Order of Operations
1.
2.
3.
4.
Parentheses: inside out [], (), {},
Exponents
Multiply or divide: Left ➔ Right
Add or subtract Left: ➔ Right
Coordinate Graphing
QII
[._ Division Ba�
-,+
+,+
-,-
+,QIII
Examples
----------------------------------------------------------------------
62 + 4 . 3
36 + 4 • 3
9·3
3 - 12 + 4
3- 3
0
42 • 2+[7- (32- 5)]
16 • 2+[7 - (32 - 5)]
32+[7- 4]
35
QI
QIV
!J
(-2, 5)
(2, 3)
++
(x, y)
Left up
-right -down
(5,-1)
(-4,-2)
Real Numbers
Integers
Natural or Counting Numbers : NE {l, 2, 3, 4, ...}
Whole Numbers: WE {0, 1, 2, 3, 4, ...}
Integers: IE {..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational Numbers (R): Any number that can be written as a
fraction alb. Natural, whole, and integers are rational.
• Irrational Number (Q): Numbers that CANNOT be written
as a fraction, like non-repeating decimals and radicals
• Real Numbers (R): All rational and irrational numbers
How to Add Integers
----------------------------------------------------------------------
•
•
•
•
Numbers are either rational or irrational
'r
[
\..
�
,"
Counting]
Whole
Integers
Rational
'
Irrational
I
Real
•• _,:.,,'J
.
-.
• Same signs: Add and keep the sign
• Different signs: Subtract large- small
Keep sign of the larger number
How to-Subtract Integers-----------------------------------------
• Change all subtraction problems to addition
a-b=a+(-b) a- (-b)=a+b
• Follow rules for addition of integers
How to Multiply and Divide Integers
----------------------------------------------------------------------
• Two Numbers
Same signs ➔ Positive
Different signs ➔ Negative
• More than two Numbers
Odd ➔ Negative
Even ➔ Positive
:-:..,j
Properties
Add-Subtract-Multiply-Divide Polynomials
• Commutative Property of Addition/Multiplication
a+b = b+a
ab=ba
• Associative Property of Addition/Multiplication
(a+b)+c=a+(b+c) (ab)c=a(bc)
• Distributive Property
a(b+c)=ab+ ac
• Identity Properties
a • 1=a and 1 • a=a
a+0=a and O+a=a
• Inverse Properties
a+ -a = 0
1
The number /a is called the reciprocal or multiplicative inverse.
Adding and Subtracting
________________________________________ _
_
• Properties of 0
a • 0=0 and O • a = 0
0/a=0 where a=0
Division by O is undefined
Algebra I Study Guide
• Combine like terms only
• Rewrite subtraction problems
• Watch SIGNS!
Examples
----------------------------------------------------------------------
(3x+4)+(2x-1)
5x +3
(x2+x +1)+(2x 2+3x +2)
3x2+4x +3
(4d-2)-(5d-3)
4d- 2-2d.+3
-d+1
(4x2+3x +2)- (2x 2 - 3x +7)
4x2+3x +2- 2x2 + 3x - 7
2x2+6x- 5
�
w w�
2x(5x- 4)=10x2 - 8x
x'+3
-2(5a- 4)=-lOa+8
5c+2
Equations
More Equations
3x = 27¢Divide for answer¢ x = 9
� = 18¢Multiply for answer¢ x = 54
3
-------------------------
Combine Like Variables
4n-2 + 7n = 20
lln-2 = 20
lln = 22
n=2
----------------------Solve in reverse order of operations Distribute to Eliminate Parentheses
■ +/-first
3(r-4) = 9
X + 5 = 12
■ then xi+
=9
3r-12
4
Watch for negatives
3r = 21
x=7
-7-13y = 32
r=7
4
-13y = 39
x = 28
y = -3
Multi-Step Equations
Two-Step Equations
1.
2.
3.
4.
Variables on Both Sides of Equation
-Move the smaller variable by adding the opposite to both sides of
the equation. (look to move negative variables)
(+3y) 5y-10 = 14 -3y (+3y)
8y-10 = 14
8y = 24
y=3
More Equations
Literal Equations
Treat variables like numbers...
T + M = R, solve for T
T = R-M
ax + r = 7, solve for x
ax = 7-r
x = 7-r
a
A = lw, solve for w
A=w
l
Big Tip: Look to get rid of fractions on variables.
Multiply all terms by the common denominator.
Proportions
Cross Products are equal
-
.....L-r.,...,7._c_
=be: � d
_3_ = _9_
4
12
9
----• P¢ D: Move decimal two to the left (2 �)
■ D¢ P: Move decimal two to the right (2➔)
• P¢ F: Write number over 100 and reduce
Hint: Write percents with one decimal over I 000, two
decimals of I 0, 000, etc.
■ F¢ D: Divide munerator by denominator
Converting Percents
_P�_rce�1t P�oblems _using Equations _____________________________
• Write the equation as you read the problem.
into, out of➔+
what➔ x
of➔ x
is➔ =
• Write percents as decimals or fraction
_Percent Change ___________________________________________________
Subtract amounts and
divide by orieinal
a·b = c·d
4 • 9 = 3 • 12
36 = 36
Examples
-------- ---------------------------------------------------------------------15
m = l3.5
10m = 135
_ 10
ill - -9-
Percent Problems
Percent = change
original
- --- •
Proportion: An equation showing that two ratios are equal.
_M_ulti-Step Example
3x-2(x + 6) = 4x - (x-10)
3x-2x-12 = 4x-x + 10
x-12 = 3x + 10
-12 = 2x + 10
-22 = 2x
-11 = X
Distribute
Combine
Move variables to the same side of the equation
Solve remaining equation
.____.I27 I
25 1
�
_I
X
_ 25
9
X
27
9x = 675
x = 75
Test Taking Tips
• Skip problems that you don't know how to do and come
back.
• Write down all your steps. In other words, don't do the work
in your head.
• When you panic, stop, and do something that relaxes you,
like close your eyes and take a deep breath.
• If you finish early, go back and do the problems again. Don't
look at your previous work.
Slope Formula
m
=
Slope
rise
run
Slope Intercept Form
Y2 -y,
Slope-Intercept Form:
■ y=mx+b...,---
X2-X1
\_ I
/ ¢ Positive Slope
\ ¢ Negative Slope
I ¢ Undefined Slope (x = 8)
Finding an Equation:
■ Find the slope
■ Use one point to find the y-intercept
Parallel Lines: Same slope
Perpendicular Lines: Reciprocal opposite of the given line
.lm
Examples
-1
3
=
(5,3); (4, -1)
(4, 8); (4, 2)
m=4
m is undefined
3-(-1)
5-4
m = 3; (2, 6)
y = mx+b
6 = (3)(2)+b
6 = 6+b
b=0
y = 3x+0
y = 3x
8-2
4-4
.-;o
3 y+6
=
3x 2y+18
3x-2y = 18
X
=
3
(1, 8); (4, 2)
y = mx+b
2 = (-2)(4)+b
2 = -8+b
10 = b
y = -2x+10
When dividing and multiplying by a negative number,
REVERSE the inequality.
.
G----------------- ---------------------------------------------------------------------Examples
15--t>10
t
4
--t>-5
4
t<20
t
�
4
4
)
Substitution Method: Use when you can easily find the value of
either the x or y variable.
Inequalities: Solve twice using+AND - and reverse the
inequality
Examples __________________________________________________________
Ix-6 I>2
I 3x -2 I = 10
x-6>2 OR x-6<-2
x>4
Substitution/Elimination
Equations: Solve twice using+AND -
3x = 12 OR 3x = -8
8
x = 4 OR x =-3
-x+4
Graphing Symbols
■ Open:
< and >0
■ Closed
� and :2:: •
Absolute Value Equations/Inequalities
3x-2 = 10 OR 3x-2 = -10
•
2 ••••......
Solving Inequalities
Standard Form of a Linear Equation
■ Ax+By = C
•rn= -A
B
• The x will always be positive
• No fractions: multiply by common denominator
-���!!!��---------------------- 2------·
y
=
Finding a Line Examples
----------------------------------------------------------------------
Standard Form
7y = -5X-35
5x+7y = -35
slope
x and y intercepts:
■ x: Substitute 0 for y and solve for x
■ y: b is the y-intercept
-¢ m = 0 ( y = S)
m=3
y-intercept,
where the line
crosses the y-
x>8 OR x<4 0
0
15x-5y = 30
y = 2x+3
15x-5(2x+3) = 30
15x-10x-15 = 30
5x-15 = 30
5x = 45
x=9
y = (2)(9)+3
y = 21
Answer: (9, 21)
Elimination Method: Use when there is no easy way to
substitute. Multiply one or both equations by a corresponding
factor. (Use opposites so you can add equations together.)
2x+3y = l<oa
10x+15y+5
xS
-10x-14y = -6
5x+7y = 3 8
y = -1
�
�
Answer: (2, -1)
2x+(3)(-1) = 1
2x-3 = 1
2x = 4
x=2
Consistent/Inconsistent Systems
Graphing Inequality Systems
Inconsistent System: No solution, Lines are parallel
2x-y=-1
4x - 2y= 4 �
Of- -6
Both lines have
the same slope
Consistent System:
■
>< +---►
>< ◄
►
Independent: lines intersect, One solution
>> above
<:S below
X
1
1
Type of line
Where to shade
y>x+3
·.,,.
·"
■ Dependent: Infinitely many solutions, Same line�
x-y=2
2X - 2Y = 4 �
0=0
'•'::,·< ...
Second equation is
2x the first equation
-1
Y<-x-1
-2
Coin Problems/Age Problems
Wind/Current and Digit Problems
Wind/Current_____________________________________________________ _
Coins
Usually involved with number of coins and monetary value
Need speed when traveling with the wind and against the wind
A coin bank has 250 dimes and quarters worth $39.25 How
many dimes and quaiters are there?
User•
(r+w)2.5=750
(r-w)2=750
r-rate; w = wind
.lOd+.25q= 39.25
d+q =250
Age _________________________________________________________________
Represent a person's age in the past, present, or future
Digit
---------------------------------------------------------------------Write value of number in expanded from (t: tens; u: ones)
A father is 32 years older than his son. In four years, the
father will be 5 times older. How old are they now?
■ Two digit number: lOt+u
■ Reverse digits: l0u+t
■ Sum of digit: t+u
s+32 = f'""" (age now)
f+ 4 = s + 4 '"""(age in 4 years)
f+4= 5(s +4)
Laws of Exponents
Product of Powers: X
111
•
11
X
= Xm+n
mn
m•n
Power of Power Property: (X ) = X
•
m
m m
Power of a Product: (xy) = x y
Hint: even powers ➔ Positive answer
odd powers ➔ Negative answer
xm=xm-n
m>n
xm
m<n
x"
[ �J
Power of Fractions:
n
11
a
= n
b
n
x- = 1
O
X =
1
1
xn-m
t=d
Scientific Notation
4
IO
6
x . x =X
6
(x )4
24
x
=
s s
(xy)5 = x y
2
7'
[ ;J
6
-6
5° = 1
Examples
----------------------------------------------------------------------
,�--
463,000,000= 4.6s X 10 8
.000597= 5.97 X 10"4
(-4) =16
(-4)3 =-64
x4 =
A number written as a product with two factors:
■ A number between 0 and 9 and
■ A power of 10
Move right for a negative
exponent; left for positive
1
--;;:-
2.5 X 106
5 X 102
= x
6
7
1
Count the number of
times you need to move
the decimal to make a
number less than 10
7
2
12 X 10"
l.2 x 10· 1
0
O
.5 X 10
4
5 X 103
0
0
0