Transcript File

Equations and
Inequalities
Algebra 2 Chapter 1
George Hardy
DO NOW: Real Numbers
 Use the number subsets to fill in the chart below
Numbers Subsets
 Whole Numbers
 0, 1, 2, 3, 4, …
 Integers
 Positive and negative whole numbers
 -∞, …, -4, -3, -2, -1, 0, 1, 2, 3, …, ∞
 Rational Numbers
 A number that can be expressed as a fraction
with an integer numerator and a non-zero
natural number denominator
 Integers and fractions: 1, -3, -50, -⅞, ½
Number Subsets
 Irrational Numbers
 A real number that CANNOT be written as a
fraction of two integers
 Decimal that never ends nor forever repeats
 EX. π, √2 ≈ 1.41421356
 Real Numbers
 Includes all of the measuring numbers, including
decimals
 Every RATIONAL and IRRATIONAL is a REAL, but
every REAL is NOT a RATIONAL
Quiz
 √5
 .333333
1
 -56
 3/9
 -7/8
 √100
 e= 2.7182818
Irrational
Rational, Real
Real, Integer, Whole, Rational
Real, Integer, Rational
Real, Rational
Real, Rational
Real, Rational, Whole, Integer
Irrational
Properties of Real Numbers
Addition
Multiplication
 Commutative
 Commutative
A+B=B+A
 Associative
(A + B) + C = A + (B + C)
 Identity
A+0=A
0+A=A
 Inverse
A + (-A) = 0
AB = BA
 Associative
(AB)C = A(BC)
 Identity
A*1=A
1 *A =A
 Inverse
A * 1/A = 1, when a ≠ 0
Identify the Properties
 (3 + 9) + 8 = 3 + (9 + 8)
 Associative Property of Addition
 14 * 1 = 14
 Identity Property of Multiplication
 2(b + c) = 2b + 2c
 Distributive Property
NOTE!
 The opposite, or additive inverse of any
number a is –a. The reciprocal, or multiplicative
inverse, of any nonzero number a is 1/a.
 Subtraction is defined as adding the opposite,
and division is defined as multiplying by the
reciprocal.
Critical Thinking
In Nordstrom, I found a sign that read “20% off the
sale price.”
 Sperry’s that originally $80 had a sale price that
was 15% less than the original price. What was the
final price of the shoes, according to the sign?
 Would the final price of the shoes be the same if
35% had been taken off the original price? If not,
which price would be lower?
Challenge
 At the register, another shopper was
buying a pair of shoes that had been
subject to the same two discounts. The
final price of the shoes was $81.60.
What was the original price?
DO NOW
 Agree or Disagree,
and support with an
example:
(a – b) – c = a – (b – c)
Order of Operations
 Which is correct?
 PEDMAS
 PEMDAS
 PEDMSA
 PEMDSA
 All are correct!
 Division or
Multiplication
 Addition or
Subtraction
 Just remember to
go left to right!
Insert Parenthesis ..
 …To make the statements true
1)8  2 3  4 14
2)72  8 6  5  64
Evaluate the Expressions
a.(25t 125t)  (t 1)for t = 3
2
2
b  b  4ac
b.
for a = 2, b = 9, c = -5
2a
2
2x 1
-1
c. 3
for x =
2x 1
2
2
Neurosis
 A test measuring
neurotic traits, such as
anxiety and hostility
indicate that people
become less neurotic as
they get older (Williams,
2006).
 The algebraic expression
23 – 0.12x describes the
average neurotic level
for people x years old.
Evaluate the expression for x = 80. Describe what it means in practical terms.
Checkpoint
 What does it mean to evaluate an
expression?
 What symbols act as grouping symbols
in the order of operations?
 Why is 3(x + 7) – 4x not simplified? What
must be done to simplify the
expression?
Homework
 Chapter 1.1 #16 – 50 even, and 1.2 #16 –
52 even
 Read 1.3 and 1.4 and take notes
Do Now: Geometry Connection
 A triangle has a base of n + 10, and a
height of n. Write and expression for the
area of the figure. Then find the area
when n = 40.
 A square has a side of x + y. Write an
expression for the area of the figure.
Then evaluate the expression when x =
12 and y = 5.
Challenge
 At the register, another shopper was
buying a pair of shoes that had been
subject to the same two discounts. The
final price of the shoes was $81.60.
What was the original price?
Equations and Inequalities
 An equation is a statement in which two
expressions are equal
 Inequalities have properties similar to
those of equations; however, there are
a few differences
 What are some of the differences?
Dry Ice
 Dry ice is solid carbon dioxide. Dry ice
does not melt – it changes directly from
a solid to a gas. Dry ice changes to a gas
at -109.3° F. What is this temperature in
degrees Celsius?
 Use the formula
9
F  (C)  32
5
Stockbroker
 A stockbroker earns a base salary
$40,000 plus 5% of the total value of the
stocks, mutual funds, and other
investments that the stockbroker sells.
Last year, the stockbroker earned
$71,750. What was the total value of the
investments the stockbroker sold?
Try These
7
1) x 1  2x  5
2
 How do we know
our answers are
correct?
2x  5 x  7 3x 1
2)


5
2
2
Graphing Calculator
 Solve the following equation without a
graphing calculator
 4(2x + 1) – 29 = 3(2x – 5)
College Tuition Costs
 The model
 T = 974x + 15,410
represents the average
cost of tuition and fees at
private four-year colleges
x years after 2000. Use the
model to determine when
tuition and fees will
average $27,098.
Note: Types of Equations
 An equation that is true for all real numbers for which
both sides are defined is an IDENTITY. The solution
set is all real numbers.
 x+3=x+2+1
 An equation that is not an identity, but that is true
for at least one real number is a CONDITIONAL
EQUATION.
 2x + 3 = 17
 An INCONSISTENT EQUATION is not true for even
one real number
 x=x+7
Homework
 Chapter 1.3 #16 – 48, 52 – 56 even,
Chapter 1.6 #14 – 52 even. Try 54 in
Fahrenheit
 Read Chapter 1.4 and take notes!
Do Now
 Solve the equations or simplify
 The bill for the repair of your car was
$390. The cost for parts was $215. The
cost for labor was $35 per hour. How
many hours did the repair work take?
 –(x + 2) – 2x = -2(x + 1)
 4x2 – 2(x2 – 3x) + 6x + 8
Graphing Calculator
 Solve the following equation without a
graphing calculator
 4(2x + 1) – 29 = 3(2x – 5)
College Tuition Costs
 The model
 T = 974x + 15,410
represents the average
cost of tuition and fees at
private four-year colleges
x years after 2000. Use the
model to determine when
tuition and fees will
average $27,098.
Note: Types of Equations
 An equation that is true for all real numbers for which
both sides are defined is an IDENTITY. The solution
set is all real numbers.
 x+3=x+2+1
 An equation that is not an identity, but that is true
for at least one real number is a CONDITIONAL
EQUATION.
 2x + 3 = 17
 An INCONSISTENT EQUATION is not true for even
one real number
 x=x+7
Solving Linear Inequalities
x 3 x 2 1
1)


4
3
4
2)2(x  4)  2x  3
Special Inequalities
 Some inequalities have one solution, no
solution, or all real solutions satisfy the set.
 No solution – eliminate the variable, and
have a false statement
 All reals – eliminate the variable, and
have a true statement.
Love
 Write an inequality
that expresses for
which years in a
relationship,
intimacy is greater
than commitment.
Applications
Car Rental
 Acme Car Rental charges $4 a day plus $.15
per mile, whereas Interstate Car Rental
charges $20 a day plus $.05 per mile. How
many miles must be driven to make the
daily cost of Acme Rental to be a better deal
than Interstate?
Phone Call
 You go online to find phone companies that have
long distance rates. You’ve chosen a plan that has
a monthly fee of $15 with a charge rate of 8 cents
per minute for all long distance calls. Of course,
there are other bills to pay; therefore, you don’t
want to spend more than $35.
 Write an inequality that represents the
situation.
 How many minutes of long distance calls can
you make?
Grades
 A professor announces that course grades will
be computed by taking 40% of a student’s
project score (0 – 100 points) and adding 60%
of the student’s final exam score (0 – 100
points). If a student gets an 86 on the project,
what scores can she get on the final exam to
get a course grade of at least 90?
Compound Inequalities
AND
 -3 < 2x + 1 ≤ 3
OR
 2x – 3 < 7 0r 35 – 4x ≤ 3
Car Repair
 Parts for a repair Dodge Charger costs $175.
The mechanic costs $34 per hour. If you receive
an estimate for at least $226, and at most $294
for fixing the car. What is the time interval that
the mechanic will be working on the car?
Connecting Ideas
Solve the compound inequality
3n + 1 > 10 and ½n – 1> 3
Classwork
 Chapter 1.6 #4 – 11 ALL
Homework Quiz: D Hour
 Close text books, show ALL work, box
answers
 Chapter 1.2
 #36, 44, 46, 56
 Chapter 1.3
 #30, 36, 42
Homework Quiz: E Hour
 Close text books, show ALL work, box answers
 Chapter 1.1
 #54, 62
 Chapter 1.2
 #36, 56
 Chapter 1.3
 #32, 44, 46
Homework Quiz: F Hour
 Close text books, show ALL work, box answers
 Chapter 1.1
 #60
 Chapter 1.2
 #46,50, 52
 Chapter 1.3
 #30, 34, 42
Homework Quiz G Hour
 Close text books, show ALL work, box answers
 Chapter 1.1
 #46, 54
 Chapter 1.2
 #46, 56
 Chapter 1.3
 #32, 44, 46
Equations Examples
Solve the following equations
6a – 4[2 – 3(4a – 3)] = -17
½ - 3(x + 1) = 8
DO NOW
 Solve the equations
1
3
x 2  3 x
4
4
1
(4 x 10)  5  3x
2
Recall
 The state sales tax in Michigan is 0.06
(or 6%). If your total bill at the music
store included $1.32 in tax, how much
did the merchandise cost?
Video Arcade
 You have $4.25 to spend at a video
arcade. Some games cost $0.75 to play
and other games cost $0.50 to play. You
decide to play 2 games that cost $0.75.
Write and solve an inequality to find the
possible number of $0.50 video games
you can play.
Love
 Write an inequality
that expresses for
which years in a
relationship,
intimacy is greater
than commitment.
Connecting Ideas
Solve the compound inequality
3n + 1 > 10 and ½n – 1> 3
DO NOW: ACT Practice
When x = 3 and y = 5, by how much
does the value of 3x2 – 2y exceed
the value of 2x2 – 3y ?
How many irrational numbers are
there between 1 and 6 ?
Critical Thinking
 Determine whether the statement makes
sense or does not make sense. Explain
your reasoning.
 If you tell me that three times a number
is less than two times that number, it’s
obvious that no number satisfies this
condition, and there is no need to write
and solve an inequality.
Intro to 1.4
 Solve for y:
1
4
Given : x  y  19
2
5
Why is it important…
 …to rewrite equations?
Rewriting Equations and
Formulas
 The formula for the perimeter of a rectangle is
P = 2l + 2w. Solve for l.
2(l  w)  P
P
lw 
2
P
l  w
2
Rewriting Equations and
Formulas
V = ⅓πr2h, solve for r
3V  r h
2
3V
2
r
h
3V
r
h
Volume
 The silo is a cylinder with a half
sphere on top. The silo can hold
576π cubic feet of grain. The radius
of the sphere is 6ft. Given the
volume of a cylinder is V=πr2h and
the volume of a sphere is V=4/3πr3,
write a formula for the volume of
the silo
Solve the formula for h; then,
determine the height of the silo.
Geometry Challenge
 The formula for the area of a circle is
A=πr2. The formula for the
circumference of a circle is C=2πr. Write
the formula for the area of a circle in
terms of its circumference.
Classwork
 Chapter 1.4 Practice Worksheet
 Copy the Table of Common Formulas
Homework
 Chapter 1.4 #12 – 32 even. 33 – 34 all,
37 – 39 all.
 Read Chapter 1.7 and Take Notes
Do Now
 Complete the Mixed Review on Page 47
 #61 - 69
DO NOW:
Homework Quiz D-Hour
Chapter 1.4
#12,16, 20, 22, 26, 28, 30
Critical Thinking
Write an inequality whose
solutions are all real numbers.
Show why the solutions are all real
numbers
Lesson Opener
 What is the definition of absolute value?
 x if x  0
x  
x if x  0
Discussion
 What is the value of x? Explain the two statements.
x 2
x  2
Absolute Value Equations
 Remember, that x can have two possible
values- a positive and a negative value!
 Therefore:
 |ax + b| = c, where c > 0 (positive) is
equivalent to the compound statement
 ax + b = c or ax + b = -c
Example 1
10  4 x  2
10  4x  2 or 10  4x  2
Example 2
3(x  4) 1
Example 3
51  4x 15  0
Critical Thinking
3x 1  x  5
Try These
2
1) x  2  10
3
2) 2n  5  7
1
3) x  3  2
2
4)21  3x  28  0
Absolute Value Inequalities
 The inequality |ax + b| < c, where c > 0,
means that ax + b is between -c and c. This is
equivalent to –c < ax + b < c.
 < can be replaced with ≤
 The inequality |ax + b| > c, where c > 0,
means that ax + b is beyond -c and c. This is
equivalent to ax + b > c or ax + b < -c.
 > can be replaced with ≥
A Way to Remember!
GREAT(OR) Than
Less TH(AND)
Example 1
4x 3
Example 2
2 3x  5  7  13
Try These
3
1)1 x  1
4
x
2)  4  7
3
Classwork
Chapter 1.7 4#4-15 all
Homework
 Chapter 1.7 #18 – 58 even
 Read chapter 1.5 and take notes
DO NOW:
Homework Quiz F-Hour
Chapter 1.4
#24, 26, 28
Chapter 1.6
#32, 36
Chapter 1.7
#32, 38
DO NOW:
Homework Quiz G-Hour
Chapter 1.4
#24, 26, 28
Chapter 1.7
#32, 38
DO NOW:
Homework Quiz D-Hour
Chapter 1.4
#12,16, 20, 22, 26, 28, 30
DO NOW:
Homework Quiz E-Hour
Chapter 1.4
#12,16, 22, 24, 26, 28, 30
DO NOW/RECALL
Solve the inequality
1) |7x + 5| < 23
2) |-½x + 3| ≥ -3
3) |x + 7| < 0
Writing Abs Value Inequalities
 Physicians consider an adult’s normal
body temperature to be within 1°F and
98.6°F. Write an absolute value
inequality that describes the range of
normal body temperatures.
Intro to Chapter 1.5
 Problem Solving Using Algebraic
Models
 Strategies
 Group Work/Discussion
 Presentation/Review
Strategies for Word Problems
 Read the problem carefully! State the problem in
your own words and state what the problem is
asking.
 Let x (or any variable) represent one of the
unknowns
 Write an equation that models verbal condition
 Solve the equation
 Check the solution
 Draw diagrams and Patterns
Homework
 Problem Solving Worksheet
 Complete the Chapter 1 Review
DO NOW
 Take out Chapter 1.4 worksheet AND
the problem solving strategies
worksheet!
 Pass them to the front
 Find your seat (in alphabetical order)