Transcript Solving Linear Equations

HELPS YOU WITH YOUR ASSIGNMENT!
CAN USE THEM ON YOUR TESTS!
1.1 Real Numbers and Number Operations
What you should learn:
Goal 1 Use a number line to graph and order
real numbers.
Goal 2 Identify properties of and use
operations with real numbers.
Whole numbers: 0, 1, 2, 3, …
Integers: …-3, -2, -1, 0, 1, 2, 3, …
Real numbers: include fractions,
decimals, whole numbers, and Integers.
positive #’s
negative #’s
origin
-3 -2 -1
0
1
2
3
Graph the numbers on the number line.
Ex 1)
Ex 2)
0, -3, 1
2, -1, -2
-3 -2 -1
0
1
2
3
Write two inequalities that compare the numbers.
Ex 1)
Ex 2)
0, -3, 1
2, -1, -2
-3 < 0 <1
-2< -1< 2
Write numbers in increasing order.
Ex 1)
0.34, -3.3, 1.12
-3.3 < 0.34 < 1.12
Ex 2)
2.23, 2.2, -2.23
-2.23< 2.2 < 2.23
Properties of Addition and Multiplication
1. a + b = b + a
ab=ba
Commutative
property
2. (a + b) + c = a +(b + c) Associative
property
(a b) c = a ( b c )
3. a + (-a ) = 0
1
a   1, a  0
a
Inverse property
4.
5.
a(b + c) = ab + ac
Distributive property
a+0=a
a 1  a
Identity property
Using Unit Analysis
Perform the given operation. Give the answer with the
appropriate unit of measure.
Reflection on the Section
When converting units with unit analysis, how do
you choose whether to use a particular conversion
factor or its reciprocal?
assignment
1.2 Algebraic Expressions and Models
What you should learn:
Goal 1 Evaluate algebraic expressions.
Goal 2 Simplify algebraic expressions by
combining like terms.
Numerical expression consists of numbers,
operations, and grouping symbols.
Expressions Containing Exponents.
Example:
4  44444
5
The number 4 is the BASE,
the number 5 is the EXPONENT, and
4
5
is the POWER.
Order of Operations
Parentheses
Exponents
Multiplication and Division
Addition and Subtraction
1. First do operations that occur within symbols of grouping.
2. Then evaluate powers
3. Then do multiplications and divisions from left to right
4. Finally do additions and subtractions from left to right.
Variable is a letter that represents a number.
Values of the variable are the numbers.
Algebraic expression is a collection of
numbers, variables, operations, and
grouping symbols.
Evaluate is to make a substitution, do the
work, and determine the value.
Value of the expression is the answer after
the expression is evaluated.
Definitions:
Terms: are the number.
Coefficient: is the constant in
front of the variable.
Constant term
4 x y  5xy  25 y  6 y  3
2
Like Terms:
Evaluate the power.
4
(2)
(2)( 2)( 2)( 2) 
Ex)
Ex)
2
 (2)( 2)( 2)( 2) 
16
4
 16
Evaluate the expression when x = 4 and y = 8.
ex)
substitute
Do the work
Get the value
4x  9 y
4(4)  9(8)
16  72
 56
Reflection on the Section
State the order of operations.
assignment
1.3 Solving Linear Equations
What you should learn:
Goal 1 Solve linear equations
Goal 2 Use linear equations to solve real-life
problems.
Using Addition or Subtraction
The key to success: Whatever operation is
done on one side of the equal sign, the same
operation must be done on the other side.
Inverse operations undo each other.
Examples are addition and subtraction.
Solving Linear Equations
Generalization:
If a number has been added to the variable,
subtract that number from both sides of the
equal sign.
ex)
x3 6
3 3
If a number has been subtracted from the
variable, add that number to both sides of
the equal sign.
ex)
x4  6
4 4
Solving Linear Equations
Generalization:
If a variable has been multiplied by a nonzero
number, divide both sides by that number.
example:
4x = - 12
4
4
Solving Linear Equations
Generalization:
If a variable has been divided by a number,
multiply both sides by that number.
example:
x
6
 2  6
6
Hint: you always start looking
at the side of the equal sign
that has the Variable.
Solving Linear Equations
3 2
3
 x4 
2 3
2
ex)
ex)
12
6
x
2
2
5
5
 4 x 
5
2
2
20
10  
x
2
Solving Linear Equations
Generalization:
First undo the addition or subtraction, using
the inverse operation.
Second undo the multiplication or division,
using the inverse operation.
Solving Linear Equations
example:
example:
w
 10  20
5
 10  10
w
5   10  5
5
subtract 10
multiple by 5
4c  2  10
2
2
4c  12
4
4
c3
Solving Multi-Step Equations
example:
2 x  4  4 x  14
6x  4  14
4
4
6 x  18
6
6
x3
Solving Multi-Step Equations
example: 3( x  4)  15
3x  12  15
 12  12
3x  27
3
3
x 9
Solving Linear Equations
example:
2( x  2)  4 x  14
2 x  4  4 x  14
6x  4  14
4
4
6 x  18
6
6
x3
Solving Multi-Step Equations
Example)
Move the
smaller
#.
 5x  2  3x  8
 5x
 5x
2  2x  8
8
8
10  2 x
2
2
5 x
Solving Linear Equations
Example)
 2x  6  2x  8
 2x
 2x
6  8
No Solution
Solving Linear Equations
Example)
6x  7  6x  7
 6x
 6x
77
All Solutions
or
All Real Numbers work
Solving Linear Equations
Reflection on the Section
How does solving a linear equation differ from
simplifying a linear expression?
assignment
Solving Linear Equations
1.4 Rewriting Equations and Formulas
What you should learn:
Goal 1 Rewrite equations with more than one
variable
Goal 2 Rewrite common formulas.
Solve this equation for x.
Ex 2)
ax  b  c
b b
ax  b  c
a
a
bc
x
a
(3.7) Formulas
Solve this equation for x.
Ex 3)
2( x  b)  c
2x  2b  c
 2b  2b
2x  2b  c
2
2
2b  c
x
2
(3.7) Formulas
Solve this equation for b.
Ex 5)
1
A  bh
2
h
2

1
h
A 1
2
 b 
h 2
1
2A
b
h
(3.7) Formulas
Example)
Solve the investment-at-simple-interest
formula A = P + Prt for t.
A = P + Prt
-P -P
A – P = Prt
Pr
Pr
A-P =t
Pr
(3.7) Formulas
How do you solve for x?
4  x  2x
now solve this one for C?
s  C  rC
(3.7) Formulas
First, substitute the given value for x,
then solve this equation for y.
Ex ) 12 x  3 y
 15; x  2
12(2)  3 y  15
 24
 24
 3 y  9
-3
-3
y3
Formulas
First, solve this equation for y, then substitute.
Ex )
12 x  3 y  15; x  2
 12 x
 12 x
 3 y  12 x  15
-3
-3
 12 x  15
y
3
 12(2)  15
y
3
=3
Formulas
Reflection on the Section
How can rewriting formulas help you solve them?
assignment
1.5 Problem Solving Using Algebraic Models
What you should learn:
Goal 1 Use general problem solving plan to
solve real-life problems
Goal 2 Use other problem solving strategies
to help solve real-life problems.
USING A PROBLEM SOLVING PLAN
It is helpful when solving real-life problems to first write an
equation in words before you write it in mathematical symbols.
This word equation is called a verbal model.
The verbal model is then used to write a mathematical
statement, which is called an algebraic model.
WRITE A
ASSIGN
WRITE AN
VERBAL MODEL.
LABELS.
ALGEBRAIC MODEL.
SOLVE THE
ANSWER THE
ALGEBRAIC MODEL.
QUESTION.
Writing and Using a Formula
The Bullet Train runs between the Japanese cities of Osaka
and Fukuoka, a distance of 550 kilometers. When it makes
no stops, it takes 2 hours and 15 minutes to make the trip.
What is the average speed of the Bullet Train?
Writing and Using a Formula
You can use the formula d = rt to write a verbal model.
VERBAL
MODEL
LABELS
ALGEBRAIC
MODEL
Distance
=
Rate
•
Time
Distance = 550 (kilometers)
Rate = r (kilometers per hour)
Time = 2.25 (hours)
d
550
t
= r •(2.25)
Write algebraic model.
= r
Divide each side
by 2.25.
244  r
Use a calculator.
550
2.25
The Bullet Train’s average speed is about 244 kilometers per hour.
Writing and Using a Formula
UNIT ANALYSIS
You can use unit analysis to check your verbal model.
244 kilometers
550 kilometers 
hour
• 2.25 hours
USING OTHER PROBLEM SOLVING STRATEGIES
When you are writing a verbal model to represent a
real-life problem, remember that you can use other
problem solving strategies, such as draw a diagram,
look for a pattern, or guess, check and revise, to help
create a verbal model.
Drawing a Diagram
RAILROADS In 1862, two companies were given the rights
to build a railroad from Omaha, Nebraska to Sacramento,
California. The Central Pacific Company began from Sacramento
in 1863. Twenty-four months later, the Union Pacific company
began from Omaha. The Central Pacific Company averaged 8.75
miles of track per month. The Union Pacific Company averaged 20
miles of track per month.
The companies met in Promontory, Utah, as the 1590 miles of track
were completed. In what year did they meet? How many miles of track
did each company build?
Drawing a Diagram
Central Pacific
VERBAL
MODEL
Total
miles
of track
LABELS
=
Miles per
month
•
Number of
months
Union Pacific
+
•
Total miles of track = 1590
(miles)
Central Pacific rate = 8.75
(miles per month)
Central Pacific time = t
(months)
Union Pacific rate =
(miles per month)
20
Union Pacific time = t – 24
ALGEBRAIC
MODEL
Miles per
month
1590 = 8.75 t + 20 (t – 24)
Number of
months
(months)
Write algebraic model.
Drawing a Diagram
ALGEBRAIC
MODEL
1590 = 8.75 t + 20 (t – 24) Write algebraic model.
1590 = 8.75 t + 20 t – 480 Distributive property
2070 = 28.75 t
72 = t
Simplify.
Divide each side by 28.75.
The construction took 72 months (6 years) from the time the
Central Pacific Company began in 1863. They met in 1869.
Drawing a Diagram
The construction took 72 months (6 years) from the time
The Central Pacific Company began in 1863.
The number of miles of track built by each company is as follows:
Central Pacific:
Union Pacific:
8.75 miles
• 72 months = 630 miles
month
20 miles
month
• (72 – 24) months = 960 miles
Looking for a Pattern
The table gives the heights to the top of the first few stories of a
tall building. Determine the height to the top of the 15th story.
SOLUTION
Look at the differences in the heights given in the table.
Story
Height to top
of story (feet)
Lobby
1
2
3
4
20
32
32
44
44
56
56
68
After the lobby, the height increases by 12 feet per story.
12
12
12
12
Looking for a Pattern
You can use the observed pattern to write a model for the height.
VERBAL
MODEL
LABELS
ALGEBRAIC
MODEL
Height to top
of a story
=
Height of
lobby
+
Height per
story
•
Story
number
Height to top of a story = h
(feet)
Height of lobby = 20
(feet)
Height per story = 12
(feet per story)
Story number = n
(stories)
h = 20 + 12 n
Write algebraic model.
= 20 + 12 (15)
Substitute 15 for n.
= 200
Simplify.
The height to the top of the 15th story is 200 feet.
Reflection on the Section
After you have set up and solved an algebraic
model for problem description, what remains to be
done?
assignment
1.6 Solving Linear Inequalities
What you should learn:
Goal 1 Solve simple inequalities
Goal 2 Solve compound inequalities.
Let’s describe the inequality
in different ways.
Verbal Phrase
All real numbers
less than 3
Inequality
x<3
Graph
-2
-1
0
1
2
3
What about this one…
Verbal Phrase
All real numbers greater than
or equal to 0
Inequality
x0
-2
-1
Graph
0
1
2
3
Solving Linear Inequalities
You solve these just like you
solved other linear equations.
Ex 1)
x 5  3
5 5
x  2
Subtract 5
Solving Linear Inequalities
Ex 2)
x  4  7
4
4
x  3
Add 4
Solving 2-Step Linear Inequalities
Beware….
 3x  8  2
8 8
 3x  6
Watch this…
3 3
Reverse the inequality!
Because you divided by a
x

2
negative.
ex)
Solving Linear Inequalities with
Variables on both sides
4 x  4  3  5x
 5x
 5x
9x  4  3
4 4
9x  1
9
9
1
x
9
Solving Linear Inequalities using the
Distributive Property
 4(2  4 x)  3
 8 16x  3
8
8
16 x  5
16 16
5
x
16
Solving Linear Inequalities using Combing Like
terms and variables on both sides of the equal sign.
4x  9  4  3  8  3x
4x  5  5  3x
 3x
 3x
x  5  5
5
5
x0
Solving Compound Inequalities
Involving “And”
A Compound Inequality consists of two
inequalities connected by the word and or
the word or.
All real numbers that are
greater than or equal to zero and less than 4.
0 x4
-1
0
1
2
3
4
Solve for x.
 2  3x  8  10
8
8
8
6  3x  18
3 3 3
2 x6
1
2
3
4
5
6
Solve for x.
 2  2  x  1
2
2
2
0  x  3
1 1 1
0  x  3
-3
-2
-1
0
Write an inequality that represents the statement.
ex 1)
x is less than 6 and greater than 2.
2 x6
ex 2)
x is less than or equal to 10 and greater than -3.
 3  x  10
x is greater than or equal to 0 and
less than or equal to 2.
ex 3)
0 x2
Write an inequality that represents the statement.
ex 4)
The frequency of a human voice is
measured in hertz and has a range of 85
hertz to 1100 hertz.
85  x  1100
What if...
x3
and
x6
What numbers make both statements true?
3
6
What if...
x3
and
x6
What numbers make both statements true?
No, just the
3
x6
6
What if...
x  5
and
x5
Can this happen??
5
-5
A number can’t be both….
Solving Compound Inequalities
Involving “Or”
Remember…
A Compound Inequality consists of two inequalities
connected by the word and or the word or.
All real numbers that are
Less than -1 or greater than 2.
x  1
-2
-1
x2
or
0
1
2
3
Solve for x and graph.
3x  1  4
1 1
3x  3
3 3
x 1
1
or
2x  5  7
5 5
2x  12
2
2
x6
6
Solve for x and graph.
x 7  3
7 7
x  10
10
or
2x  24
2
2
x  12
12
Solve for x and graph.
x  4  1
4 4
x  5
or
5x  15
5
5
x  3
Is x = -4 a solution?
-5
-3
What if...
x3
or
x6
What numbers make the statement true?
3
6
What if...
x3
or
x6
What numbers make the statement true?
All numbers greater than 3
3
6
Reflection on the Section
Compare solving linear inequalities with solving
linear equalities.
assignment
1.7 Solving Absolute Value Equations and
Inequalities
What you should learn:
Goal 1 Solve absolute value equations and
inequalities
Goal 2 Use absolute value equations and
inequalities to solve real-life
problems.
An open sentence involving absolute value
should be interpreted, solved, and graphed as
a compound sentence.
Study the examples:…
For c  0 , x is a solution of
if x is a solution of:
ax  b  c
or
ax  b  c
ax  b  c
For c  0 , x has no solution
x 2
example 1a)
x2
What can x be?
or
-2
example 1b)
x  2
2
x  8
What can x be?
Nothing…, no solution
x3  4
example 2)
x3 4
-3
-3
x 1
-7
or
x  3  4
-3
-3
x  7
1
5y  2  7
example 3)
5 y  2  7
or
5y  2  7
+2
+2 +2
+2
5y  9
5 y  5
5
5
5
5
y  1
4
y 1
5
-2
-1
0
1
2
3
example 4)
x5 7  2
7 7
1ST
get absolute value
by itself.
x5  9
x5  9
-5 -5
x4
-14
or
x  5  9
-5
-5
x  14
4
Solving Absolute Value Inequalities
An absolute-value inequality is an
inequality that has one of these forms:
ax  b  c
ax  b  c
ax  b  c
ax  b  c
3x  4  10
example 2)
3x  4  10
-4
and
3x  4  10
-4
-4
3x  14
3x  6
3
3
3
-4
3
2
x  4
3
x2
-5
-4
-3
-2
-1
0
1
2
4x  1  5
example 3)
4x  1  5
-1
4x 1  5
or
-1
-1
4x  6
4x  4
4
4
4
4
1
x  1
2
x 1
-3
-1
-2
-1
0
1
2
Solve each open sentence.
example 4)
3x  4  7
example 5)
3x  4  7
example 6)
3x  6  0
example 7)
3x  3  0
No solution
All numbers work.
No solution
1
Graph each on a number line.
x 2
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
x 2
x 2
Let’s do some examples…..
3  2x  5
example 8)
3  2x  5
-3
3  2x  5
and
 2x  8
 2x  2
-2
-3
-3
-3
-2
-2
x4
x  1
-1
-2
0
1
2
3
4
2x 1  7
example 9)
2 x 1  7
or
2x 1  7
+1 +1
+1
2x  8
2x  6
2
2
2
+1
2
x  3
x4
-3
4
Reflection on the Section
How are absolute value inequalities containing a
or
symbol solved differently from those containing a



or
assignment

symbol?