Download M60 Section 3.4b : Writing and Solving Equations

M60 Section 3.1: Scientific Notation for Positive Numbers
Review: Exponents: Short hand way of expressing _________________________ ________________
Ex. 1:
53
In the expression:
3 is the _______________ and 5 is the ______________. The exponent tells us how
many times to use the ___________ as a factor in ______________ ____________.
Use of Exponents:
Area of a Square:
Volume of a Cube:
Ex. 2: In the sentence:
2, 3, and 5 are called _____________and 30 is called the ______________
Ex. 3: In the sentence: 5 + 7 + 2 = 14
5, 7, and 2 are called ______________ and 14 is called the ____________
Ex. 4:
Ex. 5:
Ex. 6: Write in exponential form:
(-7)(-7)(-7)(-7) =
Ex. 7: Use correct notation to described the opposite of 4²
Part A: Exponents and Powers of 10
Multiplication and Division of positive numbers by powers of 10:
Ex. 8:
42.3 x 10 4 =
Ex. 9:
Ex. 10:
0.025 x 10 6 =
Ex. 11:
Ex. 12:
3, 000
=
100
Ex. 13:
6.2 x 10 9 =
40
=
10
432 100
=
1000
So, multiplication and division by powers of 10 really means keeping track of the ______________ or the
Movement of the decimal __________________.
M60, Sec. 3.1 pg.2
Part B: Standard Notation vs. Scientific Notation with Large Positive Numbers:
Standard Notation: "regular" numbers. The way we usually see numbers written:
ex:
7,340
2.6
261,000
Scientific Notation:
Useful for very large numbers where the power on 10 shows the magnitude of
a number. It consists of a number that has exactly one nonzero digit to the left of the decimal (It could
be a negative number, but we'll concentrate on positive ones first.) multiplied by some power of 10.
a x 10 b
where
(The extreme value of a is between 1 and 10)
ex: distance within the solar system, national debt (currently $14,600,000,000,000), board feet
production of lumber in a year,
ex:
4.26 x 10 3
STANDARD FORM
and
7.4 x 10 5
SCIENTIFIC NOTATION
Ex. 14: 2300
Ex. 15: 400
Ex. 16: 4270
Ex. 17: 9,273,000
Ex. 18:
2.37 x 10 4
Ex. 19:
3.1 x 10
Ex. 20:
4 x 10 5
Ex. 21:
9.6 x 10 2
M60, Sec. 3.1 pg.3
Part C: Standard Notation vs. Scientific Notation and Small Positive Numbers:
Ex: microscopic bacteria, medications, computer chip resistors
STANDARD NOTATION
SCIENTIFIC NOTATION
Ex. 22: 0.000392
Ex. 23: 0.001234
Ex. 24: 0.2376
Ex. 25: 0.00295
Ex. 26:
6.2 x 10 –2
Ex. 27:
3.72 x 10 –3
Ex. 28:
4.55 x 10 –5
Ex. 29: Write the following in scientific notation:
16 x 10⁵
In summary: For scientific notation of positive numbers:
A positive exponent on the 10 means the number is __________ than 1. Think of it as a
_____________ number. A negative exponent on the 10 means the number is less than______ . Think
of it as a number very close to ________. In other words a _________.
M60, Sec. 3.1 pg.4
Part D: Multiplying Positive Numbers in Scientific Notation:
Look at multiplying the following:
=
Product Rule of Exponents:
When multiplying the same base, ___________ the base and __________ the exponents
Ex. 29: Use the Product Rule of Exponents to multiply the following numbers.
a)
b)
c)
Ex. 30: Use the commutative property of multiplication to rewrite the following and then multiply using
your calculator to find a, and the product rule of exponents to find b.
a)
b)
c)
Part E: Applications of Scientific Notation:
A song typically uses
bytes of storage on an mp3 player. Use scientific notation and
multiplication to determine how large a player you would need to store the 1600 cover versions of the
song “Yesterday”.
M60 Section 3.3: The Distributive Property and Like Terms
Part A: The Distributive Property:
Review Order of Operations:
Ex. 1:
4 (10 + 2)
Ex. 2:
4 (10 + 2x)
We need to use the Distributive Property of Multiplication to do Ex. 2.
Notice:
4 (10 + 2) = 48
BUT
4 (10) + 4 (2) = 40 + 8 = 48
The Distributive Property
Distributive Property of multiplication over addition and subtraction:
Use the Distributive Property to write the following without parentheses:
Ex. 3:
3 (x + 4)
Ex. 4:
4 (x – 3)
Ex. 5:
3 (3 – x)
Ex. 6:
(2x – 5) 4
Ex. 7:
(3z – 6) 8
M60 Sec. 3.3 p.2
Ex. 8:
- 2 (4x + 8)
Ex. 9:
- 3 (5x – 7)
Ex. 10:
- ( - 8b – 5)
Ex. 11:
(3y + 3) ( - 7)
Part B: Combining Like Terms
In Algebra like terms have exactly the same ____________ _____________ . We can add and
subtract them just like any other "like" things.
Examples of "like" terms:
Examples of "unlike" terms:
Coefficient:
Combining like terms:
Ex. 12:
4y + 2y
M60 Sec. 3.3 p.3
Ex.13:
7x – 9x
Ex.14:
- 8b + 8b
Ex. 15:
- 8xy + 11xy
Ex.16:
3w – 4w + 2w
Ex. 17:
- 7.6x – 5.2x
Ex18:
5b – 11b – (- 2b) + 3b
Ex. 19:
7a – 4 + 6 – (- 8a)
Ex. 20:
3x – 7 – x + 4
Ex. 21:
3ab + 5a – (- 4ab) – 8a
Ex. 22:
-12.7x – (- 3.3x)
M60 Sec. 3.3 p.4
Ex. 23:
- 4x2 + 7 x2
Ex.24:
- x3 + x2 - 3x3 - 2x2
Ex. 25:
6pq – 2pq2 - 2p2q – 6pq
Part C: The Distributive Property and Like Terms
Use the distributive property and combine like terms to simplify the following:
Ex. 26:
2x - 6 (x + 1)
Ex. 27:
3w – (w – 5)
Ex. 28:
3(x – 5) – (x + 4)
Ex. 29:
-3 (2w – 5) + 2w – 5 (w + 2)
Ex. 30:
8 (1 – 2z) + 3z – 4 (2z – 2)
M60 Section 3.4a: Solving Equations with Like Terms
Part A: Combining Like Terms to Solve Equations
Solve and check the following. Show all the steps in an orderly fashion. Simplify before your solve.
1.
– 6x + 8 – 3x = 10
2.
– 2(-5 Q) – 8 = - 20
3.
-11 = 6y + 5 – 7y
4.
-6r – 3r + 9 = 4
M60 Sec. 3.4a p.2
Part B: The Distributive Property and Solving Equations
5.
–24 = 4 (6 y – 3)
6.
7 – (q + 8) = -24
7.
-3 – (6 + y) – 7y = -25
M60 Sec. 3.4a p.3
Part C: Writing and Solving Equations:
Review: Solve the following using an algebraic equation: Translations.
8. Triple the sum of a number and 4 is equal to 30. Find the number.
9. The difference between a number and 6 is multiplied by 5. The result is 60. What is the number?
Review USE WORDS strategy:
10. At your Destination Graduation gathering there were 24 more women than men. The total number
in attendance was 148. How many men and how many women were in attendance?
M60 Sec. 3.4a pg 4
Review GUESS strategy:
11. The candy jar on Vikki’s desk has peppermint candies and carmels. Having just filled it, she knows
that it contains 114 candies and that there are 28 more peppermints than carmels. How many
peppermint candies and how many carmels are in the jar?
12. George works at an appliance store. He gets a regular wage of $10.75 per hour for a 40 hour week.
In addition to that, he gets a commission of 3% on whatever sales he makes. His gross pay for one week
was $505. Assuming he worked 40 hours, how much was his sales for the week?
M60 Section 3.4b : Writing and Solving Equations
Part C: Writing and Solving Equations:
Strategy 3: Visualize the problem. Make a sketch to help you understand the situation.
This is the first time we will be solving problems that have two unknowns. However, we will
still only use one variable.
Two unknowns but one variable: one must be described “in terms of” the other.
Usually more convenient to let the variable represent “the other”:
1: Jeenie has split her rectangular horse pen into two triangular pens by stretching an electric fence
diagonally from one corner to another. The length of the diagonal is 50 feet. The width of the pen is 10
feet less than the length. The perimeter of one of the new triangular pens is 120 feet. Find the
dimensions of the original pen.
2. Sam bought 40 feet of fencing to surround his rectangular pasture. His pasture is 5 feet longer than it
is wide. What is the width of the pasture?
M60 Sec. 3.4b p.2
Word problems with two unknowns (remember to use just one variable, and describe
the second variable in terms of the first).
3. Bryan purchased two used textbooks. The literature book costs $32.50 more than the art history
book. If the total cost of the two books was $145, what was the price of each book? (Try Use Words)
4. Together a new bicycle helmet and saddlebags cost $210. The helmet cost $27 more than the
saddlebags. How much does each cost?
M60 Sec. 3.4b p.3
5. Delbert and Francine together made $60,000 last year. This year, Francine doubled her income. This
year, their combined income is $85,000. How much did each earn last year? (Try Guess)
Let x = Francine’s income last year.
6. James spent $84 for a new pair of pruning shears and a spading fork. The spading fork cost $18 less
than the pruning shears. How much did each item cost?
7. Eight times as many students as faculty members attend the Honor Society tea. If 144 people attend
the tea, how many faculty members and how many students attended the tea?
M60 Sec. 3.4b p.4
Review:
Notation for fractions:
1

3
b

2
2
x
3
3x

4
Review:
Coefficient on x is negative:
3 – 9x = -15
7-
x
=-2
2