Transcript 009
MATH 010 KEVIN JONES
BEGINNING ALGEBRA
CHAPTER 1 REAL NUMBERS
1.1 Intro to Integers
:inequalities
< < > >
:opposites
(-)
:absolute values |x|
1.2 Add/Subtract Integers
SAME SIGNS:
Add their absolute values and keep the
same sign.
DIFFERENT SIGNS:
Find the difference of their absolute
values and keep the sign of the largest
absolute value.
1.3 Multiply & Divide Integers
Same Signs:
Signs are the same the answer
will be positive.
Different Signs:
Signs are different the answer
will be negative.
Properties of Zero and One in
Division
> 0 divided by any number =0
>any number divided by the
same number equals one.
>any number divided by one
equals the number
>any number divided by zero is not
defined.
a0
0 a
a
0
RATIONAL NUMBERS:
A number that can be written
a
b
Where a and b are integers and
B cannot = 0
WHY?
Simplest Form: reduce so the
numerator and the denominator have no
common factor.
12
2* 2*3
32
2*2*2*2*2
=
3
8
Decimals: also rational numbers
Repeating -
2
3
Terminating -
3
4
Addition of Fractions
3 5 Find the LCM of 8 & 6
8
6
8=2*2*2
LCM= 2 *2 *2*3=24
6=2*3
9 + 20 = 29
24
24
24
Reduce if possible
Exponents (powers)
3•3•3=3³
b•b•b=b³
b is the base
3 is the exponent
4
Evaluate (-2) =(-2)(-2)(-2)(-2)
= 16
4
Evaluate –2 = -2•2•2•2
= -16
even
(-a)
=positive
odd
(-a)
=negative
ORDER OF OPERATIONS
Please Excuse My Dear Aunt Sally
P►parentheses, do what inside
grouping symbols first
( ), { }, | |, [ ], and Fraction Bar
Excuse: simplify exponents
My Dear: multiply or divide
left to right.
Aunt Sally: add or subtract
left to right
Example:
evaluate 12 - 24(8-5) ÷ 2²
12 – 24(3) ÷ 2²
12 – 24 (3) ÷ 4
12 – 72 ÷ 4
12 – 18
-6
Example (2)
6 ÷ [4 - (6 – 8)] + 2²
6 ÷ [4 – (-2)] + 2²
6 ÷ 6 + 2²
6 ÷ 6 +4
1+4
►= 5
CHAPTER 2 VARIABLE
EXPRESSION
Variable expression is an expression
that contains one or more variables
3x² - 5y + 2xy – x – 7
5 terms
Variable terms: 3x², -5y, 2xy, -x
Constant term: -7
Evaluating Variable Expressions
Evaluate:
x² - 3xy when x = 3 & y = -4
(3)² - 3(3)(-4) substitute
9 – 3(3)(-4)
9 + 36
45
Simplify Variable Expressions
Variable Term; a term with a
variable 3x² or 3xy or 3x²y³
Parts of a Term: 3x²
3 called the coefficient
x called the variable
2 called the exponent
The variable and the exponent are
called the VARIABLE PARTS
LIKE TERMS: terms with the same
variable parts.
Combining like terms to simplify.
2x + 3x add the coefficients
5x
Simplify: 7y – 10y +5
-5y + 5 Combine 7y and –10y,
5 is not a like term.
Steps to simplify variable
expressions:
1. Remove all grouping symbols.
2. Look to collect the like terms.
Ex(1) Simplify
7(4 + 2x)
Distribute the 7
28 + 14x
No like terms
Ex(2) Simplify
2n – 3(2n – 7r)
2n – 6n + 21r
-4n + 21r
Distribute the -3
Collect like terms
Ex. 3 Simplify
-7x + 3[x – (3 – 2x)]
-7x + 3[x – 3 +2x]
-7x +3x – 9 + 6x
2x - 9
Parentheses
Bracket
Collect like
terms
Translate Verbal Expressions
into Variable Expressions
Key words:
Addition added to
more than
the sum of
increased by
the total of
Subtraction: minus
*less than*
decreased by
the difference between
Multiplication: times
of
the product of
multiplied by
Division: divided by
the quotient of
the ratio of
always use a fraction bar not ÷
Power (exponent):
the square of
the second power of
the cube of
the third power of
the fifth power of
Ex: 1 Translation
“The total of 3 times n and n”
When a sentence starts with an
operation, such as the total, then the
and is where you would place that
operation.
“The total of 3 times n and n”
3n + n
Ex: 2 Translations
“A number added to the product of
four and the square of the number”
Number → n
Added → +
Product of → and connects what
will be multiplied.
Four, square of the number→4n²
n + 4n²
Ex: 3 Translation
“a number multiplied by the total
of six and the cube of the number”
multiplied by the total > this is
what I call back to back operations and you must use a
parentheses
n(6 + n³)
*first ( is for the
multiplication.
Objective C Ex: 1 Translate
“The height of a triangle is 10 ft
longer than the base of the triangle.
Express the height of the triangle in
terms of the base of the triangle.”
We are comparing the height to the
length of the base, so let the length
of the base be your variable.
Base = x, so then the height = x + 10
Ex 2 Translate
A rope 12 ft long was cut into two
pieces of different length. Express
The length of each piece.
Smaller = X
Larger = 12 - X
This is called a sum of two
unknowns