Transcript Document 7199631
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Welcome!
This is Math 108 Intermediate Algebra Kathy Stover
Chapter 1
Set
– a collection of
objects called elements
Special sets of numbers:
Real Numbers Irrational Numbers Rational numbers Integers Whole numbers Natural numbers
Additive inverse (opposite) Additives inverses are on opposite sides of 0 on the number line…so just change the sign to get the additive inverse.
Examples: 3 and –3 x and –x
Absolute Value Absolute value measures the distance from 0 on the number line. It is always + Examples:
5
= 5
6
= 6 Inequality symbols
Graph on a number line 3 ways to write sets: Roster method Set-builder method
Interval notation is not a number
Operations on sets: Union Intersection See p. 14, # 73-106
Know all the rules for + and – signs
Rules to note for division:
Fractions: add, subtract, multiply, divide
Decimals: add, subtract, multiply, divide
Complex fractions
Do operations above and below the main fraction bar first
Exponential notation
Order of operations: 1.
Within P arens or other grouping symbols 2.
E xponents 3.
M ultiply and D ivide from left to right 4.
A dd and S ubtract
PEMDAS
Properties of real numbers
The most important to remember by name are:
Commutative
Associative
Distributive
Evaluate variable expressions
Remember the order of operations!
Geometric formulas
Look inside the front cover of your book.
Geometric formulas You’ll need to add this: Volume of pyramid is V = 1/3 (base area)(height) Use appropriate labels on all answers!
Translating verbal expressions into symbols See p. 38 - 39
Solving equations in one variable The answers to an equation are called roots or solutions. They are values which will make the given equation true.
Combine like terms on each side of the = , then do “opposite operations” to put the equation into the form: variable = constant If the terms contain fractions, multiply each side of the = by the common denominator to clear the equation of fractions, then solve.
• Stamp and Coin problems Make a table ; let x = the number of one type of coin or stamp • Write the number of coins or stamps as well as the value of those coins or stamps.
Stamp and Coin problems • Write an equation using the relationships given in the problem. • Solve and check.
Integer Problems The sum of two integers is given…… for example If sum is 25, let one # = x the other = 25-x
Integer Problems Consecutive integers: n, n+1, n+2, etc.
Consecutive even integers: n, n+2, n+4 Consecutive odd integers are also: n, n+2, n+4
1. Let a variable represent one of the integers, then express the others in terms of that same variable. 2. Write an equation using the relationships given in the problem.
3. Solve.
Value Mixture Problems In these problems, you need to combine ingredients to make one blend
Make a table !
Write an equation: Sum of values of ingred. = value of mix Amount of the ingredient Unit cost Value of the ingredient
Uniform Motion Problems Use the formula: Rate • Time = Distance Again, a table will help you organize the information.
Problems involving % For investment problems, 1. Use the formula: Principal • Rate = Interest 2. Make a table Note: change % rate to a decimal!
3. Write an equation using info from the problem.
4. Solve
For % Mixture Problems 1. Make a table. 2. Write an equation.
Amount of solution (A) % of its concentration as a decimal ( R ) Quantity of that substance A R
Solving Inequalities The solution is a whole set of numbers:
If you add any #, subtract any #, multiply by a +, or divide by a +, the inequality symbol stays the same…..
But if you divide (or multiply) both sides by a negative, you must reverse the inequality symbol!!!
In compound inequalities remember that ‘and’ means intersection ‘or’ means union
If the compound inequality has ‘x’ in the middle section, do opposite operations to isolate it in the middle.
Important concepts and formulas for section 3.1
1.
Rectangular coordinate system 2.
Ordered pair solutions to equations
3. Pythagorean Theorem 4. Distance formula 5. Midpoint formula 6. Scatter diagrams
Main terms and processes of section 3.2
Relation
Function
Function notation and evaluating functions
Domain and range
Excluding values from the domain
Linear equations and functions
Linear equations have x and/or y with + or –, but never x n , never ‘xy’, and never division by x or y
Every nonvertical line is a function and the equation can be put into the form Y = mx+b or f(x) = mx+b
Ways to graph lines: 1. Table of random x
y values 2. x-intercept and y-intercept 3. Recognize special lines
4. Using slope and y intercept…..
next section!
Slope
Slope is a measure of the steepness of a line. There are several ways to remember the formula:
To graph the slope, Top # go up if + , down if – Bottom # go to right
Draw graphs of lines using slope
Finding the equation of a line Two ways: 1.
Using y = mx+b 2.
Using y – y 1 = m(x – x 1 ) Put your answer as y =
Parallel lines same slope (if not vertical) Perpendicular lines Slopes are negative reciprocals (if not vertical) (product of the slopes is -1)
Chapter 4
System of equations: 2 or more equations considered together
The solution will usually be an ordered pair (or triple) which satisfies all the equations.
Ways to solve systems of equations 1. Graphing a. Graph line #1 b. Graph line #2 c. Name the point where the lines cross
Special cases 1.
If lines are parallel ….
No solution or Ø The system is called inconsistent
2. If both equations make the same line …. Write a generic ordered pair (x, expression) to represent all the points The system is called dependent
Another way to solve a system 2. Substitution method a. Rewrite one equation as x = or as y = b. Put this expression into the other equation
c. Solve to get the value of one variable d. Put this value into the x = or y = equation to find the the second from step 1 value of variable
3. Addition Method a. Multiply one or more equations by constants to make the coefficients of one variable equal …but opposite in sign …like 6x and –6x, or 12y and – 12y b. Add the equations
c. Solve to get the value of one variable d. Put this value into either equation to find the value of the other variable.
Some systems have 3 equations and 3 variables See handout!!!
Matrix
Rectangular array of numbers (any size)
The numbers are called elements of the matrix
Size: rows x columns
Every square matrix has a numerical value called its determinant .
Evaluate 2x2 and 3x3 determinants
Cramer’s Rule uses determinants to solve systems of equations. (This is our 4 th method for solving systems of equations!)
Cramer’s Rule is very useful when you don’t have to solve for all the unknowns….. circuits, chemistry, physics, etc.
5 th way to solve a system of equations:
Augmented matrices and “row operations” See handout!
Chapter 5
Monomial – a number, a variable, or the product of numbers and variables
Degree of a monomial – add the exponents on the variables
A constant is degree 0
Rules for Exponents
See summary on p. 257
Scientific notation p. 261-262
Scientific Notation
Used to write really large and very small numbers in compact form
1. 2.4 x 10 –4 2. 1.7 x 10 5
Polynomials
Monomial
Binomial
Trinomial
Degree of a polynomial : the greatest of the degrees of any of the terms
Leading coefficient: coef. of the term with highest degree (not necessarily the first term)
Constant
Evaluate polynomial functions
Graph polynomial functions
Add and subtract polynomials
Multiplying Polynomials
Monomial • trinomial
Binomial • binomial
Binomial • trinomial
Multiplying Polynomials
Special products: (a + b)(a – b) (a +b) 2 (a – b) 2
Application problems
Multiplying Polynomials
Monomial • trinomial
Binomial • binomial
Binomial • trinomial
Multiplying Polynomials Special products: (a + b)(a – b) (a +b) 2 (a – b) 2 Application problems
Division of polynomials
Long division
Synthetic Division
Remainder Theorem: If you divide P(x) by (x – a)… so that “a” is outside the division box, the remainder will always = P(a)
constant is + signs are both + if middle term is + ; signs are both – term is – if middle constant is – one binomial has +, the other has –
An important fact is that if the terms of the trinomial do not have a common factor, then you cannot have a common factor within either binomial.
Special Factoring
a 2 + b 2 is not factorable a 2 – b 2 = (a – b) (a + b) a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) a 3 + b 3 = (a + b)(a 2 – ab + b 2 )
Special Factoring
Some trinomials are factorable, even though not degree 2.
Checklist for the factoring process (p. 304): 1. Is there a common factor ? 2. Only 2 terms ? Is it a 2 - b 2 a 3 - b 3 or a 3 + b 3 ?
or Use the memorized patterns!
3. Trinomial ? Make 2 binomials! Check by FOIL 4. 4 terms ? Make 2 groups of 2 terms then factor each group watching for a common factor to pull out in front.
5. Are all factors prime or can they be factored more ?
Using Factoring to Solve Equations Method is based on the “principle of zero products”. If ab = 0, then a = 0 or b = 0
Important: Equation must be set = 0 Then factor Then set each factor = 0 and solve
Application problems: 1.The sum of the squares of two consecutive odd integers is 130. Find the two integers.
Remember For consecutive even or consecutive odd integers use x, x + 2, x + 4 For consecutive integers, use x, x + 1, x + 2
2. The length of a rectangle is 5 inches longer than the width. The area of the rectangle is 66 in 2 . Find the width and the length of the rectangle.
3. If f(x) = x 2 – x – 2, find two values of c in the domain of f(x) for which f(c) = 4 (see p. 312)
Rational Expression: a fraction with polynomials in the top and bottom
Some problems are review… ☻
☺
☻
☺
Function notation
Evaluate rational functions
Find the domain
Operations on Rational Expressions (see handout) 1. Simplify – Factor first!
– Divide out (cancel) common factors
2. Multiply – Factor everything!
– Divide out common factors from top and bottom – Multiply straight across leaving in simplified factored form
3. Divide – Factor everything!
– Invert second fraction and multiply by recip – Divide out common factors – Multiply straight across... again leave parens
4. Add and Subtract
– Factor denom and find common denom!
– Change all fractions to common denom
– Add / subtract tops and put that answer over the common denom – Simplify (factor then cancel)
Important!!!
Never cancel across +, –, ÷ , =
Rational Expression: a fraction with polynomials in the top and bottom
Some problems are review… ☻
☺
☻
☺
Function notation
Evaluate rational functions
Find the domain
Operations on Rational Expressions (see handout) 1. Simplify – Factor first!
– Divide out (cancel) common factors
2. Multiply – Factor everything!
– Divide out common factors from top and bottom – Multiply straight across leaving in simplified factored form
3. Divide – Factor everything!
– Invert second fraction and multiply by recip – Divide out common factors – Multiply straight across... again leave parens
4. Add and Subtract
– Factor denom and find common denom!
– Change all fractions to common denom
– Add / subtract tops and put that answer over the common denom – Simplify (factor then cancel)
Important!!!
Never cancel across +, –, ÷ , =
Complex Fractions have at least one fraction within a fraction.
To simplify: 1. Find the smallest common denominator (LCD) of all the denominators in the top and bottom of the fraction
2. Multiply the entire numerator and the entire denominator by the LCD.
This should clear the fractions from the top and the bottom of the “main” fraction.
3. Factor the new top and bottom, then cancel common factors to simplify.
Rational equations (equations containing fractions): A. Multiply both sides of the = by the common denom. of all the fractions . This should clear all the denominators!
If you have just one fraction on each side of the = you can just cross multiply.
B. Solve this “fraction free” equation C. Check your answer into the original equation and reject any answer that makes a denom = 0
Work Problems See handout!
Uniform Motion Problems See handout!
Proportion – equation with 2 rates or 2 fractions set =
Set up the pattern in words, then create the equation!
To solve, do the 2 cross products and set them =
Be sure that both fractions have the same set up: lb = lb tax = tax cost cost total total etc.
Variations K = constant of proportionality
Direct variation: y = kx So y = constant = k x
In a direct variation, both values increase or both decrease.
weight & postage cost of item & sales tax income & income tax
In an inverse variation, y = k x So xy = constant = k
In every inverse variation, one quantity increases but the other decreases.
volume of gas & pressure light intensity & distance from bulb rate of speed & time
In every variation, determine the pattern first and write it down, then plug in the given numbers.
In a joint variation, one quantity varies directly as a product....
But there are none in exercises!!
Literal equations have more than one variable.
You will have to rewrite the given equation so that a specified variable is isolated.
To do this:
1.
Clear fractions
and
get rid of parens.
2. Do opposite operations to isolate the specified variable on one side of the =
Note: If 2 terms have the needed variable, put them on the same side of the =, then factor before dividing.
Chapter 7
Our same rules for exponents from Chapter 5 apply even if the exponents are fractions.
Remember a negative exponent does not cause a negative # …and do not leave any negative exponent!
Using fraction exponents
Apply the rules
Evaluate expressions
Change to radicals
Change radicals back to fraction exponents
Facts about radicals
Principal
√
n
0
n a n
n p a b
Can be simplified! Start with innermost or use fraction exponents.
Operations with radicals
Simplify
Multiply (need same index)
Add / Subtract (need same index and same radicand
Divide (need same index) Do not leave a fraction under any radical Do not leave any radical in the denominator! Monomial in denom.
Binomial in denom.
Complex Numbers
1
i
i
2
= – 1
Complex Numbers
a + b
i
real imaginary part part
Complex Numbers
Simplify
Always rewrite the with “i” first thing!
Add / subtract
neg
Complex Numbers
Multiply
Divide Conjugate of a + bi is a – bi
31. One printer can print the paychecks for the employees of a company in 54 min. A second printer can print the checks in 81 min. How long would it take to print the checks with both printers operating? 32.4 min
32. A mason can construct a retaining wall in 18 h. The mason's apprentice can do the job in 27 h. How long would it take to construct the wall if they worked together? 10.8 h
33. One solar heating panel can raise the temperature of water l' in 30 min. A second solar heating panel can raise the temperature l' in 45 min. How long would it take to raise the temperature of the water l' with both solar panels operating? 18 min
34. One member of a gardening team can landscape a new lawn in 36 h. The other member of the team can do the job in 45 h. How long would it take to landscape a lawn if both gardeners worked together? 20 h
35. One member of a telephone crew can wire new telephone lines in 5 h. It takes 7.5 h for the other member of the crew to do the job. How long would it take to wire new telephone lines if both members of the crew worked together? 3 h
36. A new printer can print checks three times faster than an old printer. The old printer can print the checks in 30 min. How long would, it take to print the checks with both printers operating? 7.5 min
37. A new machine can package transistors four times faster than an older machine. Working together, the machines can package the transistors in 8 h. How long would it take the new machine, working alone, to package the transistors? 10 h
38. An experienced electrician can wire a room twice as fast as an apprentice electrician. Working together, the electricians can wire a room in 5 h. How long would it take the apprentice, working alone, to wire a room? 15 h
39. The larger of two printers being used to print the payroll for a major corporation requires 40 min to print the payroll. After both printers have been operating for 10 min, the larger printer malfunctions. The smaller printer requires 50 more minutes to complete the payroll. How long would it take the smaller printer, working alone, to print the payroll? 80 min
40. An experienced bricklayer can work twice as fast as an apprentice brick layer. After they worked together on a job for 8 h, the experienced bricklayer quit. The apprentice required 12 more hours to finish the job. How long would it take the experienced bricklayer, working alone, to do the job? 18 h
41. A roofer requires 12 h to shingle a roof. After the roofer and an apprentice work on a roof for 3 h, the roofer moves on to another job. The apprentice requires 12 more hours to finish the job. How long would it take the apprentice, working alone, to do the job? 20 h
42 . A welder requires 25 h to do a job. After the welder and an apprentice work on a job for 10 h, the welder quits. The apprentice finishes the job in 17 h. How long would it take the apprentice, working alone, to do the job? 45 h
43 . Three computers can print out a task in 20 min, 30 min, and 60 min, respectively. How long would it take to complete the task with all three computers working? 10 min
44. Three machines fill soda bottles. The machines can fill the daily quota of soda bottles in 12 h, 15 h, and 20 h, respectively. How long would it take to fill the daily quota of soda bottles with all three machines working?
5 h
45. With both hot and cold water running, a bathtub can be filled in 10 min. The drain will empty the tub in 15 min. A child turns both faucets on and leaves the drain open. How long will it be before the bathtub starts to overflow? 30 min
46. The inlet pipe can fill a water tank in 30 min. The outlet pipe can empty the tank in 20 min. How long would it take to empty a full tank with both pipes open? 60 min
47. An oil tank has two inlet pipes and one outlet pipe. One inlet pipe can fill the tank in 12 h, and the other inlet pipe can fill the tank in 20 h. The outlet pipe can empty the tank in 10 h. How long would it take to fill the tank with all three pipes open? 30 h
48. Water from a tank is being used for irrigation at the same time as the tank is being filled. The two inlet pipes can fill the tank in 6 h and 12 h, respectively. The outlet pipe can empty the tank in 24 h. How long would it take to fill the tank with all three pipes open? 4.8 h
49. An express bus travels 320 mi in the same amount of time it takes a car to travel 280 mi. The rate of the car is 8 mph less than the rate of the bus. Find the rate of the bus. 64 mph
50. A commercial jet travels 1620 mi in the same amount of time it takes a cor porate jet to travel 1260 mi. The rate of the commercial jet is 120 mph greater than the rate of the corporate jet. Find the rate of each jet.
Commercial: 540 mph; corporate: 420 mph
51. A passenger train travels 295 mi in the same amount of time it takes a freight train to travel 225 mi. The rate of the passenger train is 14 mph greater than the rate of the freight train. Find the rate of each train.
Passenger: 59 mph; freight: 45 mph
52. The rate of a bicyclist is 7 mph more than the rate of a long-distance runner. The bicyclist travels 30 mi in the same amount of time it takes the runner to travel 16 mi. Find the rate of the runner. 8 mph
53. A cyclist rode 40 mi before having a flat tire and then walking 5 mi to a service station. The cycling rate was four times the walking rate. The time spent cycling and walking was 5 h. Find the rate at which the cyclist was riding. 12 mph
54. A sales executive traveled 32 mi by car and then an additional 576 mi by plane. The rate of the plane was nine times the rate of the car. The total time of the trip was 3 h. Find the rate of the plane. 288 mph
55. A motorist drove 72 mi before running out of gas and then walking 4 mi to a gas station. The driving rate of the motorist was twelve times the walking rate. The time spent driving and walking was 2.5 h. Find the rate at which the motorist walks. 4 mph
56. An insurance representative traveled 735 mi by commercial jet and then an additional 105 mi by helicopter. The rate of the jet was four times the rate of the helicopter. The entire trip took 2.2 h. Find the rate of the jet. 525 mph
57. An express train and a car leave a town at 3 P.m. and head for a town 280 mi away. The rate of the express train is twice the rate of the car. The train arrives 4 h ahead of the car. Find the rate of the train. 70 mph
58. A cyclist and a jogger start from a town at the same time and head for a destination 18 mi away. The rate of the cyclist is twice the rate of the jogger. The cyclist arrives 1.5 h before the jogger. Find the rate of the cyclist. 12 mph
59. A single-engine plane and a commercial jet leave an airport at 10 A.M. and head for an airport 960 mi away. The rate of the jet is four times the rate of the single-engine plane. The single-engine plane arrives 4 h after the jet. Find the rate of each plane. Jet: 720 mph; single-engine: 180 mph
60. A single-engine plane and a car start from a town at 6 A.m. and head for a town 450 mi away. The rate of the plane is three times the rate of the car.
The plane arrives 6 h before the car. Find the rate of the plane. 150 mph
61. A motorboat can travel at 18 mph in still water. Traveling with the current of a river, the boat can travel 44 mi in the same amount of time it takes to go 28 mi against the current. Find the rate of the current. 4 mph
62. A plane can fly at a rate of 180 mph in calm air. Traveling with the wind, the plane flew 615 mi in the same amount of time it took to fly 465 mi against the wind. Find the rate of the wind. 25 mph
63. A tour boat used for river excursions can travel 7 mph in calm water. The amount of time it takes to travel 20 mi with the current is the same amount of time it takes to travel 8 mi against the current. Find the rate of the current. 3 mph
64. A canoe can travel 8 mph in still water. Traveling with the current of a river, the canoe can travel 15 mi in the same amount of time it takes to travel 9 mi against the current. Find the rate of the current. 2 mph
Solving Equations Containing Radicals
1. Isolate the radical.
If there are 2 radicals, isolate one of them.
2. Raise both sides of the = to the same power as the index on the radical. Use FOIL as needed!
3. If you still have a radical, repeat steps 1 and 2.
(Isolate the radical and square again!) 4. Solve 5. Check results into the original equation.
Application Problems Use Pythagorean Theorem (see pages 412-413) Use formula given in the problem (see p. 414)
Quadratic Equations Standard form: ax 2 + bx + c = 0 Quadratic equations are always degree 2
Quadratic Equations Solve by factoring: 1. Set equation = 0 2. Factor 3. Set each factor = 0 and solve
Quadratic Equations Given solutions, write the quadratic equation with integral coefficients
Solve by taking square roots: 1.
2.
3.
Write ( ) 2 = # Take of both sides (Remember ) Isolate the variable
Ways to Solve Quadratic Equations
1.Factoring
2.Taking
3. Completing the square
Solve by completing the square 1. Divide all terms by the coef of the x 2 so that you have just x 2 2. Move constant to the right of =
3. Add (1/2 coef.of x) 2 to both side of = to “complete the square” 4. Rewrite as ( ) 2 = # 5. Take ….Remember 6. Isolate the variable
Ways to Solve Quadratic Equations
1.Factoring
2.Taking
3. Completing the square 4. Using the Quadratic Formula
Use the Quadratic Formula 1. Write the equation = 0 ax 2 + bx + c = 0 2. Use the formula X = –
b
b
2 4
ac
2a 3. Simplify
Use the discriminant to decide the types of solutions Discriminant = b 2 – 4ac
If b 2 – 4ac < 0 , there are 2 complex (
i
) solutions If b 2 – 4ac 0 , there are 2 real (no
i
) solutions If b 2 – 4ac = 0 , there is 1 real solution (called a double root)
Equations that are in Quadratic Form The exponent on one variable will be ½ of the exponent on the other variable term.
Set = 0 Factor Set factors = 0 and solve
Radical Equations
Isolate the radical
Square both sides of the = (Use FOIL when needed)
If there is still a repeat steps 1 and 2
Set = 0 and solve using factoring, the quad. formula, or completing the square
Reject any solution that causes
even neg
Equations with fractions
Multiply by the common denominator to clear fractions
Set = 0 then solve by factoring, completing the square, or quad. Formula
Reject any solution that makes denom = 0
Nonlinear Inequalities Two types: See handout!
1. Polynomial inequalities 2. Rational (fraction) inequalities
Quadratic function f(x) = ax 2 + bx + c
y = ax 2 + bx + c or
The graph is a parabola opens up if a
0 opens down if a
0 axis of symmetry is x = -b/2a
Vertex Y – intercept (let x = 0) X – intercepts: let y = 0 and solve the equation by factoring, quadratic formula or completing the square
The parabola will have 0, 1, or 2 x – intercepts.
You can tell how many x – intercepts from the graph, from the solving process, or from the discriminant (b 2 – 4ac)
Domain: all real numbers
Range: y values used (look at graph!)
Graphing functions
Knowing the general shape will help you graph it!
1. straight line: y = mx + b 2. parabola: y = ax 2
3. Cubic: f(x) = ax 3 4. Absolute value 5. Radical
Domain and range can be estimated from the graph and named in set notation or interval notation
Some graphs are functions… and some are not.
Use the vertical line test to determine whether a graph is a function.
Operations on Functions
Add Subtract Multiply Divide
Operations on Functions
Composition of functions Inverse of a function 1. Interchange x and y 2. Solve for the “new” y if it is an equation 3. Notation for inverse is f -1
One-to-one functions
Every 1 – 1 function passes the horizontal line test as well as the vertical line test.
If a function is 1 – 1, it has an inverse that is also a function.
Test today... or tomorrow!!!!
Are you ready???
Test today... or tomorrow!!!!
Don’t forget!!!!
Today is the last day for the Chapter Test….
Don’t forget!!!!
The Chapter 4 Test opens today…..
Are you ready????
The Chapter 6 Test opens tomorrow…..
Are you ready????
The Chapter 6 Test opens tomorrow…..
Are you ready????
The Chapter 7 Test opens tomorrow…..
Are you ready????
Chapter 10….our last chapter!
Exponential functions:
Format
Evaluate
Graph
Special function, f(x) = e x
Logarithmic Form
For b > 0, y = log b equivalent to b y x is = x
Rewrite log form and exponential form
Logarithmic Form
Evaluate logs
Solve for x in various positions
Special logs
log x means log 10 x
ln x means log e x
Notice that
If b
p
= b
n
, then p = n
This can be used to solve equations.
Properties of logs 1. If log b x = log b y, then x = y 2. log b xy = log b x + log b y
Properties of logs 3. log b x/y = log b x – log b y 4. log b x p = p log b x
Properties of logs 5. log b 1 = 0 6. log b b = 1 7. log b b x = x
Evaluate logs using a calculator
Use the “change of base” formula:
log a n = log n
or
ln n log a ln a
In any y = log b must have x , you x > 0 and b > 0
log
7
–49 is undefined! log
10
0 is undefined! log
-2
8 is undefined!
2 ways to graph f(x) = log b
1. Rewrite as b y
x
= x and make an x/y roster of points.
2. Use calculator and the change of base formula to get points.
Solving exponential equations
Two types: A. Sometimes you can rewrite as b x = b y and then x = y
Solving exponential equations
B. Sometimes you need to take the log or ln of both sides and move the exponents down.
Then solve.
Solving log equations
Two types: A. Logs on just one side of = 1.
2.
Use properties to rewrite with one log on left Rewrite in exponential form 3.
Solve
Solving log equations
B. Logs on both sides of the = 1.
Use properties to make one log on each side of the = 2.
3.
Drop the logs Solve
Know what your instructor requires!
Read your syllabus; keep it for future reference.
Don't fall behind! Math skills must be learned immediately and reviewed often. Keep up-to date with all assignments.
Most instructors advise students to spend two hours outside of class studying for every hour spent in the classroom. Do not cheat yourself of the practice you need to develop the skills taught in this course!
Take the time to find places that promote good study habits.
Find a place where you are comfortable
and can concentrate.
(library, quiet lounge area, study lab)
Survey each chapter ahead of time.
Read the chapter title, section headings and the objectives listed to get an idea of the goals and direction for the chapter.
Take careful notes and write down examples .
The book provides material to read and examples for each objective studied. It also has answers to the odd-numbered exercises in the back of the book so that you can check your answers on assignments.
Be sure to read the Chapter summary and use the Chapter Review and Chapter Test exercises to prepare for each Chapter exam. (All answers are in the back for these)
Spaced practice is generally superior to massed practice. You will learn more in 4 half-hour study periods than in one 2 hour session.
Review material often because repetition is essential for learning. You remember best what you review most.
Much of what we learn is soon forgotten unless we review it.
Attending class is vital if you are to succeed in any math course.
Be sure to arrive on time…. and stay the entire class period!
You are responsible for everything that happens in class, even if you are absent.
If you must be absent :
1. Deliver due assignments to instructor as soon as possible (even ahead of time if you know in advance).
2. Copy notes taken by a classmate while you were absent.
3. Ask about announcements, assignments or test changes made in your absence.
If you have trouble in this course – seek help!
1. Instructor 2. Tutors 3. Video Tapes 4. Computer Tutoring
Study Tips: Preparing for Tests Try the Chapter Test at the end of each chapter before the actual exam. Do these exercises in a quiet place and pretend you are in class taking the exam.
Study Tips: Preparing for Tests If you missed questions on the practice test, review the material, practice more problems of the same type, get help as needed.
Try these strategies of successful test takers: 1. Skim over the entire test before you start to solve any problems.
2. Jot down any rules , formulas or reminders you might need.
3. Read directions carefully.
4. Do the problems that are easiest for you first.
5. Check your work to be sure you haven't made any careless errors.