Download 2.6 Notes

Mathematician
Honors Algebra 2: 2.6 Theorems and Proofs
Properties of Equality
•
•
•
•
•
Reflexive property
Symmetric property
Transitive Property
Addition Property
Multiplication Property
If a = b, then b = a
If a = b and b = c, then a = c
If a - thena+c=b+c andc+a=c+b
If a - then ac = bc and ca = cb
Field Properties of Real Numbers
•
•
Closure Properties
Commutative Properties
Associative Properties
Identity Properties
Addition
Multiplication
a + b and ab are unique real numbers
a+b=b+a and ab=ba
(a + b)+c = a + (b + c) and (ab)c = a(bc)
There are unique real numbers 0 and 1 such that
a+0=a and0+a=a
a•l=a
and 1•a = a
Inverse Properties
of Opposites
of Reciprocals
Distributive Property
and
1
a
I and —•a=l
1
a
+ c) = ab + ac and (b +
= ba + bc
Substitution Principle
an expression may be replaced by another expression that has the
same value.
Definition of Subtraction
Va,beÐR
Definition of Division
a
Va,be Yt,b 0,
b
1
b
or a 7•b=a•—
1
b
Multiplication Property of 0
Multiplication Property of— 1
Cancellation Properties
of Addition
of Multiplication
Va e YRa(—l)
If a + c —b + c then a = b; if c+a = c + b then a = b
e YR;c # 0 If ac = bc then a = b; if ca = cb then a = b
1
1
1. If a # 0, b # 0 and — —then a=b
a b
1
b.
1
1
1
ab • —ab•
a
b
1
1
c. ba • —ab•
a
b
d.
ÀSSOCIC+IVC Pro
e.
Invcrçc ?rcpertg
f.
Id-cnft
Reap
mmÜTlO
2. Prove: If a O, then
a
b.
b
a
1
a
a
1
c.
a
d.
a
a
19tT1bvhvc Pro
Inverse Prop
Reciprocals
b
a
b
a
a
Pro cre
3. Prove: (a-b) + b = a
a. (a-b) + b = [a+(-b)] + b
b.
c.
d.
(a-b) + b = a
4. Prove For every real number a, a •0=0 and O •a = O
a. a is a real number
Cl'v¿lñ
b.
c.
Mvlft
.O=a.O
e.
QancÐÐo.+1TA
g.
and O
Cornm :
5. Prove. If 2 (3x+5) + 8x = 38 then x=2
a. 2 (3x+5) +
38
Given
Prcp
c.
38
d.
10 = 38
e. [6x+
+ 10 = 38
f.
DI-cp
+ 10 = 38
10 = 38
g.
h.
+ 10 + (-10) = 38+(-10)
i.
+ [10+(-10)] = 38+(-10)
k.
= 38+(-10)
l.
= 28
m.
n.
Commc+a-hv-c.Prc
1
1
14
14
•
14
o.
1.x=A.28
14
p.
1
—
• 28
14
28
• 28
kgoC\0hve
PCDO O-è
V?D/.-Pro
Prop o?
S obsHå-v€lOu
q.
Txp as
Rem rocals