Transcript 2.1 Algebra

Ordering numbers, absolute value, and
opposites.
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Turn everything into a decimal
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Fractions: divide top by bottom/mixed fraction place the
whole number before the decimal ex: 2 = 2.
Negative Numbers the greater the absolute
value the smaller it is.
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- ,-
, - , -1
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-1, -1.7, - , -1
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Graph -4 and -5 on a number line. Then
write 2 inequalities that compare the two
numbers.
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The table to the right shows the
low temperatures recorded in
Nome, Alaska, each day for five
days in December.
Which low temp reading was the
coldest?
Which dates had low
temperatures above 10˚F?
Which dates had low
temperatures below -5˚F
Date
Temp
Dec. 18
-10˚F
Dec. 19
-11˚F
Dec. 20
16˚F
Dec. 21
3˚F
Dec. 22
2˚F
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Opposite numbers: are the same distance
from 0 with different signs
What is the opposite of the following
numbers?
4
-25
-9.31
5.7
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Absolute Value tells the distance a number is
from 0 and is shown by 2 straight bars on
either side of the number
-
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|x| = 7
◦ “What numbers are 7 units from the origin?” Both 7
and -7 are 7 units from the origin so there are two
solutions 7 and -7
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|x| = -5
◦ The absolute value of a number is never negative,
so there is no solution
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Velocity: Indicates both speed and direction
(up is positive and down is negative). The
speed of an object is the absolute value of its
velocity.
A space shuttle launch pad elevator drops at
a rate of 10 feet per second. What are its
velocity and speed?
◦ Velocity = -10 ft/second
◦ Speed = |-10| = 10 ft/second
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To prove that a statement is true, you need to
show that it is true for all examples. To
prove that a statement is false, you need to
show that it is not true for only one example
which is called a counterexample?
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The opposite of a number is always negative
The absolute value of a number is never
negative
The expression –a is never positive
The expression –a is sometimes greater than
a
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Graph the numbers on a number line. Then
write two inequalities that compare the two
numbers.
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1. -6.4 and -6.3
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2. -1 and -1
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When you have 2 negatives or 2 positives:
Add and keep the same sign
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When you have a negative and a positive:
Subtract and take the sign of the larger
-15 -18 =
-15 + (-18) =
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-9+2 =
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5-7-12 =
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5 + (-7) + (-12)=
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Associative Property of Addition: The way you
group three numbers when adding does not
change the sum.
◦ Example: (-4 + 8) + 6 = -4 + (8+6)
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Commutative Property of Addition: the order in
which two numbers are added does not change
the sum
◦ Example: 3 + (-2) = -2 + 3
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Identity Property of Addition: The sum of a
number and 0 is the number
◦ Example: -4 + 0 = -4
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Property of Zero (Inverse Property): The sum of a
number and its opposite is 0.
◦ Example: 5 + (-5) = 0
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Replace the variable with the number it
represents.
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9 + x + (-10), x = 3
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-12 + 5 + x, x = -13
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-12 + x + x, x = -6
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5-3 is the same thing as 5 + (-3) the sign
before the number goes with the number
If you subtract a negative the two signs
become a positive example: -3 – (-6) = -3+6
Remember: the absolute value of a number is
the distance from 0 example: 4= 4-2
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When you add and subtract fractions you
have to have a common denominator
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- +
-(- )=
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6-(- ) -
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Terms: When an expression is written as a
sum the parts that are added
◦ -9 – 2x
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When substituting a – number in for the
variable make sure the sign with the variable
stays
example: y = -x + 1, x = -2, y = - (-2) + 1
Evaluate the function for these values of x: 2, -1, 0, 1 and Organize your results in a
table
y = -6 – (-x)
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Matrix: a rectangular arrangement of
numbers into horizontal rows and vertical
columns. Each number in the matrix is called
an entry or an element.
The size is describes by :
(# of rows) x (# of columns)
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Write a matrix to organize the following
information about your CD collection
◦ Country: 4 groups, 6 solo artist, 0 collections
◦ Rock: 8 groups, 3 solo artist, 3 collections
◦ Blues: 1 group, 5 solo artist, 2 collections
+
+
-
=
=
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Even number of negatives when multiplying
the answer will be positive
Odd number of negatives when multiplying
the answer will be negative.
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Commutative Property: The order in which 2
numbers are multiplied does not change the
product
Associative Property: The way you group three
numbers when multiplying does not change the
product.
Identity Property: The product of a number and 1
is the number
Property of Zero: The product of a number and 0
is 0
Property of Opposites: The product of a number
and -1 is the opposite of the number
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3x2 – 5x when x = -2
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-4 (|y-12|) when y = 5
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-2x2 + 3x – 7 when x = 4
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AREA MODEL:
3
x
2
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2(x + 5)
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(x – 4)x
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(1+ 2x)8
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y(1-y)
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-3(x + 4)
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(y+5)(-4)
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-(6 – 3x)
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(x-1)(-9x)
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Coefficient: the number in a term
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-x + 3y2
coefficients are:
Like Terms: Terms that have the same
variable raised to the same power
Constant Terms: Numbers with no variables
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9x3 – 4x3 – 2
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-3 + y + 7
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2xy + 4x – 7xy + 7 - x
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3t(t-5) + 6t2
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-x3 + 2x(x – x2)
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-4(y+2) – 6y
X-7
x
X+11
X-2
2x+3
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Reciprocal:
◦ Zero is the only real number that has no reciprocal
◦ The product of a number and its reciprocal is 1
(Inverse property of multiplication)
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10 ÷ -2
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-39 ÷
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The quotient of two numbers with the same
sign is positive.
The quotient of two numbers with opposite
signs is negative
42y ÷
-7 • (÷
)
when x = -3 and y =
when a = -
and b =
when x = 2 and y = ½
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You are descending in a hot-air balloon, You
descend 500 ft in 40 sec. What is your
velocity?
Velocity =
Velocity = v (ft/sec)
Displacement = -500 feet
Time = 40 seconds
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When a function is defined by an equation, its
domain is restricted to real numbers for
which the function can be evaluated.
Division by zero is undefined, so input values
that make you divide by zero must be
excluded from the domain
◦ For instance, the function y =
x = 2 in its domain
does not have
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y=
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y=
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You launch a model rocket that rises 550 ft in
2.75 seconds. It then opens a parachute and
falls at a rate of 11 feet per second
◦ What is the rocket’s average velocity going up?
◦ What is the rocket’s velocity coming down?
◦ In how many seconds after the launch will the
rocket reach the ground?
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Probability of an event is a measure of the
likelihood that the event will occur. It is a
number between 0 and 1, inclusive.
P=0
P = 0.25
Impossible
Unlikely
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P = 0.5
P = 0.75
Occurs half the time Quite likely
P=1
Certain
Outcomes: The different possible results
Event: a collection of outcomes
Theoretical Probability =
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You toss two coins. What is the probability P
that both are heads?
An algebra class has 17 boys and 16 girls.
One student is chosen at random from the
class. What is the probability P that the
students is a girl?
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Experimental Probability =
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Odds =