
Give an example of the Multiplicative Inverse
94. State the Multiplication Property of Zero.
95. Explain how the Addition Property of Equality can
be used to al low you to subtract the same number
from each side of an equation.
96. Investigation You define a new mathematical operation using the symbol . This operation
(a) Is this operation commutative? Explain.
(b) Is this operation associative? Explain.