EQUATIONS & INEQUALITIES
Properties
WHAT ARE EQUATIONS?
Equations are mathematical sentences that state
two expressions are equal.
 Example: 2x – 5 = 3(x + 4)

WHAT ARE THE PROPERTIES OF EQUALITY?

In order to solve
equations in, you
must perform
operations that
maintain equality
on both sides of the
equation using the
properties of
equality.

Properties of Equality

Reflexive property of equality

Symmetric property of equality

Transitive property of equality

Addition property of equality

Subtraction property of equality

Multiplication property of
equality

Division property of equality

Substitution property of
equality
PROPERTIES OF EQUALITY
Reflexive Property of
Equality
Symmetric Property of
Equality
a=a
 A number is equal to
itself.
 -5 = -5


If a = b, then b = a.
 If numbers are equal,
they will still be equal
if the order is
changed.
 If x = 2, then 2 = x
PROPERTIES OF EQUALITY
Transitive Property of
Equality

If a = b and b = c, then a = c.

If numbers are equal
to the same number,
then they are equal to
each other.
Addition Property of
Equality

If a = b, then a + c = b + c.
Adding the same
number to both sides
of an equation does
not change the
equality of the
equation.
x=6
x+2=6+2

PROPERTIES OF EQUALITY
Subtraction Property of
Equality

If a = b, then a − c = b − c.
Subtracting the same
number from both
sides of an equation
does not change the
equality of the
equation.
x=6
x−2=6−2

Multiplication Property
of Equality
If a = b and c ≠ 0, then
a • c = b • c.
 Multiplying both sides
of the equation by the
same number, other
than 0, does not
change the equality of
the equation.
x=6
x•2=6•2

PROPERTIES OF EQUALITY
Division Property of
Equality

If a = b and c ≠ 0, then
a ÷ c = b ÷ c.

Dividing both sides of the
equation by the same
number, other than 0, does
not change the equality of
the equation.

If x = 6 then
x 6

2 2
Substitution Property of
Equality
If a = b, then b may be
substituted for a in any
expression containing a.
 If two numbers are
equal, then
substituting one in for
another does not
change the equality of
the equation.

PROPERTIES OF OPERATIONS (REVIEW)
Property
Rule
Example
Commutative property
a+b=b+a
of addition
3+5=5+3
Commutative property
a•b=b•a
of
multiplication
3•5=5•3
Associative property of
(a + b) + c = a + (b + c)
addition
(3 + 5) + 6 = 3 + (5 + 6)
Associative property of
(a • b) • c = a • (b • c) (3 • 5) • 6 = 3 • (5 • 6)
multiplication
Distributive property
of
multiplication over
addition
a • (b + c) = a • b + a • c
3 • (5 + 6) = 3 • 5 + 3 • 6
WHAT PROPERTY?????
Equation
What property ?
3x – 12 = 15
original
3x = 27
x=9
WHAT PROPERTY?????
Equation
x
 5  4
2
x
 9
2
x = -18
What property ?
original
WHAT ABOUT INEQUALITIES?
Property
How it works
If a > b and b > c, then a > c.
If 10 > 6 and 6 > 2, then 10 > 2.
If a > b, then b < a.
If 10 > 6, then 6 < 10.
If a > b, then –a < – b.
If 10 > 6, then –10 < –6.
If a > b, then a ± c > b ± c.
If 10 > 6, then 10 ± 2 > 6 ± 2.
If a > b and c > 0, then a • c > b • c.
If 10 > 6 and 2 > 0, then 8 • 2 > b • 2.
If a > b and c < 0, then a • c < b • c.
If a > b and c > 0, then a ÷ c > b ÷ c.
If 10 > 6 , then 10 • –1 < 6 • –1.
If you multiply by a negative on both
sides, switch the inequality sign
If 10 > 6 and 2 > 0, then 8 ÷ 2 > 6 ÷ 2.
If 10 > 6 , then 10 ÷ –1 < 6 ÷ –1.
If a > b and c < 0, then a ÷ c < b ÷ c.
If you divide by a negative on both
sides, switch the inequality sign
Laws of Exponents/Review
- Multiplication of Exponents
- Power of Exponents
- Division of Exponents
- Exponents of Zero
- Negative Exponents
Multiplication of Exponents
General Rule:
b b  b
m
n
Specific Example:
m n
Power of Exponents
General Rule:
b 
m n
b
mn
bc 
b c
n
n n
Specific Example:
4  2
3
4 2
3 3
Division of Exponents
General Rule:
m
b
mn
b
n
b
Specific Example:
Exponents of Zero
General Rule:
b 1
0
Specific Example:
Negative Exponents
General Rule:
b
n
1
 n
b
and
Specific Example:
and
1
n
b
n
b