Quick study academic pre algebra 600dpi

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DIVISION
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~~~--------~--------~==--------,m

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-
-

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The f ollowing sets are infinite; that is, there are n~ last numbers.
The three dots indicate continuing or never-endmg patterns.
Counting or natural numbers = {1, 2, 3, 4,..., 78, 79,... }.
Whole numbers = {O, 1,2,3,4,...,296, 297,... }.
Integers = { ••• , -4, -3, -2, -1, 0, 1, 2, 3, 4... }.
R ational num bers = {all numbers that can be written as fractions, p/q,
where p and q are integers and q is not zero} . RatlOna.1 .numbers mclude all
counting numbers, whole numbers and integers, 10 addlti.on to all proper and
improper fraction numbers, and endmg or repeat 109 deCimal numbers.

Exs: 4/9, .3, -1 8.75, .Jj6, - .J25 .
Irrational num bers = {all numbers that cannot be expressed as rational num­
bers}. As decimal numbers, irrational numbers do not end nor repeat.
Exs: 3.171171117..., f"i, -.J2, 1t.
Real numbers = {all rational and all irrational numbers} .

OPERATIONS
ABSOLUTE VALUE
Absolute val ue is the distance (always positive) between a number and zero on
the number line; the positive value ofa number. Exs: 131 = 3; 1-31 = 3; 1-.51 =.5.

ADDITION
I. Integers: When adding integers, follow these rules .
.
..
a. If both numbers are positive, add them; the Sign of the answer WIll be posItive.
b. lfboth numbers are negative, add them; the sign of the answer WIll be negative.
c. Ifone number is negative and the other is positive (in any order), subtract the two
numbers (even though they are joined by a plus sign); the sign ofthe answer will
be the same sign as the sign of the number that has the larger absolute value.
Exs: 4 + (-9) = -5; (-32) + (-2) = -34; (-12) + 14 = 2.
2. Rational numbers:
a.When adding two mixed number s, fractions, or decimal numbers, fol­
low the same sign rules that are used for integers (above), but also fol­
low the rules of operations for each type of number. .
b.For mixed numbers and fractions, make sure the fractIOns have a com­
mon denominator, then add the numbers. Mixed numbers and fractIOns
can also be changed to decimal numbers and then added.
c.For decimal numbers, line the decimal points up, then add the numbers,
bringing the decimal point straight down.

Exs: (-4 1/2) + (5 3/4) = (-4 2/4) + (5 3/4) = 1 1/4; 5.667 + (-.877) = 4.79.
3. Irrational num bers:
a. When adding irrational numbers, exact decimal values cannot b~ used.
If decimal values are used, then they are rounded and the answer IS only
an approximation. Instead, if the two irrational numbers are multiples of
the same sq uare root, radical expression, or p.i (n), then simply add the
coefficients (numbers in front) of the roots or pi (n).
Exs: 4.)3 +5.)3 =9 f 3;(-61t)+91t=31t ; 3f"i +3.J2 cannot be added any
further because the two square roots are different.

SUBTRACTION
I. Subtraction of all categories of numbers can be accomplished by adding the
..
opposite of the number to be subtra.cted.
2. After changing the sign ofthe number ill back of the mInUS Sign, follow the rules
addition as stated above. Exs: 8- (-3)=8+(+3)=11;(-15)-(9) =(-15)+(-9) = - 24.

MULTIPLICATION

1. Integers: When dividing integers, follow these ru.le ~:
a. If the signs of the numbers are the same, divide th em and make the
answer positive.
b. If the signs of the numbers are different, divide the m and make the
answer negative.
c. The sign ofthe answer does not come from the number with the larger absolute
value as it does in addition.

Exs: (-30)/(5) = - 6; (-22)/(-2) = 11; (70)/(-10) = -7.

2. Rational numbers:

a. When dividing rational numbers, follow the sign rules that are used for di­
viding integers (listed above) and the rules for dividing ea~h type of number.
b. For mixed numbers, change each mixed number to an Improper fractIOn,
invert or flip the number behind the division sign and follow the rules for
multiplying fractions.
.... .
c. For decimal numbers, first move the deCimal polOt 10 the diVisor to the
back of the number, then move the decimal point the same nU'!lber of po­
sitions to the right in the dividend. Divide the numbers, then brmg the dec­
imal point straight up into the quotient (answer).. Additional zeros can be
written after the last digit behind the decimal pomt 10 the dIvidend so the
division process can continue if needed.
3. Irrational numbers:
a. When dividing irrational numbers, follow the same sign rules that are
. .
used for dividing integers (listed above).
b. Ifradical expressions are divided and they have the same mdlce ~ , ~hen the
numbers (radicands) under the root symbols (radicals) can be divi ded.
Exs: (-M)/ (.)3)=-f5; (-J30 )/(--v6 )=-f5; W/.J2 cannot be
divided, only simplified as demonstrated in the Quick Study®Algebra Part
One study guide.

EXPONENTS/POWERS
I. Definition: an

=~.

, that is, the number written in the upper right-hand

n

corner is called the exponent or power,
and it tells how many times the other number (called the base) is mul tiplied
times itself. If an exponent cannot be seen, it equals l. Exs: 56. = 5 - 5.- 5 ­
5 - 5 - 5 = 15,625; notice that the base, 5, was multiphed times Itself 6 times
because the exponent was 6.
2. Rule: an_ am = a m+ n ; that is, when multiplying the same base, the new exponent
can be found quickly by adding the exponents of the bases that are multJphed.
Exs: (53) (54) = 57; (3 2) (7 3) (72) (3 S) = (3 7) (7 S) .
3. Rule: ant am= a n- m; that is, when dividing the same base, the new exponent
can be found quickly by subtracting the exponents of the bases that are di­
vided. The new base and exponent go either in the numerato r or 10 the de­
nominator, wherever the highest exponent was located 10 the ongmal prob­
lem. Exs: (7S) / (7 2) = 73; (3 4) / (3 6) = 1 / (3 2).
4. Rule: a-I = lIa; and lIa-1 = a; that is, a negative exponent can be changcd to
a positive exponent by moving the base to the other section of the fraction ;
numerator goes to denominator or denominator goes to numerator.
Exs: 7-3 = 11(7 3 ); 11(5-2 ) = 52; 3(2-4) = 3/(24); notice the 3 stayed in the nu­
merator because the invisible exponent is always positive l.
5. Rule: (aO)m = a om ; that is, when there is a base with an exponent raised to an­
other exponent, then the short cut is to multiply the exponents.
Ex: (-3Z 2)3 = (_3)3(Z2)3 = -27z 6.

ORDER OF OPERATIONS

I. Integers: When multiplying integers, follow these rules.
..
When a problem has many operations, the order in which the operations are

a. If the signs ofthe numbers are the same, mul~ply $1d make the answer poslJ;ive.
completed will give different answers; so, there is an order ofoperatio,!s rules.

b. Ifthe signs ofthe numbers are different, mulnply and make the answer negative.
1. Do the operations in the parentheses (or any enclosure symbols) fIrst.
c. NOfE: The sign.of the answer does not come from the number WIth the larger ab­ 2. Do any exponents or powers next.
..
solute value as it does in addition. Exs: (- 4)(5)= - 20;(-3)(-2) = 6; (7)(-10)= -70.
3. Do any multiplication and division, go~g I~ft to nght 1I1 th~ or~er they appear
2. Rational numbers:
(this means division is done before multlphcatJon If It comes first 1I1 the problem).
a. When multiplying rational numbers, follow the sign rules that are used for 4. Do the addition and subtraction, going left to right in the order they appear (thiS
multiplying integers (above) and the rules for multiplying each type ofnum.ber.
means subtraction is done before addition ifit comes flTSt in the problem).
b. For mixed numbers, change each mixed number to an Improper fractIOn,
Exs:4+2 (3+7)=4+2 (10)=4+20=24; 4075-2 +474=8-2+ 1 = 16+ 1 = 17.
and then multiply the resulting fractions .
.
SCIENTIFIC NOTATION
c. For fractions, multiply the numerators and the denommators, then reduce
A fo rm ofa decimal number where the decimal point i.~ always behind exthe answer.
actly one non-zero digit and the number is multiplied by a power often.
d. For decimal numbers, multiply them as though they were integers, then put the
Exs: 4.87 x 108 ; 3.981 X W -<>.
decimal point in the answer so there is the same number of di~ts behind the dec­
imal point in the answer as there are behind both decunal POillts ill the problem. 1. It is a method for representing very large or very small numbers :-vitho ut
writing a lot of di gits. Ex: 243,700,000,000,000 would be wntten as
3. Irrational num bers:
2.437 x 10 14 ; .000000982 would be written as 9.82 x 10-7.
a. When multiplying irrational numbers, follow the same sign rules that are
2.
A

positive or zero exponent on the 10 means the number value is more than or equal
used for integers (listed above).
. .
to one. A negative exponent on the 10 means the number value 1S less than one.
b. If radical expressions are multiplied and they have the same mdlces? then the
Ex: 5.29 x 10-10 = .000000000529 and 5.29 x 10 14 = 529,000,000,000,000.
numbers (radicands) under the root symbols (radicals) can be multiplied.
3. Operations with very large or very small numbers can ~e completed using
Exs: (-$)( f"i )=- J35; (3v1)(-4)=-12f"i .
the scientific notation form of the numbers,
WIth calculators.

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ALGEBRA CONCEPTS

PROPERTIES

DEFINITIONS

I. Add ition/Subtraction Property of Equality: If a = b, th en a + c = b + C
and a - c = b - c; that is, you can add or subtract any number or term to or

from an equation as long as you do it on both sides of the equal sign.
2. Multip lication/Division Property of Equality: If a = b, then ac = bc
and a/c = b/c (when c "# 0); that is, you can multiply or divide by any
number or term as long as you do it on both sides of the equal sign.
Remember, do not divide by zero because it is undefined .
3. Symmetric Property: rfa = b, then b = a; that is, two sides of an equa­
tion can be exchanged without changing any signs or terms in the equation.
Ex: 3n + 7 = S - 2n becomes S - 2n = 3n + 7.

1. A variable is a letter that represents a number.

2. A coefficient is a number that is multiplied by the variable. It is found in
front of a variable, but the multiplication sign is not written. If the coeffi­
cient is one, the one is not written . Exs: 5n = 5 x n; if n were 3, then 5n
would equalS x 3 or 15.
3. A term is a mathematical expression involving multiplication or division.
Terms are separated by an addition or subtraction sign. Exs: 7a is one
term; 3k + 9 is two terms; 4m 2 - Sm + 3 is three terms.

SOLUTION METHODS


4. Like (or similar) terms are terms that have the same variables and expo­
nents, written in any order. The coefficients (numbers in front) do not have
to be the same.
Exs: 4m and 9m are like terms ; 5a 2c and -7a 2c are like terms; 3r3 and -9r2
are not Iike terms because the exponents are not the same; 15z4and St4 are
not like terms because they do not have the same variables.

FIRST DEGREE, ONE VARIABLE

I. Solving an equation means you are findin g the one numerical value that makes
the equation true when it is put into the equation in place of the variable.
2. Using inverse operations is the best method for first-degree equations. Using in­
verse operations means you do the operation opposite to the one in the equation.
3. One-step eq uations: Equations having only one operation (+, -, x, or ..;.)
with the variable require only one inverse operation. If the equation has ad­
dition, then you do subtraction; if subtraction, you do add ition; if multipli­
cation, you do division; if division, you do multiplication.
Ex 1: n + 7 = -3 ~ 7 is added to n, so,
~ Subtract 7 from both sides

n + 7 - 7 = -3 - 7
n = -10 ~ giving the solution of-1O


OPERATIONS & PROPERTIES
I. Addition and Subtraction: Only like (or similar) terms can be added or
subtracted. Once it is determined that the terms are like terms, only the co­
efficients (numbers in front of the terms) are added or subtracted.
Exs: 3n + 7n - 11n = IOn - 11n = IOn + (-11n) = -In or simply -n;

Ex 2:

j

= 9 ~ a is divi ded by 3, so,

1· 3 = 9 • 3 ~ Multiply by 3 on both sides

14k2 + 5n - 10k2 - n = 4k2 + 4n.
2. Multiplication: Any terms can be multiplied. They do not have to be like
terms. When multiplying terms, multiply the coefficients (numbers in front)
and the matching variables. Ex: (-3m2n)(5m4n) = -15m 6 n 2; remember, when
multiplying, make sure the bases are the same, then add exponents.

a = 27 ~ giving the solution of27.

4. Two-step eq uatio ns:
a. Equations that have two operations connected to the variable require two
operations that are the opposites of the ones that are in the equation. It is
much easier to do addition or subtraction before doing multiplication or
division. This is the opposite of the order of operations because you are
doing inverse or opposite operations to solve the equations.
Ex 1: 3x + 4 = - S ~ 4 was added, so,
3x + 4 - 4 = -S - 4 ~ subtract 4 on both sides
3x ..;. 3 = -12 ..;. 3 ~ 3 was mUltiplied, so divide by 3 on both sides
x = - 4 ~ giving the solution of - 4.

3. Division: Any terms can be divided. They do not have to be like terms.
Division is usually written in fraction form. When dividing terms, divide
or reduce the coefficients (numbers in front) and the matching variables.
Remember that, to divide with exponents, you must subtract the exponents
once you match the same bases. Ex: (30a7c2)/(-6a4c3d2) = (-5a 3)/(cd 2) be­
cause 30 divided by -6 is -5, a 7divided by a 4 is a\ c3divided by c2 is c, and
there is no other variable d to divide by the d 2, so it remains the same .

Ex 2: ~ - 7 = 3 ~ 7 was subtracted, so,

4. Commutative Property: a + b = b + a and a - b = a + (-b) = (-b) + a ; there­
fore, terms can be moved as long as you take the proper sign (negative or
positive) with the term . Exs: 4p2 + Sp3 = Sp3 + 4p2; 14c - 3f= (-3t) + 14c.

~- 7+7= 3+7

~ add 7 to both sides

.!l. • 2 = 10 • 2 ~ n is divided by 2, so multiply by 2 on both sides

5. Associative Property: (a + b) + c = a + (b + c) and (a - b) - c = a + (-b + -c);
therefore, terms can be added in any order as long as all subtraction is first
changed to addition. Ex: (5j - Sj) - 12j = 5j + (-8j + -12j).

2

n = 20 ~ giving the solution of 20.
b. If the equation has the variable on the right side of the equal sign, then it
can be solved, leaving the variable on the ri ght side, or it can be turned
6. Distributive Property: a(b + c) = ab + ac and a(b - c) = ab - ac; therefore,
around by simply taking everything on each side of the equal sign and put­
if the terms inside the parentheses cannot be added or subtracted, multiply
ting it on the opposite side without chang ing any signs or terms in any way
them BOTH or ALL by the value located in front of the parentheses. Exs:
(symmetric property).
2
2
2
4
2

2
2
3
3
3n(5n + 6) = 15n + ISn; a c(5a + 2ac - c ) = 5a c + 2a c - a c .
5. More tha n two-step equations: Eq uations sometimes req uire sim plifyin g
each side of the equation separately before beginning to do inverse operations.
7. Double Negative Property: - (-a) = a; therefore, if there is a negative of
Ex: 3(2n + 1) + 9 = 4n - 10 ~ di stribu te 3
a negative, it becomes positive, just like a negative number times a nega­
6n + 3 + 9 = 4n - 10
~ add like terms
tive number equals a positive number.
6n + 12 = 4n - 10 ~ now begin inverse operations
6n + 12 - 40 = 4n - 10 - 4n ~ subtract 4n on both sides
TRANSLATING
2n + 12 - 12 = -10 - 12
~ subtract 12 on both sides
2n ..;. 2 = -22 ..;. 2
~ divide by 2 on both sides
PUTTING WORDS INTO ALGEBRAIC STATEMENTS
~ giving the sol ution of -II.
n = -11
There are several key words or phrases that often help in
6. Proportions: Equations in which both sides of the equa l sign are fractions.
converting words into algebraic statements.
The cross-multiplication rule can be used to solve such equations.

I . Addition: Plus; add; more than; increased by; sum; total. Exs: "4 more than
The

rule is that if .!!. = !l , then ad = bc.

a number" becomes 4 + n; "a number increased by 3" becomes n + 3.
e
d

2. Subtraction: Minus; subtmct; decreased by; less than; difference. Exs: "6 less
Ex
1:1
=1


than a number" becomes n - 6. It cannot be written 6 - n, because the 6 is being
x 7

taken away from the number, not the other way around; "a number decreased by
5 • 7 = 3 • x ~ cross multiply

5" becomes n - 5 and not 5 - n; always consider which value is being subtmcted.
35 ..;. 3 = 3x ..;. 3 ~ inverse operation, divide by 3

3. Multiplication: Times; multiply; product; of (when used with a fraction);
11 = x ~ solution

doubled; tripled. Exs: " 2h of a number" becomes (2/3)n; "the product of 7
and a number" becomes 7n.
Ex 2:.! = _ 3_

5 (x +2)

4. Division: Divided by; divided into; quotient; a half (divide by 2); a third (divide by 3).
Exs: "A number divided by 2" becomes nl2; "the quotient ofS and a number"
4(x + 2) = 5 • 3
~ cross multiply

becomes SIn.
4x + S = 15
~ distribute the 4

5. Inequality and equality symbols:
4x + S - S = 15 - S
~ inverse operation; - S

a. > comes from "is greater than" or "is more than" and not "more than,"
~ inverse operation; divi de by 4

4x ..;. 4 = 7 ..;. 4
x = 1.75
~ solution

which is addition .
b. < comes from " is less than" and not " less than," which is subtraction.
7. Grap hing solutions: Since equations have only one solution, the graphs of their
c. :::: comes from "is more than or equal to" or " is greater than or equal to."
solutions are simply a solid dot on the number on the real number line. Ex: If you
d. ::; comes from "is less than or equal to."
solved the equation 4k-7 = -15 and found the answer
e. "# comes from "is not equal to."
k = -2, then you would draw a real number line and "_~
I -\

f. = comes from "is equal to" or " equals."
put a solid dot on the line above -2, such as at right.

t

.f

2


GRAPHING LINES
There are many ways to graph a linear equation.

ALGEBRAIC INEQUALITIES
Algebraic inequalities are statements that do not have an equal sign
but rather one of these symbols: >, < , ~ , ~ , or 7=.
PROPERTIES

I. Pick any number to be the value of the x variable. Put it into the equation for
the x and then solve the equation for the y. This gives one ordered pair (x,y),
with the number you picked followed by the nwnber you found when you
solved the equation. You should pick at least three different values for x and
solve, giving 3 points on the x
x + 2y = 4
Y
line. If the 3 points don't t--+-...,,--"---+-'--.
form a line, a mistake has 0
0 + 2y = 4
2
y=2

been made on at least one of 1--+-...,,-"'--:,--:-+-,

the equation solutions.
1
I + 2y - 4
1.5

y =1.5
Ex: The linear equation I--+---'~--+_,

x + 2y = 4 can be put into 2
2 + 2y = 4

y=l
a chart like the one at right: .........._....;:._;.......1._.....


2. Find the points where the line crosses the x-axis (called the X-intercept)
and the y-axis (called the y-intercept).
a. This can be done by putting a zero into the equation for the x variable and
solving for the y. This gives the point where thc line crosses the y-axis
because all points on the y-axis have x numbers of zero.
b. Next, put a zero into the equation for the y number and solve for the x.
This gives the point _-,....----....,.-..

where the line crosses x
3x - y = 5
y
y
the x-axis because all t--+-------+--f

(1%,0)
3' 0 - y = 5 -5

P oints on the x-axis 0
Y=-5
have y numbers of zero. t-1""'1-+--'----+----f
Ex: The linear equation
3
3x - 0 = 5
0
(0, 5)
3x - Y= 5 could be put into
x = ,1.
a chart like the one at right: - -......- -....' .- .....- ..
c. Find one point on the line by:
(I) Putting a number into the equation for the x and solving for the y.
(2) Next, use the slope of the line. The slope can be found in the equation.
Look at the coefficient (nwnber in front) of the x variable, change the
sign of this number and divide it by the coefficient of the y variable.
This is the slope of the line.
(3) Then, graph the point you found and count the slope
from that point using (rise)/(run). Ex: The linear
equation 2x - Y = 7 goes through the point (3, -1).
(3,-1)
The slope is -2/ -] because you change the sign of
rise =-2
the number in front ofthe x variable and divide it by
run =- 1
the coefficient of the y variable, which is -]. Graph

these values as at right.

I. Addition/Subtraction Property of Inequality: If a > b, then a + c > b + c
and a - c > b - c. Also, if a < b, then a + c < b + c and a - c < b - c. This
means that you can add or subtract any number or term to or from both sides
of the inequality.
2. MultiplicationlDivision Property of Inequality: If a> b, then ac > bc
and a/c > b/c only if c is a positive number. If c is a negative number,
then a > b becomes ac < bc or a/c < b/c. (Notice that if 8 > 5 and you
multiply each side by -2, then you get -16 > -10, which is false, but if
you tum the symbol around, getting -16 < -10, it becomes true again.)
Caution: Tum the symbol around only when you multiply or divide by
a negative number.

SOLUTION METHODS
FIRST DEGREE, ONE VARIABLE
1. Solving inequalities is exactly the same as solving equations, as discussed on
page 2, with only one exception. The exception is when you multiply or di­
vide by a negative number, the inequality symbol turns around to keep the
inequality true, so you will get a true solution. The symbol does not tum
around when you are adding or subtracting any terms or numbers or when
you are multiplying or dividing by a positive number.
Ex: 3(x + 2) > -15

3x + 6 > -15 f- distribute the 3

3x + 6 - 6 > -15 - 6 f- inverse operation (- 6)

f- inverse operation (-+- 3)

3x -+- 3 > -21 -+- 3
x> -7
f- solution

Note: The > symbol did NOT turn around because the division was by + 3, not - 21.
2. Graphing solutions: Inequalities have many solutions or answers, so the
graphs of the solutions look very different from the graphs of equations.
a. Graphs of equations usually have only one solid dot, but the graphs of in­
equalities have either solid dots with rays or open dots with rays.
Ex: If the solution to an inequality is x > -7, the graph at right is with an
•I
E9
I ••
open dot because -7 does NOT make the in-8
-7
-6
-5
equality true, only numbers more than -7 do.
b. The solid dot shows that the number is part of the answer, but an open dot
shows that the number is not part of the answer but only a beginning point.
Ex: If. the solution is n ~ 3, the graph at • I
I
•
I
I '"
right shows a solid dot.
1
2
3
4

5

I


It

A

-

COORDINATE PLANE


GRAPHING INEQUALITIES
On the coordinate plane, linear inequalities are lille graphs
with a shaded region included, either above or below the line.

POINTS

I. The coordinate plane is a grid with an x-axis and a y-axis.
2. Every point on a plane can be named using an ordered pair.
3. An ordered pair is two numbers separated by a comma and enclosed by
parentheses (x,y). The first number is the x nwnber and the second number
is the y number. Ex: (3, -5), where x = 3 and y = -5.
4. The point where the x-axis and the y-axis intersect or -+
cross is called the origin and has the ordered pair (0,0).
5. The x number in the ordered pair tells you how far to
go to the right (if positive) or to the left (if negative)
from the origin (0,0).

6. The y number in the ordered pair tells you how far to go
up (if positive) or down (if negative), either from the ori­
gin or from the last location found by using the x nwnber.

1. Graph the line (even if the inequality does not include the equal sign, you
must graph the corresponding equality).
2. Pick a point above the line.
3. Put the number values for x and for y into the inequality to see if they make
the inequality true.
4. If the point makes the inequality true, shade that side of the line.
5. Tfthe point makes the inequality false, shade the other side of the line.
6. The actual line is drawn as a solid line if the inequality includes the equal sign.
7. The actual line is drawn as a dashed line if the inequality does not include
the equal sign. Ex: Graph x + y < 2.
x
0

LINES & EQUATIONS
I. The coordinate plane and ordered pairs are used to name all ofthe points on a plane.
2. When the points form a line, a special equation can be written to represent
all of the points on the line.
3. Since points are named using ordered pairs with x numbers and y numbers
in them, equations of lines, called linear equations, are written with the vari­
ables x and/or y in them. Exs: 2x + Y = 5; Y =. x - 6; x = -2; Y = 5.
4. Lines that cross both the x-axis and the y-axis have equations that contain
both the variables x and y.
5. Lines that cross the x-axis and do not cross the y-axis have equations that con­
tain only the variable x and not the variable y.
6. Lines that cross the y-axis and do not cross the x-axis have equations that con­
tain only the variable y and not the variable x.

2

slope = Yt -Yl
X,-Xl

= 0-(-2) =~=_1
-2-1

-3

or slope = rise
3
run

=

x+0=2
x=2

0

r------.
Test (0,0) in

x+y<2

0+0<2
is true, so shade

that side

of the line.

2

TIT
x-Bxis

-5

up 2
over-3

2
-3

(0,0j"

."­

FINDING LINEAR EQUATIONS

.

5


d. Ex: b = 2 and m =2' The equatIOn: y =- zX+2.

2

3.

Or, in standard form: 5x + 2y = 4.
3

r


y

I. Some linear equations can be found by obscrving the relationship between
the x numbers and the y numbers. Exs: A line with the points (3,2), (5,4),
(-2, -3), and (-5, -6) has the equation y = x-I , because every y number is
one less than the x number.
2. Some linear equations require methods other than simple observation, such
as the slope-intercept form of a linear equation (see #4 below).
3. The standard form oflinear equations is ax + by = c, where a, b, and c are integers.
4. The slope-intercept form of a linear equation is y = mx + b, where the m rep­
resents the slope and the b represents the y-intercept ofthe line. One way to tind
the equation of a line is to find the y-intercept (where the Iine crosses the y-ax­
is) and the slope, then put them into the slope-intercept fonn of a linear equation.
a. Find the y-intercept, then use it to replace the b in y = rnx + b.
b. Next, find the slope of the line; use the slope to replace ...-....-----......,
the rn in y = rnx + b.
c. The result is the equation of the line with the number
values in place of the rn and the b in the form y = lUX + b.

SLOPE OF A LINE
I. Every line has a slope, except vertical lines (have no

slope). The slope can be thought of as a kind of slant

to the line.
2. Slope is found by comparing the positions of

any two points on the line.

3. Slope is (y, -Yl)/(X, - Xl). It is also described as

the (rise)/(run) or (the change in y)/(the

change in x). Exs: On the graph at right, the

slope of line "//' can be found using the formula:


x+y=2
0+y=2
y=2


POLYGONS

GEOMETRY

1. Polygons are closed plane figures whose sides
are line segments joined at the endpoints.

2. Regular polygons have all angles equal in de­
grees and all sides equal in length. This is not true if the
polygon is not a regular polygon.
G::i~'-._--t
3. Special Polygons: Triangles are 3-sided t-_-=-;:B':1Y,;..:A.:...N-,,;LEs
Right one 90° angle
polygons.
a. The sum of the angles of a triangle is 180·.
Obtuse one angle> 90'
b. The sum of the lengths of any 2 sides of a t-__A...;..:...
cu t.:.J
e ...11
. ..,.~
.n..!:g;;;.
les.:...<.:...9__
0• --t
triangle is greater than the length of the t----,---=,;-::.:="-----I
Sc.lene no sides equ.1
third side.

PLANE GEOMETRY
1. Plane geometry refers to geometry of flat surfaces (planes).
2. Lines are always straight and continue forever in two opposite directions.
3. Points are always named using capital letters.
4. Lines are named using any two points on the line. A line with an arrow
on each end is drawn on top of the two capital letters that name the

M.

points, like

5. Definitions
a. A line segment is two points on a line (the endpoints) and all of the
points between them. The notation for a line segment is the two end­
points (in capital letters) with a bar over them. Ex: Line segment .PQ,
written PQ, is shaded in blue on the line •
:
~


c. Triangles can be classified in two ways,
by their angles and by their side lengths.


THE PYTHAGOREAN THEOREM
The Pythagorean Theorem can be used to find the length of one side of

any right triangle when given the length of the other two sides. It is usu­
ally written as a 2+ b 2 = c 2, where a and b are the lengths of the two legs
and c is the length of the hypotenuse. Ex:

b. A midpoint is the point in the center of a line segment. It separates
a line segment into two equal parts.

c. A ray is one point on a line (the endpoint) and all of the points on the line
continuing on from the endpoint and going in one direction forever. The no­
tation for a ray is the endpoint followed by any other point on the ray, with
a ray pointing always to the right, drawn on top of the two capital letters.
~

~

H

Ex: Ray AB, written AB , is shaded in blue on the line AB.
B

•

Isosceles 2 or more sides equal
Equil.t....1 all sides equ.1

a=?

A

•

poo-----­

150.

To find length of leg "a"
a2 + b1 = c1


.2 ++(5)'25 == 49(7),

.2 = 49 - 25
32

• = .J24
b=5
3 = 4.899
d. Intersecting lines cross or touch in exactly one point. They are al­

ways coplanar (in the same plane).

TRIG FUNCTIONS
e. Perpendicular lines intersect or cross, forming 90-degree angles at
Trigonometric (trig) functions of an angle can be used to find the
the point of intersection.
measures of the angles and the lengths o/the sides ofright triangles.
f. Parallel lines go in the same direction and never touch. They are al­
I. There are 6 trig functions that are ratios, but only 3 will be discussed here.
ways coplanar (in the same plane).
.
.
opposite leg
a. sme of an angle = sm A = -'.-'---~~.....£
g. Skew lines go in different directions and never touch; as a result,
hypotenuse
they are not coplanar (not in the same plane).
h. An angle is the union of two rays with a common endpoint. The com­
.
adjacent leg
b. cosme of an angle = cos A = h
.
mon endpoint of the two rays becomes the vertex of the angle. The
ypotenuse

two rays become the sides of the angle. Angles are measured with

opposite leg

protractors and in degrees.
c. tangent of an angle = tan A =
(I) Acute angles have measures less than 90 degrees.
adjacent leg
(2) Right angles have measures that equal 90 degrees.
2. Some things to note:
(3) Obtuse angles have measures greater than 90, but less than 180 degrees.
a. The A represents an acute angle in the right triangle.
(4) Straight angles have measures equal to 180 degrees.
b. The leg of the right triangle is considered either the opposite leg or
the adjacent leg changes, depending on which of the acute angles of
I. Complementary angles are two angles whose measures total 90 degrees.
the right triangle is being evaluated in the trig function.
j. Supplementary angles are two angles whose measures total 180 degrees.
c. The opposite leg of a right triangle is the leg that does not touch the

k. Adjacent angles are angles that share a common vertex and one
vertex of the angle named in the trig function.

common side with no common interior points (points in the region
located between the two sides of an angle). Ex: 4ABC and
d. The adjacent leg of a right triangle is the leg that does touch the ver­

tex of the angle named in the trig function.
4CBD are adjacent angles because they share
and no interior

e. Ex: When evaluating the trig functions for angle A in
points. 4ABD and 4CBD are not adjacent angles
this right triangle, leg a is the opposite leg for angle A A

~
because 4CBD has side BC inside 4ABD. Note:

because it does not touch point A; however, leg b is the ~
adjacent leg for angle A because it does touch point A.
The symbol 4 means angle and the vertex must

The hypotenuse is side c. In the same right triangle, leg
b c
be the point written in the middle.
b is the opposite leg for angle B because it does not
\. Vertical angles are angles that have the same ver­
B
touch point B; however, leg a is the adjacent leg for an- c
tex and whose sides form lines. Ex: 4KMN and
gle B because it does touch point B.
4RMP are vertical angles. 4KMR and 4NMP
f. NOTICE: The opposite leg for angle A is also the adjacent leg for angle
are also vertical angles.
B, and the adjacent leg for angle A is also the opposite leg for angle B.
m. Corresponding angles of intersecting lines are two
3. Since trig functions are ratios (fractions), they are often converted to
angles, with one of the angles having its vertex and

decimal numbers.

one side on one of the sides of the other angle.
4. Using the trig function decimal number values to find or use angle

~EX: 4ABC and 4BDE are corresponding angles.
measures requires either a trig function chart or a calculator with trig
function options.

n. Alternate interior angles are two angles that
Exs: Using a chart or a calculator to find the sin 30" gives the decimal

have one side of each angle overlapping and
number .500. Using a chart or a calculator to find the angle, K, that has

forming a line while the other sides of the angles
tan K = 1.483 gives an angle measure of 56".
go in opposite directions from different vertices

5. Triangle Trig Applications:

(plural ofvertex). Ex: 4MNP and 4NPQ are al­
There are 2 basic ways in which trig functions are used

ternate interior angles. CIRCLES
with triangles: To find angle measures and to find side lengths.

Circles are points in a plane that are all equidistant (same distance)

a. Finding acute angle measures
from the center point. The center point is not part of the circle.
(I) To find the 2 acute angle measures when given 2 sides of a right

I. Chords are line segments with endpoints on a circle.
triangle, it is easiest to find the length of the third side first.
Ex: In the right triangle (at right), if you know the length of any
2. Diameters are chords that go through the center of the circle.
two sides, then you may use the
3. Radii (plural of radius) are line segments with endpoints in the center
Pythagorean Theorem (leg2 + A ~
3 2 + b 2 = c2
of the circle and a point on the circle. They are half the
leg2 = hypotenuse2) to find the
a322++64
(8)2 = (10)2
length of the diameter of the same circle or of congruent
= 100

length of the third side.
8
8 2 = 36
circles.

(2) Once the three side lengths are found
10
• = .J36

4. Circumference is the distance around a circle. The
(not necessary, but it is easier), use the C a B
a =6
formula is C = 7t d.

5. Area = 7t r2.

trig functions to find the degree measure of one acute angle.

per
hen

r. If
that
r.

it

~

~
~

4





.....

(3) Using the same right triangle above, r-----~_:"the measure of A can be found using tan A = opposite leg
any of the trig functions, so just pick
adjacent leg
one of them (see right).
tan A =
(4) The measure of the second acute angle tan A = .750

may be found by simply subtracting
4- A = 37° ~~:~.:;~:;:r
the measure of the acute angle just ....- - - - - - - -..
found from 90°.
b. Finding side lengths
K~
sin K = opposite leg
(l)To find the side r 50 ' 16
. <;00 ~ypotenuse
lengths of a right
SIn = 16
triangle when given
40 '
,-766 k
only one side length N
k
R
I k : A~256
and one acute angle, first subtract the - - - - - - - -....
given acute angle measure from 90°; second, use the trig functions
to find the length of another side of the triangle.

5. Similar polygons are the same shape, but not necessarily the same size
(they can be). They have matching angles that have the same degree
measures. Congruent polygons are also similar. Similar polygons can be
placed in each other so the matching equal angles are one inside the oth­
er. Corresponding or matching sides have lengths that are proportional.
Ex: Below.
L\MNP -L\SOR, so
4M=4S

4N=4Q
4P=4R

j

MN = NP =PM

SQ

AREAS & PERIMETERS

r--~~~-"

SQUARE AREA: A = hb
Ifh = 8, then b = 8; also,
h
as all sides are equal, then:
A = (8)(8) = 64 square units

A = 1/, (8)(12)
A = 48 square units

...II1II
"'I11III

There are a few special relationships about polygons in general.
1. To find the sum of the measures of all angles in a polygon, take the num­
ber of sides, subtract 2, and multiply by 180 degrees. The formula may be
written as 180(n - 2), where n = the number of sides of the polygon.
Ex: A nonagon has 9 sides, so 180 (9 - 2)=180 (7)=1,260 degrees in total.

2. To find the measure of each angle of a regular polygon (angles all have
the same measure), find the measure of the sum or total of all of the an­
gles and divide by the number of angles in the polygon (same as the num­
ber of sides). Ex: If the total of all angles ofa nonagon is 1,260 degrees,
then divide by 9, equaling 140 degrees for each angle.
3. To find the total number of diagonals (segments whose endpoints are
vertices of the polygons, excluding sides of the polygon) of a poly­
gon, take the number of sides of the polygon, subtract 3, multiply by
the number of sides, and divide by 2. The formula may be written as
n(n - 3)/2, where n = the number of sides of the polygon. Ex: An oc­
tagon has 8 sides, so 8 (8 - 3)/2 = 8 (5)/2 = 40/2 = 20 diagonals.
4. Congruent polygons are polygons that are the same size and the same
shape. The symbol is ~ . They can be placed on each other (using flip,
slide, or rotate), so equal parts match. Ex: Below.

DfJ
A Be

P
UN

G

CIRCLE AREA: A - rrr'
Ifrr = 3.14 and r = 5, then:

A = (3.14)(5)'= (3.14)(25)
~
A = 78.5 square units
~

CIRCUMFERENCE: C = 2rrr
b
C = (2)(3.14)(5) = 31.4 units

...~

.


SPATIAL SHAPES

ANGLES, DIAGONALS, CONGRUENT, SIMILAR

III

TRAPEZOID AREA: A -1/, h(bl+b,)
Ifh = 9 and b l = 8 and b,= 12, then:
A = II, (9)(8+12)
~
b
A = 1/, (9)(20)
:h 1

A = 90 square units
.
b


1. Polyhedrons: A polyhedron is a closed, 3-dimensional, spatial shape
made with 4 or more polygons that have common sides.

a. The polygons are the faces of the polyhedron.
(I)The line segments of the polygons are the edges of the polyhedron.
(2) The vertices of the polygons are the vertices of the polyhedron.
b. Prisms are polyhedrons with two parallel, congruent bases and sides
that are parallelograms or rectangles.
c. Pyramids are polyhedrons with one base and sides that are triangles.
2. Cylinders, Cones, Spheres
a. Cylinders are spatial shapes with 2 parallel bases (congruent circles)
and a curved surface joining the bases.
b. Cones are spatial shapes with one base (a circle) and a curved sur­
face (joining the base) that comes to a point.
c. Spheres are spatial shapes made of all points equidistant (the same
distance) from one central point.
3. Volume and Total Surface Area
a. The volume (V) is the number of cubes needed to fill a spatial shape;
therefore, it is measured in cubic units.
b. The total surface area (TSA) is the sum of the areas of all of the faces
or surfaces ofa spatial shape.; measured in square units because it is area.

~V

--

.il~7
. . . . ,


TRIANGLE AREA: A = II, bh

(RECTANG~

a.

Iw

PARALLELOGRAM AREA: A = hb
Ifh = 6 and b = 9, then:

A = (6)(9)

A = 54 square units
b

Ifh = 8 and b = 12, then:

~

Z

=

II
L---.J h

Quadrilaterals are 4-sided polygons.
I. Parallelograms are quadrilaterals with opposite sides parallel and
equal in length. Opposite angles are equal in measure and any two
consecutive angles (next to each other) are supplementary.
2. Trapezoids are quadrilaterals with one pair of opposite sides parallel
and the other pair of opposite sides not parallel. Trapezoids are not

parallelograms. Parallelograms are not trapezoids.
3. Rectangles are parallelograms whose angles each measure 90 de­
grees. Sides mayor may not be equal. Some rectangles are squares.
4. Rhombi (plural of rhombus) are parallelograms whose sides arc all equal
in lenb>th. Angles mayor may not be equal. Some rhombi
are squares.
5. Squares are rectangles whose sides are all equal. 'QUADRILATERALS
Squares are also rhombi whose angles are each
TRAPEZOIDS
90 degrees. This means that all squares are both
rectangles and rhombi.
6. Pentagons are 5-sided polygons.
(PA~\
7. Hexagons are 6-sided polygons.
8. Heptagons (or septagons) are 7-sided polygons.
9. Octagons are 8-sided polygons.
10. Nonagons are 9-sided polygons.
II. Decagons are I O-sided polygons.
12. n-agons are polygons with n number of sides.

iii

0

RECTANGLE AREA: A = hb, or A
Ifh = 4 and b = 12, then:
A = (4)(12)
A = 48 square units
b

QUADRILATERALS

"'I11III

RS

I. The perimeter (P; distance around) of any polygon can be found by
adding the lengths of all of the sides.
2. The area (A) of each polygon is found using the formula listed in the
following chart. Area is measured in square units because it is the
number of squares that are needed to cover the surface of a region.

a 2 +b 2 =c 2
(2) Once 2 sides of the right triangle are r2 +(12.256)2 =(16)2
known, use the Pythagorean Theorem to
r2 =105.79
find the length of the third side.
r=10.285

...II1II

QR

RECTANGULAR PRISM VOLUME:
TRIANGULAR PRISM VOLUME:
Y = (area oftriangle)h
~
Y=lwh;lfl=12
If triangle has an area
I

w = 3, h = 4, then:
Y = (12)(3)(4)
equal to 1/, (5)(12), then:
5 h
V =144 cubic units
w
Y = 30h and if h=8, then:
\
..C- U
- B-E- V
-O
- L-U-M- E
- :- Y
- =- e-3- - - - - t Y = (30)(8), V=240 cubic units

hCP?

RECTANGULAR PYRAMID VOLUME:

V=I/, (area of rectangle)h
~
:--1
w h

If 1= 5 and w = 4,
\ I

the rectangle area is

A = (5)(4) = 20, then:

V = 1/,(20)h and ifh = 9

V = 11,(20)(9)

V = 60 cubic units

SPHERE VOLUME:
)

If radius, r = 5, then: V = 4~r

~CO
~~NE~~~LU~
~ VO ~~ME~Y~-~
: ~ = I"-rr~
~:==-I-I V = 4(3.14)(5)'
If r = 6 and h = 8, then:
:
3
Y = II, 1[ (6)' (8)
:
h
V = 1570
' ••r
Y = II, (3.14)(36)(8)
3
Y = 301.44 cubic units
!.,
V= 523.3 cubic units

Each edge length, e, is equal O l J
to the other edges in a cube.
If e = 8, then:
e
Y = (8)(8)(8)
V = 512 cubic units
e
e
CYLINDER VOLUME: Y = rrr'h
If radius, r = 9, h = 8, then:
V = 1[ (9)'(8)
h
Y = 3.14 (81)(8)
V = 2034.72 cubic units
r..

Cd

r!'D~A~C
~B~D-="':"D~K"!'!M~P-~~
N~ with""
4A=4K AB =KM
4B=4M BC=MN
4C=4N CD = NP
4D=4P AD= PK

'@

NOTICE TO STUDENT: This QUICKSTUDY'" guide outlines the major

topics taught in Pre-Algebra courses. For further deta il, see Algebra Part­
I and Algebra Part-2. Due to its condensed format, however, use it as a
Pre-Algebra guide and not as a replacement for assigned course work.
5


% INCREASE
% increase amount of increase
100
original value
or (original value) x (% increase) = amount of increase
If not given, the amount of increase may be found through this subtraction:
(new value) - (original value) = amount of increase.
Ex: The Smyth Company had 10,000 employees in 1992 and 12,000 in
1993. Find the % increase. Amount of increase = 12,000 - 10,000 = 2,000
.F ORMULAS:

I . Ratios are comparisons of two numerical values or quantities. Forms
include 3­ 7, 3 :7, 317 or

l

E 2. Percents are comparisons of numerical values to 100, so they are values "out

of 100." Percents can be changed to equivalent fractions or decimal numbers,
and vice versa.

a. To change a percent to an equivalent fraction,
30
3

take the numerical percent value and put it over Ex: 30%=100 =10­
III
100; then, reduce to lowest terms.
rl
b. To change a fraction to a percent, change the .
2 2 20 40
denominator (bottom) to 100; the numerator Ex: 5 =5- 20- = 100 =40%

(top) IS the percent value. FractIOns can also

be changed to decimal numbers, and then changed to percents.

c. To change a percent to a decimal number, move
the decimal point two places to the left and re- Ex: 85% = 85.% = .85
move the percent sign. It is moved two places beW
cause two decimal places is hundredths, just as percents are hundredths or "out
of 100."
d. To change a decimal number to a percent,
move the decimal point two places to the right Ex: .375 = .~5 = 37.5%
and put the percent sign after the number.
3. Proportions are equalities between two ratios or fi-"dctiollS.

Forms include: 3 is to 4 as 12 is to 16; 3:4::12:16; and 1 g

4 16
a. There are basically two ways to solve a proportion:
(I ) Get a common denominator for the two fractions.
II

•

L

% increase:

,

n = 20 and the % increase = 20%
(because % means "out of I ~O'').

% DISCOUNT
FORMULAS: % discount amount of discount
100
original price
or (original price) x (% discount) = $ discount

If not given, ($ discount) = (original price) - (new price).

Ex: The Smyth Company put suits that usually sell for $250 on sale for $150.

Find the percent discount.
% discount: ~ = $100 , so n = 40 and the % discount = 40%.
100 $250

SIMPLE INTEREST
FORMULAS: i = prt or total = p + i
Where i = interest
p = principal; money borrowed or lent or saved
r = rate; percent rate

t = time; expressed in the same period as the rate, i.e., if rate is pcr
year, then time is in years or part of a year. I f rate is per month, then
time is in months.
Ex: Carolyn borrowed $5,000 from the bank at 6% simple interest per year.
she borrowed the money for only 3 months ('/4 year), find the total amount that
she paid the bank.
$ interest = prt = ($5,000)(.06)(.25) = $75

Notice that the 6% was changed to .06 and the 3 months to .25 of a year.


Ex: l = ...!!.. becomesJL=...!!..so n=9.
7 21
21 2L
(2)

1~0 =10~0000' so

Use cross-multiplication, which states that if !=~, then ad = bc.
3 4
Ex. - = - becomes3-5 =4 en, then 15 =4n and 3.75 =n .
·n 5

1


~1"""""""~;:~~~~~~~~~"""""""~
..........~$~t0~~~~+
~tal p ;!~
i $~,0~~

5~ 00 +~~5~
$7~ $:5~~
,07!5. . . . . . . . . . . .
COMPOUND INTEREST

I
iI

% MARKUP

(

)ot
FORMULA: A=p l+ ­ nr
II
•
Where: A= total amount
,
p = principal; money saved or invested
ill
r = rate of interest; usually a % per year
III
t = time; expressed in years
rl
n = number of periods per year
EX4MPLE: John put $ 1()() into a savings account at 4%
compounded quarterly for 8 years.
How much was in the account at the end of 8 years?

Z

$
markup _
markup
100
- original price
or (original price) x (% markup) = $ markup

If not given, ($ markup) = (new price) - (original price).

Ex: The Smyth Company bought blouses for $20 each and sold them for

FORMULAS:

A= (1 +~)ot
p
n
(04)
A = 100 1 + ~4 4x
A = 1 00 (1.01)32
A =100(1.3749)
A 137.49

01
10

$44 each. Find the percent markup.

$ markup: S44-S20 = $24, % markup: ~ = 24, son= 120, the % markup = 120%.

100 20

% PROFIT

% DECREASE
% decrease amount of decrease
100
original value
or (original value) x (% decrease) = amount of decrease
If not given, the amount of decrease = (original price) - (new value).
Ex: The Smyth Company had 12,000 employees in 1993 and 9,000 in 1994.
Find the percent decrease. Amount of decrease = 12,000 - 9,000 = 3,000

FORMULAS: % profit
$ profit
100
total $ income
or (total $ income) x (% profit) = $ profit
lfnot given, $ profit = (total $ income) - ($ expenses).
Ex: The Smyth Company had expenses of $150,000 and a profit of $10,000.
$10000
Find the % profit.
n
total $ income = $150,000 + $10,000 = $160,000, % profit: 100 = $160,000

% decrease: ~= 3,000 , so n = 25 and the % decrease = 25%.
100 12,000

or ($160,000) x (n) = $10,000. In either case, the % profit = 6.25%.

FORMULAS:

% EXPENSES OR COSTS

% COMMISSION
FORMUL AS: % commission ~_ c.()!I1~i.s~~

FORMULAS. % expense.s = $ expenses
.
100
total $ income

100
$ sales
or ($ sales) x (% commission) = $·commission
Ex: Missy earned 4% on a house she sold for $125,000. Find her dollar commission.
4
$ commission
% commission: 100
$125,000 or

or (total $ income) X (% expenses) = $ expenses
Ex: The Smyth Company had a total income of $250,000 and $7,500 profit last
month. Find the percent expenses. $ expenses: = $250,000 - $7,500 = $242,500.
01
10

($125,000) x (4%) = $ commission, so $ commission = $5,000.

ISBN-13: 978-157222726-2

IS BN-10: 157222726-5

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01
10

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/0.

"IS" & "OF"
Any problems that are or can be stated with percent and the words "is " and
"of" can be solved using these.formulas:
FORMULAS: %
"is" number

100 "of" number

or "of" means multiply and "is" means equals.

Ex. 1: What percent of 125 is 50?

_11. . =~- or n X 125 = 50. In either case the percent = 40%.

100 125
'

Ex. 2: What number is 125% of 80?

125 =-'!.. or (1.25) (80) = n. In either case, the number = 100.

100 80


u.s. $5.95 CAN. $8.95

Author: S. Klzlik

0 200 2

expenses.• ~
100 -- .!~42,~00
$250,000 ' so n -- 97 an d th e

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