Download Algebra 3 Stugent Notes 1.2 Properties of Equality

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Algebra 3 Stugent Notes 1.2
Properties of Equality & Inequal ity
The Seesaw Problem
Suppose two people of equal weight are sitting
distance from the fulcrum
on opposite ends of a seesaw and the same
(the support for the seesaw). Experience suggests that the
seesaw is balanced. What happens when one person is given a bag of sugar? Is the seesaw
balanced? How can you make it balance again without
one person moving?
Properties of Equality
When two expressions have the same numerical value, they are said to be equal. If the
two expressions are written
with an equal-symbol between them, the statement
can be
called an equation.
2 + 3 = 12 - 7
Consider the equation:
2
+
3
?
+
9:: 12 - 7
The point of the example is that you cannot perform
equation; if you do, the resulting
an operation on just one side of an
equation normally becomes a false statement.
On the other hand, if the same operation
is done to both sides of an equation, the
resulting equation remains a true statement.
4 x 3
=
10 + 2
?
4 x 3 + 5 :: 10 + 2
12 + 5 = 12 + 5
17
=
+
5
17
Properties
Addition Property of Equality
of Equality
If
a=h , then a+-c
x
b+:«:
I
Subtraction
Multiplication
property
Property
of Equality
of Equality
Division Property of Equality
If
a= h , then a-e=b-e
If
a
If
a b
a = b , then - = e e
= b , then
ae
= be
,nf?~ - pRoP
0F
E9 .
Examples: Gi~e reasons why each s atement is true.
1.
Since 12-5=7,
then 12-5+5=7+5
Reason:
2.
Since 6+8=10+4,
then 6 + 8 - 8 = 10 + 4 - 8
Reason:
3.
Since 3 x 2 = 5 + 1,
then 3x2 = 5+1
Reason: DIU/Sib..,
2
4.
Since 2+1 = 3
Using the properties
5.
5+3=10-2
-~
isolate the number 5 on one side of the
6.
7.
5(-2)=(-1)(3.2+4)
-3
-'Z.
_j_
_.
s-
9.
2.+"'1
3
30
5" ---
-
'"3.2-
3"
~(
Symbols:
2=3+5-6
-8,+b
s:
s-
~5
>
means
L, R I:?fTb7(.
<
means
L
>
means
<
means
Rewrite the English statements
A.)
t:~s, fl+l/-v
(? ~ el4f1:;Y(
t
rrHA
sxs
as mathematical
0,<
THlfu
17.f1Jv 01<
2.
3 is larger than -15
3.
-6 is smaller than or equal to -6
4.
1 is greater
5.
the number x is less than zero
7.
the number x is positive
than or equal to -9
12-9
6.
TO
statements ..
4 is less than 13
-b
E(Jvll'h._
~vv1l
1.
-t !:_
symbol.
3·2·5
10
3=
~
Fa £" •
10
Properties "'of Inequality
Inequality
=
-3+6
-..
-7_
EQ I
P. O. c,
2 - 3+ 6
30 =2+4
5
0 F-
Reason: . M lJ('T/fI,ICIfTlOv
S~/O-2.-.3
8.
S (.I B. Pf!of'
2
then 4x(2+1)=4x3
of equalities,
AvO
the number x is at most zero
ro
Properties of Inequality
a <b , then a+e <b +e
Addition Property of Inequality
If
Subtraction Property of Inequality
If a < b , then a +e
Multiplication Property of Inequality
If a <b , then ae <be
< b +e
If a >b , then ae »-bc
a b
If a «b , then - <c c
DivisionProperty of Inequality
If
Ex1: If -5<3,
a
c
Ex 2: If 3>-15,
Is -5+7<3+7?
b
c
a> b , then - > -
Is (-2).3>(-2)(-15)?
/VO
3
-15
Ex 3: If 3> -15, Is -3 > -3 ?
What statement can be made about multiplying or dividing an inequality by a negative
number?
';1 0 u,
MuST
FI.,I
P
THr;-
X-N~vtA-- •...'T~
Examples: Using the properties of inequality, isolate the number 5 on the left side of the
symbol.
-
<
5.
5+3<10-2
30 <10-4
5 -
~
~
,
-I
--s-~~
-1
•. 2_
-
s
~~
{;
5(-2) < (-1)(3.2+4)
-~
-.3
-3
8.
6.
57
9.
3<
3·2·5
10
.(.'2..
5" !:-
s:
s
J..'Z..
7.
2<3+5-6
~3
~~
+3
~