Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations
Properties of Inequalities
If a < b then a + c < b + c and a c < b c
a b
If a < b and c > 0 then ac < bc and <
c c
a b
If a < b and c < 0 then ac > bc and >
c c
b ab
a =
c c
ab + ac = a (

Common Algebraic Errors
Error
Reason/Correct/Justification/Example
2
2
0 and 2
0
0
Division by zero is undefined!
32 9
32 = 9 ,
(x )
(x )
2 3
2 3
x5
a
a a
+
b+c b c
1
x 2 + x 3
2
3
x +x
a ( x 1) ax a
x2 + a2 x + a
x+a x + a
( x + a)
x n + a n and
n

Functions and Graphs
Constant Function
y = a or f ( x ) = a
Graph is a horizontal line passing
through the point ( 0, a ) .
Line/Linear Function
y = mx + b or f ( x ) = mx + b
Graph is a line with point ( 0, b ) and
slope m.
Slope
Slope of the line contai

Logarithms and Log Properties
Definition
y = log b x is equivalent to x = b y
Logarithm Properties
log b b = 1
log b 1 = 0
log b b x = x
b logb x = x
log b ( x r ) = r log b x
Example
log 5 125 = 3 because 53 = 125
log b ( xy ) = log b x + log b y
Special