Download Properties of Real Numbers

Chapter 1 Section 1
What does the term tools of algebra
refer to and why are these terms
important to our study of algebra?
What are the two branches of
the Real Number system and
how are they distinguished
from each other?
Natural Numbers: aka the counting numbers
Ex. 1, 2, 3, 4,...
Whole Numbers: natural numbers and 0
Ex. 0, 1, 2, 3, 4,...
Integers: the natural numbers, their opposites, and 0
Ex. …,-2, -1, 0, 1, 2,…
Rationals: numbers that can be written as quotients
of integers with NON-ZERO DENOMINATORS
Can either be written as terminating or
repeating decimals
1 4
Ex. 3, - 2.45, 0, ,
3 5
Irrationals: can’t be written as quotients of integers
Their decimal representations neither
repeat nor terminate
If a positive rational number is not a perfect
square such as 25 or 4/9, then its square root
is irrational
2
Ex : 2 , 7 , ,  ,1.476547857...
3
Real Numbers
Ex. Fast Eddie is a former math teacher gone concert
promoter. Your favorite band, Steve Subset and the
Irrationals is playing tonight at his club. Although the
show is sold out, Eddie says you can get in if you
answer these questions about what numbers best
describe the data, he’ll think about letting you in.
The cost cprofit
dollars
per
Eddies’s
or loss
ticket
for xfor
tickets:
per week
the year:
Week: one of first 52
c: rational
natural numbers
Profit: rational
x: whole
Graph a.  2
b. 0.3
9
c. 4
c.
-2
a.
b.
-1
0
1
2
Opposite or Additive Inverse: of any number a is –a
so that the sum of opposites is 0.
Ex. The additive inverse of 6 is -6
6+(-6)=0
Reciprocal s or Multiplica tive Inverse : of any non zero
1
number a is such that the product of reciprocal s
a
is 1.
1
Ex. The multiplica tive inverse of 8 is
8
Find the opposite and the reciprocal of 3/7
Opposite:
-3/7
Reciprocal:
7/3
Find the opposite and the reciprocal of –1/2
Opposite:
½
Reciprocal:
-2
Ex. -6+(2-4)=(-6+2)-4
Associative property of addition
Ex. 14 * 1=14
Identity property of multiplication
Ex. -6 * 4= 4 * -6
Commutative property of
multiplication
Absolute value: the distance a number is from 0
on a number line.
 b if b  0
| b | 
- b if b  0
Ex. |6|= 6 because 6 > 0
Ex. |-7|= -(-7)=7 because -7<0
Ex. |-2 * 3 + 5|= |-6+5|=|-1|= -(-1)=1
Page 8-10
2-30 even
36, 40, 44, 48, 52, 56, 60,
64, 68, 72, 76, 80, 92, 93
If you still have questions on today’s lesson visit
www.PHSchool.com
web code: age 0775
chapter 1, lesson 1
Homework quiz
web code: aga 0101
Chapter 1 Section 2
What does the term tools of algebra
refer to and why are these terms
important to our study of algebra?

How does the order of operations help us
evaluate algebraic expressions?
Variable: a symbol, usually a letter that represents one or
more numbers
Algebraic/Variable Expression: a mathematical statement
containing numbers, one or more variables, and often
mathematical operators.
Ex. 3x+y
4xy
Evaluate: to substitute numbers for variables in an
expression and follow the order of operations
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
Multiplication and Division
have the same priority
working from left to right!
Addition and Subtraction
have the same priority
working from left to right!
Evaluate a+3b-ab when a=1 and b=-2
Ex. Evaluate - x  2( x  1) for x  3
2

If you want to square a negative, make sure
you put the negative in parentheses:
In calculator terms:
 3  9, but (3)  9
why ?
2
2
Evaluate 4a+7b+3a-2b+2a when a=-5 and b=3
The expression -0.3y+61 represents the percentage
of eligible voters who voted in the presidential
elections from 1960-2000. y represents the
number of years since 1960.
Find the approximate percentage of eligible voters
that voted in 1988.
Step 1: find a value for y
1988-1960=28
Step 2: substitute y into the expression
-0.3(y)+61=-0.3(28)+61
=-8.4+61
=52.6, which is about 53
Term: a number, variable, or a product of a number
and one or more variables
Coefficient: the numerical factor in a term.
AKA: the large number in front of a
variable
Ex. identify the terms and variables in a-2b
terms:
a and -2b
coefficient:
1 and -2
Like Terms: terms that have the same variables
raised to the same power. THE VARIABLE
PORTIONS MUST BE EXACTLY THE SAME!!
Ex. Are these terms like or not?
3x and 3 y
3x and 2 x
3x and  6 x
2
2
3
2
Ex. 3x  2 x
2
2
#33 from book: 7b-(3a-8b)
Pages 15 to 17
2- 52 even
64, 68-76 even
If you still have questions on today’s lesson visit
www.PHSchool.com
web code: age 0775
chapter 1, lesson 2
Homework quiz
web code: aga 0102
Chapter 1 Section 3
What does the term tools of algebra
refer to and why are these terms
important to our study of algebra?

What does it mean to solve an equation and
how do you use the properties of equalities to
solve the equations?
The solution of an equation is a number, that
when substituted in for a variable, makes an
equation true.
Ex. 2x+3=13
A solution to this equations is x=5
2x+3=2(5)+3
=10+3
=13
1. Apply the distributive property
2. Clear fractions by using the common
denominator
3. Combine like terms on each side of the =
sign
4. Move numeric terms to one side and variable
terms
to the other using additive inverses.
5. Remove the coefficient from the variable
using the
multiplicative inverses.
6. State any restrictions
7. Check your answer
8. Note steps 3-5 are using PEM DAS in
reverse.
Ex. Solve 14x+28=6x-4
Check x=-4
14(-4)+28=6(-4)-4
-56+28=-24—4
-28=-28
6x-3(4x+6)=-3x-2(x-5)
What if you know you will be doing some problems
where you’re given the area and the base of some
triangles and are asked to find the height?
1
A  bh
2
The length of a rectangle is 3 cm greater
than its width. The perimeter is 24 cm.
Find the dimensions of the rectangle.
x
x
Solve  1  for x
a
b
Are there any restrictions on a and b?
Yes!!! If a=b, then x will have 0 as a denominator
which we know can’t happen!!!
pages 21 to 23
4 – 48 the multiples of 4 (4, 8 ,12, 16, . . .)
55, 57, 66a
If you still have questions on today’s lesson visit
www.PHSchool.com
web code: age 0775
chapter 1, lesson 3
Homework quiz
web code: aga 0103
Chapter 1 Section 4
What does the term tools of algebra
refer to and why are these terms
important to our study of algebra?

Compare the differences in the type of
solutions generated from equalities and
inequalities?



Kind of like equations, but instead of having an
equals sign, it has alligators (<,>, ≤,≥ ).
To remember which is less than: Make an ‘L’
with your finger and squash it. That’s less than.
Examples:
4<5
x≥y+2
3≤3
How many solutions are there to 4x=12?
One: x=3
How many solutions are there to 4x>12?
When x =‘s
4x’s
True or False
3
12
False
4
16
True
44
176
True
So any number greater than 3 makes the inequality true.
More or less the same as equations
 One very important thing! When multiplying or
dividing by a negative number, you must switch
the way the inequality is facing.
 Ex. -3x>15
x<-5
Why is this?
Because when multiplying or dividing by a negative
number, you change the sign of the original
number, so in an equality or inequality we change
all the signs.

If you are working with < or >, put an open
circle around the spot on the number line that
represents the value you solved for and shade
the numbers that make the inequality true.
Ex. x>3

1
2
3
4
Hint: If the variable is on the left ,
follow the direction of the arrow.
5
If you are working with ≤ or ≥, put a shaded
circle around the spot on the number line that
represents the value you solved for and shade
the numbers that make the inequality true.
Ex. x≤3

1
2
3
4
5
Solve and graph:
-4x+6<30
-8
-7
-6
-5
-4
Solve and graph:
2x+4≤-8
-8
-7
-6
-5
-4
What if we are solving inequalities and
something like this happens:
6x≥6x-3
0 ≥-3
There is no ‘x’ in the simplified form,
but it is a true statement, so all real
numbers make this inequality true.
What if we are solving inequalities and
something like this happens:
5x≤5x-7
0 ≤ -7
There is no ‘x’ in the simplified form,
but it is a false statement, so there
are no solutions to this inequality.
Ex. 3 page 28:
The pictured band agreed to play Solanco fest for
$200 plus 25% of the ticket sales. Find the ticket
sales needed for the band to receive at least
$500.






Compound Inequality: a pair of
inequalities joined by and or or.
And means both parts must be true
(intersection).
Or means either part could be true
(union).
Ex. -1<x and x≤3, which can also be
written as -1<x ≤3.
Ex. x>2 or x≤-4

Graph the solution of -6<2x-4<12

Graph the solution of 5a-4>16 or -3a+2>17

Ex. Graph the solution of 2y-2≥14 or 3y-4≤-13
pages 29 to 31
4 to 48 multiple of 4 ( 4, 8, 12, . . .)
56, 60
If you still have questions on today’s lesson visit
www.PHSchool.com
web code: age 0775
chapter 1, lesson 4
Homework quiz
web code: aga 0104
Chapter 1 Section 5
What does the term tools of algebra
refer to and why are these terms
important to our study of algebra?

Why does absolute value generate multiple
solutions in equations and why does it generate
compound inequalities?
Absolute value: the distance a number is from 0
on a number line.
 b if b  0
| b | 
- b if b  0


Let’s look at |3x-4|=11. How many
solutions does this equation have?
Two! Because the EXPRESSION 3x-4
could 3x-4 or –(3x-4), so we must do
both!
|3x-4|=11
if positive: 3x-4=11
if negative: -(3x-4)=11
Then solve each part as normal!

Solve |3x-4|=11

Solve 3|2x-1|-5=10

Solve |2x-6|=-17

Solve |2x+5|=3x+4

An Extraneous Root is a solution found by
solving an equation for a variable that turns
out to not be a true solution.
*ISOLATE the absolute value
expression.
*Split the Ab-val equation into
a set of “or” equations.
*Solve the equations.
*Check Your answers for
extraneous solutions!




Let k represent a positive real number.
|x|≥k: > , ≥ imply ‘or’
|x| ≤k: <, ≤ imply ‘and’
So Ab-val Inequalities can be written as
compound Inequalities.
Ex. |3x+6|≥12
Ex. 3|2x+6|-9<15
pages 36 to 38
4 to 52 multiple of 4 ( 4, 8, 12, . . .)
55, 62, 69
If you still have questions on today’s lesson visit
www.PHSchool.com
web code: age 0775
chapter 1, lesson 5
Homework quiz
web code: aga 0105