Download Complex Numbers (a + bi)

2.1: Linear Equations
Algebra
• Representing real-world situations with mathematical expressions & statements
• Solving real-world and/or mathematical problems
involving unknown quantities
SEARCHING FOR THE UNKNOWN: A VARIABLE
A variable is a symbol, usually a letter (like X, Y, T, P) that is used to represent an
unknown number.
If Sam ate many tacos and we don’t know how many, we might say
Sam ate X tacos. Tomorrow if Sam eats 3 more than he did today,
we could say Sam eats X+3 tacos tomorrow.
X and X+3 are algebraic expressions representing the number of tacos eaten.
If Sam ate 10 tacos today, then he will eat X + 3 = (10) + 3 = 13 tomorrow.
If Sam ate 4 tacos today, then he will eat (4) + 3 = 7 tacos tomorrow.
An equation is the equality of two algebraic expressions.
3+6=9
x + 3 = 13
x + 4 = 20 – 3x
Linear Equations in 1 Variable
A linear equation in one variable is an equation that can be written as:
Ax + B = C where A, B, C  R, A  0
To solve a linear equation, find all variable values that make the equation true.
These values are called the solution set. Steps to solving symbolically:
Step 1: Locate both sides of the equation (separated by the ‘=‘ sign)
Step 2: Clear any fractions or decimals
Step 2: Simplify each side separately : Use distribution & combine like terms
Step 3: Move the ‘variable terms’ to one side and ‘number terms’ to the other
Step4: Reverse/Inverse what is happening to X until you have X = __
Step5: Check your answer by plugging X back in
Examples:
2x –1 = 0
+1 +1
2x
=1
2
2
x
= ½
-5x = 10 + x
-x
-x
-6x = 10
3x + 8 = 2
-8 -8
3x = -6
-6
-6
3
3
x
= -5/3
x
= -2
Distribution & Clearing Fractions
2 (x – 1) = 4 – ½ (4 + x)
6
1 (2x - 3) –1 x = - 2
3
2
2x –2
= 4-2 -½x
[Multiply by Common Denominator]
2x – 2
= 2 -½x
2 (2x –3) –3 x = -12
+2
+2
2x
=
+½x
2½x
=
(2/5) (5/2) x =
x
=
4–½x
+½x
4x - 6 - 3x
x-6
= -12
+6
= +6
x
= -6
4
4 (2/5)
8/5
= -12
Clearing Decimals
100
.06x + 0.09(15 – x) = 0.07(15)
6x + 9(15 – x) = 7(15)
6x + 135 –9x = 105
-3x + 135
= 105
-135
-135
-3x
-3
=
-30
-3
x
=
10
3 Types of Linear Equations
5x - 9 = 4(x – 3)
5x – 9 = 4x – 12
-4x
-4x
---------------------X – 9 = -12
+9
+9
----------------------X = -3
5x – 15 = 5(x – 3)
5x – 15 = 5x – 15
+ 15
+ 15
---------------------------5x
= 5x
-5x
-5x
---------------------------0
=
0
5x – 15 = 5(x – 4)
5x – 15 = 5x – 20
+ 15
+ 15
---------------------------5x
= 5x -5
-5x
-5x
---------------------------0
=
-5
1 Solution
Infinite Solutions
(All Real Numbers)
NO Solutions
(Null Set : O )
CONDITIONAL
IDENTITY
CONTRADICTION
2.2-2.3 Formulas & Equations
A formula is an equation that can calculate one quantity by using a known
value of another quantity. Formulas usually involve real-world applications.
D = RT
A = LW
I = PRT
D – distance
R – rate
T – time
A – Arearectangle
L – Length
W – Width
I - Interest
P – Principal ($$ borrowed/invested)
T – Time (years)
If Anna travels 50mph for 15 hours, how far did she travel?
D = RT
D = (50)(15) = 750 miles
Formulas can be solved for a specific variable
P = 2L + 2W (solve for W) Solve for W:
-2L -2L
P = 2(L + W)
P – 2L
= 2W
2
2
Solve for B
N=A+B
2
Percentages
Change a Percent to a Decimal
 Move the decimal point two places to the left
45% = .45
5% = .05
120% = 1.2
3.2% = .032
500% = 5
Change a Decimal Number to a Percent
 Move the decimal point two places to the right
.45 = 45%
.05 = 5%
1.2= 120%
.032= 3.2%
5 = 500%
A class has 50 students. 32 are males. What is the percent of males in the class?
Partial amount = percent
32 = .64 = 64%
Whole amount
50
What is 25% of 70?
X
= .25 • 70
X
= 17.5
16 is what percent of 50
16 = X • 50
X = 16/50 = .32 = 32%
A man weighed 150 lbs last year. This year
the same man weighs 175 lbs. What was the
percent increase from last year to this year.?
Difference
Original
= 25
150
= .167 = 16.7% increase
Word Problems/Applications
Tips on word problems:
1. Read the problem through once entirely, then go back and
read it again noting the important information. You may
have to read it more times too as you work the problem & you may
wish to organize your thoughts with pictures or charts.
2. Assign variables for unknown quantities & anything you need to find.
3. Write equation(s) related to the problem using your variables.
(Translate words/sentences in the problem into an algebraic equation)
4. Solve the equation & check your solution to see if it is reasonable.
Examples
Find the number:
Twice a number, decreased by 3 is 42 : 2x –3 = 42
The quotient of a number and 4 plus the number is 10: x + x = 10
4
Classic Problems
#1 Geometric Dimensions:The length of a rectangle is 1cm more than twice
the width. The perimeter of the rectangle is 110 cm. Find the length and the
width of the rectangle.
#2 Percent Interest: Mark had $40,000 to invest. He puts part of the money in
the bank earning 4% interest and the rest in stocks paying 6% interest for an
annual income of $2040. Find the amounts in the bank and in stock.
#3 Acid Mixture: a chemist mixes 8 L of 40% acid solution with some
Unknown quantity of 70% solution to get a 50% solution. How much 70%
Solution is used?
#4 Coins: A bill is $5.65. The cashier received 25 coins (all nickles & quarters).
Howmany of each coin did the cashier receive?
Investment Formula/Problem
(P. 70 – Example 4)
Karen Estes just received an inheritance of $10000
And plans to place all money in a savings account
That pays 5% compounded quarterly to help her son
Go to college in 3 years. How much money will be
In the account in 3 years?
Use the formula:
A = P(1 + r/n)nt
A = amount in account after t years
P = principal or amount invested
T = time in years
R = annual rate of interest
N = number of times compounded per year
2.4-2.5 Inequality Set & Interval Notation
Set Builder Notation
{1,5,6}
{ }

{x | x > -4}
x such that
x is greater than –4
{x | x < 2}
x such that x is less
than or equal to 2
Interval (-4, )
Notation
Graph
-4
{6}
{x | -2 < x < 7}
x such that x is greater
than –2 and less than
or equal to 7
(-, 2]
(-2, 7]
2
-2
0
Question: How would you write the set of all real numbers?
7
(-, ) or
R
Inequality Example
Statement
7x + 15 > 13x + 51
Reason
[Given Equation]
-6x + 15 > 51
[-13x from both sides]
-6x > 36
[-15 from both sides]
x
[Divide by –6, so must ‘flip’ the inequality sign
< -6
Set Notation: {x | x < -6}
Interval Notation: (-, -6]
Graph:
-6
Three-Part Inequality
-3 < 2x + 1 < 3
-1
-1 -1
-4 < 2x < 2
2
2
2
Set Notation: {x | -2 < x < 1}
Interval Notation: (-2, 1]
Graph:
-2 <
x
< 1
-2
0
1
An Inequality Word Problem: (P. 107) : Average Test Score
Martha has scores of 88, 86, and 90 on her 1st 3 tests. An average score of 90
Will earn her an A in the course. What does she need on her 4th test to have
An A average?
88 + 86 + 90 + x  90
4
Set Operations and Compound
Inequalities
Intersection () – “AND”
A  B = {x | x  A and x  B}
Set Notation: {x | X  8 and X  5}
X+ 1  9 and X – 2  3
Interval Notation: (- , 8]  [5, )
X 8
and
X 5
0
[
]
5
8
Union () – “OR”
A  B = {x | x  A or x  B}
Set Notation: {x | X  -2 or X  -3}
-4x + 1  9 or
X  -2
or
Interval Notation: (- , -2]  (- , -3]
5X+ 3  12
X  -3
-2