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PREALGEBRA REVIEW DEFINITIONS
ORDER OF OPERATIONS (PEMDAS)
Please: do all operations within parentheses and other grouping symbols (such as [ ], or operations in numerators and denominators of
fractions) from innermost outward.
Excuse: calculate exponents
My Dear: do all multiplications and divisions as they occur from left to right
Aunt Sally: do all additions and subtractions as they occur from left to right.
Example:
20 – 2 + 3(8 - 6)2 Expression in parentheses gets calculated first
= 20 – 2 + 3(2)2 Next comes all items with exponents
=
20 – 2 + 3(4) Next in order comes multiplication. Multiplication and Division always come before addition or division, even if to the
right.
= 20 – 2 + 12 Now when choosing between when to do addition and when to do subtraction, always go from left to right, so do 20-2
first, because the subtraction is to the left of the addition.
= 18 + 12 Now finally we can do the addition.
= 30
The Test of Reasonableness
In general applying a test of reasonableness to an answer means looking at it in relation to the numbers operated upon to determine if
it’s “in the ballpark.” Put in simple terms, you look at the answer to see if it makes sense. For example, if you determine 10% of $75
to $750, you should immediately notice something is very wrong, because 10% of something is much smaller than the original
amount.
The test of reasonableness comes very much into play in word problems. For example, questions asking for length, dollars, or time
should never give negative answers because those things would not make sense if they were negative (what is -3 feet?). Also, think
about what the answer should be. If I invest $400 dollars in a bank at 8% interest for 3 years, I would expect the balance to be larger
than the original amount that I invested (that’s why we put money in the bank!), so an answer that is smaller than $400 is obviously
wrong.
VOCABULARY DEFINITIONS
Factor - One of two or more quantities that divides a given quantity without a remainder. For example, 2 and 3 are factors of 6; a and
b are factors of ab.
Factor X Factor = Product
Product - The number or quantity obtained by multiplying two or more numbers together.
Factor X Factor = Product
Dividend - A quantity to be divided.
Divisor - The quantity by which another quantity, the dividend, is to be divided.
Quotient - The number obtained by dividing one quantity by another. In 45 ÷ 3 = 15, 15 is the quotient.
Dividend/ Divisor = Quotient
or Dividend ÷ Divisor = Quotient
quotient
divisor dividend
Remainder - The number left over when one integer is divided by another: The remainder plus the product of the quotient times the
divisor equals the dividend.
If there is a remainder, then
PREALGEBRA REVIEW DEFINITIONS
Dividend = Quotient X Divisor + Remainder
A set is a collection of objects. The objects in the set are called the elements of the set. The roster method of
writing sets encloses a list of the elements in braces.
Example: The set of even natural numbers less than 10 can be written like this: {2,4,6,8}.
The set of natural numbers is {1,2,3,4,5,6,7,….}. These are basically the “counting numbers.”
The set of whole numbers is the set of natural numbers and the number, 0. {0, 1, 2, 3, 4, 5, ….}
The set of integers is the set of whole numbers and their opposites. {…. -4,-3,-2,-1,0,1,2,3,4,….}
The number 0 is an integer, but it is neither negative nor positive. For any two different places on the number
line, the integer on the right is greater (>) than the integer on the left.
The absolute value (using the | | symbol) of a number is its distance from zero on the number line. The
absolute value of a number is ALWAYS POSITIVE (or 0).
| 5| = 5,
|-5| = 5,
|3-8| = |-5| = 5, |3-3| = |0| = 0
|-4|=4
-4 is 4 units away
from 0
|4|=4
+4 is 4units away
from 0
PREALGEBRA REVIEW DEFINITIONS
INTEGERS
Adding and Subtracting Integers
When adding two integers with the same sign, just ignore the signs, then attach them on the answer.
-3 + -5 = - (3 + 5) = -8
When adding two integers with different signs, take the absolute values (make both numbers positive), and subtract the smaller one
from the larger one. The sign of the integer with the larger absolute value will be the sign of your answer.
7 + (-8)
The absolute value of 7 is |7| = 7
The absolute value of -8 is |-8| = 8 larger
8–7=1
Remember in the original problem, the integer whose absolute value was 8 was -8, so our answer is negative.
7 + (-8) = -1
Subtracting a negative integer is the same as ________ a _______ integer.
3 – (-5) = 3 + 5 = 8
-2 – (-3) = -2 + 3 = 1
Subtracting a positive integer is the same as _____ a _____ integer.
5 – 3 = 5 + (-3) = 2
-2 – 3 = -2 + (-3) = 1
Multiplying and Dividing Integers
(positive integer) X (positive integer) = (_____ integer)
(positive integer) ÷ (positive integer) = (______ integer)
(negative integer) X (negative integer ) = (______ integer)
(negative integer) ÷ (negative integer ) = (_______ integer)
(positive integer) X (negative integer) = (______ integer)
(positive integer) ÷ (negative integer) = (_______ integer)
PREALGEBRA REVIEW DEFINITIONS
(negative integer) X (positive integer ) = (negative integer)
(negative integer) ÷ (positive integer ) = (negative integer)
Memory Tip:
When something bad happens to a good person, that’s bad.
When something good happens to a bad person, that’s bad.
When something good happens to a good person, that’s good.
When something bad happens to a bad person, that’s good.
When a negative integer is raised to an even power, the result is _____.
(-2)4= (-2)(-2)(-2)(-2) = 16
When a negative integer is raised to an even power, the result is ______.
(-2)5= (-2)(-2)(-2)(-2)(-2) = -32
Exponents are only applied to the number directly diagonally left of it. If a negative integer is raised to a power, it must be in
parentheses () in order for the exponent to apply to the negative number.
(-2)2 ≠ - 22
(-2)2 = (-2)(-2) = 4
- 22 = - (2)(2) = -4
EXPONENTS
Exponent - A number or symbol, as 3 in (x + y)3, placed to the right of and above another number, variable, or expression (called the
base), denoting the power to which the base is to be raised. Also called power.
In this example: 82 = 8 × 8 = 64
The exponent (or power) tells how many times the base is to be multiplied by itself.
Example 1:
(x + y)3 = (x + y)(x+y)(x+y)
Example 2:
(-3)4 = (-3)(-3)(-3)(-3) = 81
PREALGEBRA REVIEW DEFINITIONS
VARIABLE EXPRESSIONS AND LIKE TERMS
Expressions (contain no “=” sign) :
An expression is one or numbers or variables having some mathematical operations done on them.
Numerical Expressions:
3+5
3(4)
6/2
5-1
4
Expressions can be evaluated or simplified: 3 + 5 can be simplified to 8
This just means, “Whatever you see, do”
Algebraic Expressions:
x+5
3x  in Algebra, it is implied that 3x means 3 times x. The “proper” way to write the product of a number and a variable is to always
write the number to the left of the variable. x times 5 = 5x
When numbers are multiplied by variables, they are given a special name, “coefficient”.
5 is the coefficient of 5x.
Algebraic expressions can be simplified by using the associative and distributive properties.
-4m(-5n) can be simplified by rearranging the terms (we can do this when the only operation is multiplication) so that all the constants
are grouped together and all the variables are grouped together(alphabetically))
-4m(-5n) = (-4)(-5)mn = 20mn
2(-4z)(6y)=2(-4)(6)yz = -48yz
3(s+7) can be simplified by using the distributive property
3s + 3(7) = 3x + 21
-6(-3x -6y + 8) can also be simplified. Change any subtraction to adding a negative.
-6(-3x + -6y + 8)
Now distribute the -6 and make sure to glue the negative sign on that -6 wherever you distribute it!
-6(-3x) + -6(-6y) + -6(8) = 18x + 36y + -48 = 18x + 36y + 48.
Like terms are terms with exactly the same variables raised to the exactly the same powers.
considered like terms. Terms that are not like terms are called unlike terms.
Like Terms
Unlike Terms
2x, 3x, -4x
Same variables,
each with a power
of 1.
2x, 2x
Different powers
3, 5, -1
Constants
3, 3x, 3x
2
2
5x , -x
Same variables and same
powers
2
2
Different powers
2
2
5x , 5y
Different variables
Any constants in an expression are
PREALGEBRA REVIEW DEFINITIONS
Only Like Terms can be combined!
Combining like terms is to add or subtract like terms. Combine like terms containing variables by combining their coefficients and
keeping the same variables with the same exponents.
Example: 3x2 – 8x2 = (3 - 8)x2 = -5x2
Example: 5x + 2x = (5 + 2)x = 7x
Example:
3x2 + 5x + 2 + x2 – x - 3
=3x2 + x2 + 5x-x + 2-3
= 4x2 + 4x – 1
Algebraic expressions can only be simplified if they have “like terms,” which are terms with the same variable and same exponent.
x+5+2
Only 5 + 2 can be simplified. x + 5 + 2 becomes x + 7
3x + x + 2
can be simplified to 4x + 2
2
x +x
cannot be simplified because they don’t have the same exponents and are therefore not like terms.
Algebraic expressions can be EVALUATED if you are given the value for the variables.
 x  15
for x = 3. Substitute 3 for x in the expression.
6
 (3)  15  3  15  3  (15)  18



 3
6
6
6
6
Example: Evaluate
Example:
2
If a=-2, evaluate -3a + 4a
2
-3(-2) + 4(-2) Do exponents first
= -3(-2) + 4(4) Multiplication left to right
= 6 + 4(4)
= 6 + 16
Addition
= 22
PREALGEBRA REVIEW DEFINITIONS
Translating Verbal Expressions into Mathematical Expressions
Verbal Expressions Examples
Math Translation
Addition
added to
more than
the sum of
increased by
the total of
6 added to y
8 more than x
the sum of x and z
t increased by 9
the total of 5 and y
6+y
8+x
x+z
t+9
5+y
Subtraction
minus
less than
subtracted from
decreased by
the difference
between
How much more /
How many more?
x minus 2
7 less than t
5 subtracted from 8
m decreased by 3
the difference between y and 4
x-2
t-7
8-5
m-3
y-4
Multiplication times
of
the product of
If Kay is 48 and Leo is 54,
54-48
how much older is Leo than Kay?
10 times 2
one half of 6
the product of 4 and 3
10 X 2
(1/2) X 6
4X3
multiplied by
y multiplied by 11
11y
divided by
the quotient of
x divided by 12
the quotient of y and z
x/12
y/z
the ratio of
the ratio of t to 9
t/9
Power
the square of
the cube of
squared
the square of x
the cube of z
y squared
x2
z3
y2
Equivalency
equals
is
is the same as
1+2 equals 3
2 is half of 4
½ is the same as 2/4
1+2 = 3
2 = (½)X4
yields
represents
3+1 yields 4
y represents x+1
3+1 = 4
y=x+1
greater than
less than
greater than or equal
to
at least
no less than
less than or equal to
at most
no more than
-3 is greater than -5
-5 is less then -3
x is greater than or equal to 5
-3 > -5
-5 < -3
x≥5
x is at least 80
x is no less than 70
x is less then or equal to -6
y is at most 23
y is no more than 21
x ≥ 80
x ≥ 70
x ≤ -6
y ≤ 23
y ≤ 21
Division
1
2
 24
Comparison
PREALGEBRA REVIEW DEFINITIONS
Solving Application Problems
Problem-Solving Strategy:
•
Analyze the problem. What are you trying to find? What’s the given info?
•
Work out a plan before starting. Draw a sketch if possible. Look for indicator words (e.g. gained, lost,
times, per) to know which operations (+,-, x,÷) to use.
•
Estimate a reasonable answer.
•
Solve the problem.
•
Check your work. If the answer is not reasonable, start over.
Example:
Each home in a subdivision requires 180 ft of fencing. Find the number of homes that can be fenced with 5760
ft of fencing material.
•
What are we trying to find? Total number of homes that can be fenced.
Given info: 5760 ft of material, 180 ft of fencing per home.
•
Work out a plan. We want to know how many homes can be fenced with 5760 ft of fencing. If we divide
180 feet per home into 5760 feet we will get the total number of homes ( The feet units cancel out).
•
Estimate a reasonable answer.
Round 5760 to 6000
and 180 to 200.
6000÷200 = 30
4) Solve the problem.
5) Check your work.
Check division. Does 32x18=5760 ? Yes.
Is 32 close to our estimate, 30? Yes.
So our answer is 32 homes. (Make sure the units “homes” is in the answer!)
5760 ft 
1 home 5760

 32 homes
180 ft
180
32
180 5760
- 540
360
- 360
0