Transcript Power Point

CHAPTER 1
• Equations & Inequalities
1.1
Graphs & Graphing Utilities
Objectives
• Plot points in the rectangular coordinate
system
• Graph equations in the rectangular
coordinate system
• Interpret information about a graphing
utility’s viewing rectangle or table
• Use a graph to determine intercepts
• Interpret information from graphs
Rectangular coordinate system
•
•
•
•
X-axis: horizontal (right pos., left neg.)
Y-axis: vertical (up pos., down neg.)
Ordered pairs: (x,y)
Graph of an equation: infinitely many
ordered pairs that make a true statement
• X-intercept: point where y=0, (x,0)
• Y-intercept: point where x=0, (0,y)
Graphing an Equation
• List ordered pairs that make your equation
true (plug values in for x and find the
resulting y’s). Include the x-intercept & yintercept among your points.
• Plot several points and look for a trend.
• Use the graphing utility on your calculator
and compare graphs. Do they always
match? Why or why not?
Intercepts
• Intercepts are key points to plot when
graphing an equation. Remember, an
intercepts is a POINT (x,y), and not just a
number!
• Look at the x-axis: What is true of EVERY
point on the axis? (the y-value’s always 0)
• Look at the y-axis: What is true of EVERY
point on the axis? (the x-value’s always 0)
Horizontal and Vertical Lines
• What if the y-value is ALWAYS the same, regardless
of the x-value (it could be anything!). (i.e. (4,2),
(5,2), (-14,2), (17,2)) It’s a horizontal line! The x isn’t
part of the equation, because its value is irrelevant:
y=k (k is a constant)
• What if the x-value is ALWAYS the same, regardless
of the y-value (it could be anything!). (i.e. (2,4),
(2,5), (2,-14), (2,17)) It’s a vertical line! The y isn’t
part of the equation, because its value is irrelevant:
x=c (c is a constant)
1.2
• Linear Equations and Rational Equations
Objectives
• Solve linear equations in one variable
• Solve linear equations containing fractions
• Solve rational equations with variables in
the denominators
• Recognize identities, conditional
equations, inconsistent equations
What is a linear equation in one
variable and what is “solving it”?
Only one variable (x or y, generally) is in the
equation and it is NOT squared or raised
to a power other than 1.
To “solve” the equation means to find the
value (or values) that would make the
equation true.
How do we solve an equation?
•
•
•
•
Eliminate parentheses (distribute!)
Collect like terms (additive identity)
Isolate the variable (multiplicative identity)
Remember: it’s an EQUATION to start
with, meaning the left equals the right. It
will no longer be equal, if something is
done to one side and not the other!
• CHECK your solution in the original
equation: does it make it true?
EXAMPLE
• 4(2x-3) = 2(x+3)
• 1)Distribute to eliminate parentheses
8x-12 = 2x + 6
2)Collect x’s on one side & constants on
the other (use additive identity)
8x(-2x) – 12(+ 12) = 2x(-2x) + 6(+ 12)
6x = 18
3)Isolate the x (use multiplicative identity)
4) Check your solution in the original
4(2(3)-3) = 2(3+3)
4(3) = 2(6) YES!!
6 x 18

6
6
x3
Rational Equations
• Equations that involve fractions!
• The variable (x) could be in the numerator of the
denominator. IF the x is found in the
denominator, we must consider values x
canNOT take on. (i.e. zero denominator)
• EVEN after you’ve simplified an equation to
eliminate the fractions, you haven’t eliminated
the original restriction that may have been
present.
• With fractions, EITHER eliminate the fraction OR
get a common denominator (if denominators are
EQUAL, so are numerators)
Solve by getting a common
denominator
3
1
1
  , x  2,...WHY ??
x2 8
2
3(8)
1( x  2)
1( x  2)( 4)


( x  2)(8) 8( x  2)
2( x  2)( 4)
24  x  2
4x  8

8( x  2)
8( x  2)
 26  x  4 x  8
26  x  x  8  4 x  8  x  8
18  3 x, x  6
CHECK !!
Types of Equations
• Conditional: True under certain conditions
(could be one or several solutions)
• Inconsistent: Inconsistencies between the
2 sides (never true – NO solutions)
• Identity: One side of the equation is
identical to the other (doesn’t matter what
x is, infinitely many solutions)
Example
• Solve 3x – 6 = 3(x – 2)
Notice, after distributing on the
right, 3(x – 2) = 3x – 6
The left side is identical to the
right. No matter what values you plug in
for x, it will always be true.
The solution set is: {all reals}
THIS IS AN IDENTITY.
Example
• Solve: 4x – 8 = 4(x – 5)
• Distribute on right = 4x – 20
• Think: Can 4 times a number minus 8
possibly equal 4 times the same number
minus 20??? NO!!
• If you continue to solve, you get:
• 0x = -12 (Can 0 times a number ever equal -12?
NO!
• INCONSISTENCIES!! Solution: { }
1.3
• Models & Applications
Objectives
• Use linear equations to solve problems
• Solve a formula for a variable
Solving Word Problems
• 1) Carefully read the problem
• 2) Determine what do you know and
what do you want to know
• 3) Identify variables
• 4) Develop equation relating what you
know & what you want to know
• 5) Solve the equation & check (correct?)
• 6) Make certain you answered the
question you were being asked!
EXAMPLE
• You need to drive from Chicago to your
cousin’s house in Omaha (a distance of
550 miles) at an average 65 mph on the
Interstate highway. What time should you
leave if you have to be at your cousin’s at
3:30 pm?
• What do you want to know?
– How long will it take you to drive? (x = time)
– What time must you leave?
• What do you know?
– Total distance you’ll travel (550 miles)
– Speed (65 miles per hour)
• What is the relationship between known &
unknown?
– Distance = Rate x Time
– 550 miles = 65 mph x (X) (cont. on next slide)
550mi
x
 8.5hr
mi
65
hr
Did you answer the question?
NO – WHEN should you leave?
In order to arrive at 3:30pm, you
leave 8.5 hrs earlier, which would
be at 8:00 am.
EXAMPLE
• You have been asked to make an
aluminum can (cylindrical shape) to hold
300 ml of your product. The can is to be
10 cm high. How much aluminum (in
square cm) do you need?
• What do you know?
– Can holds 300 ml (the volume!)
– Height = 10 cm
What do you want to know?
-How much material you will need (surface area).
What relates the known & unknown?
For cylinders:
V  r h
2
SA  2r  2rh
2
(example continued)
V  300ml  300cm  r (10cm)
3
3
300cm
2
r 
 9.55cm
 10cm
r  3.1cm
2
2
Now find surface area!
(answer the question!)
(remember, a cylinder is just
2 circles and a rectangle)
SA  2r  2rh
2
SA  2 (3.1cm)  2 (3.1cm)(10cm)
2
SA  255cm
2
Ava purchased a new ski jacket, on
sale for $66.50. The coat had been
advertised as 30% off! What was
the original cost?
1.
2.
3.
4.
$95
$86.50
$90
$96.50
1.4
• Complex Numbers
Objectives
•
•
•
•
Add & subtract complex numbers
Multiply complex numbers
Divide complex numbers
Perform operations with square roots of
negative numbers
i = the square root of negative 1
• In the real number system, we can’t take
the square root of negatives, therefore the
complex number system was created.
• Complex numbers are of the form, a+bi,
where a=real part & bi=imaginary part
• If b=0, a+bi = a, therefore a real number
(thus reals are a subset of complex #)
• If a=0, a+bi=bi, therefore an imaginary #
(imaginary # are a subset of complex #)
Adding & Subtracting Complex #
• Add real to real, add imaginary to
imaginary (same for subtraction)
• Example:
(6+7i) + (3-2i)
•
(6+3) + (7i-2i) = 9+5i
• When subtracting, DON’T FORGET to
distribute the negative sign!
• (3+2i) – (5 – i)
• (3 – 5) + (2i – (-i)) = -2 + 3i
Multiplying complex #
• Treat as a binomial x binomial, BUT what is i*i?
It’s -1!! Why??
• Let’s consider i raised to the following powers:
i  (  1)  1
2
2
i  i  i  (1)i  i
3
2
i  (i )  (1)  1
4
2 2
2
EXAMPLE
(2  3i )  (3  6i )
 6  12i  9i  18i
 6  3i  18  (1)
 6  3i  18  24  3i
2
Dividing Complex #
• It is not standard to have a complex # in a
denominator. To eliminate it, multiply be a wellchosen one: ( conjugate/conjugate)
• The conjugate of a+bi=a-bi
• We use the following fact:
(a  bi)  (a  bi)  a  abi  abi  b i
2
 a  b  (1)  a  b
2
2
2
2 2
2
EXAMPLE
3  8i (4  3i )
(3  8i )  (4  3i ) 

4  3i (4  3i )
12  9i  32i  24i

16  9
 12  41i  12 41i



25
25 25
2
1.5
• Quadratic Equations
Objectives
Solve quadratic equations by:
a) Factoring
b) Using the square root property
c) Completing the square
d) Using the quadratic formula
(WHEN TO USE WHICH METHOD?)
Use discriminant to determine # & type of
solutions
Solve application problems involving
quadratics.
What is a quadratic equation?
ax  bx  c  0
a, b, c  {Re als}
2
Zero-Product Rule
• If the product of two or more numbers is
zero, at least one of the numbers must
equal zero!
• If AB=0, then A=0 and/or B=0
– One or both of the terms must equal zero
Why is this important?
It allows us an easy way to solve an equation,
but FIRST make certain the expression is a
product that equals zero.
• A product involves FACTORS
• (2x-3)(x+2)=0
• 2x – 3 is a factor of the expression, as is
2+x
• Set each factor = 0
• 2x – 3 = 0, thus x = 3/2
• x + 2 = 0, thus x = -2
• SO, if EITHER x = 3/2 or x = -2, the
original expression = 0
• SO, solve by FACTORING if equation,
once equal to 0, is FACTORABLE
Often, you must get expression into
factored form FIRST:
Solve : 6 x  11x  4
2
6 x  11x  4  0
Factor : (3x  4)( 2 x  1)  0
3x  4  0
2
2x 1  0
4 1
x ,
3 2
Solving with square root property
• When would you use this approach?
– When one side of the equation is a perfect square
EXAMPLE:
(3  2 x) 2  5
(3  2 x ) 2    5 (Why  ??)
3  2 x  5i (WHY ??)
 2 x  3  5i
 3  5i 3 5
x
  i
2
2 2
Solve by Completing the Square
• When can you use this method? ALWAYS
– However, if the expression is factorable or is
already a perfect square, those methods may
be more desirable
HOW does it work?
If you don’t have a perfect square, you create
one by adding a “well-chosen” zero (adding
the same term to both sides)
Decide what to add by determining what
additional term would create a perfect square
EXAMPLE
Solve : 2 x 2  4 x  7  0
7
2( x 2  2 x  )  0
2
7
2
x  2x  
2
( Note : ( x  1) 2  x 2  2 x  1)
7
5
x2  2x 1   1  
2
2
5
2
( x  1)  
2
5
( x  1) 2   
2
5
5
x  1   i, x  1  i
2
2
Completing the square generalized
to any quadratic equation results in
the quadratic formula.
• When can you use it? ALWAYS. (However, it still
may be easier to factor & use zero-factor property
or take the square root if it’s already a perfect
square.)
2
ax  bx  c  0
 b  b  4ac
x
2a
2
Solve:
2x  4x  6
2
1.
2.
3.
4.
x = -1, 3
x = 1, -3
x = 2,3
x=2
What is the discriminant and why is
it useful to us?
• The discriminant is the part of the
quadratic equation that is under the
radical.
• Based on what is under that radical, we
can determine if our solution will be an
integer (is what’s under there a perfect
square?), an irrational (is what’s under
there a positive number that is NOT a
perfect square), or complex (is what’s
under there a negative number?)
1.6
• Other Types of Equations
Objectives
•
•
•
•
•
Solve polynomial equations by factoring.
Solve radical equations.
Solve equations with rational exponents.
Solve equations that are quadratic in form.
Solve equations involving absolute value.
Solving by factoring
• First: Set equation equal to zero
• Next: Check for a common term to factor
out of all terms
• Next: Proceed with factoring, as with
quadratics
• Remember: The degree of the equation
will indicate the maximum number of
solutions. (if you now have a 4th degree
polynomial, you may have 4 distinct
solutions)
What if your equation involves a
variable under a radical?
• In order to eliminate an nth root, you must
raise both sides of the equation to the nth
power.
• Be CERTAIN that you isolate the radical
(have it on one side of the equation by
itself) before you raise both sides to the
nth power.
What if the variable is found under
a radical twice in an equation?
• Isolate one radical and raise both sides to
the nth power.
• Then, isolate the other radical (it will not
have disappeared from the other side),
and raise both sides to the nth power
again.
What is x is raised to an exponent
that is NOT an integer?
• If the variable (or expression involving a
variable) is raised to the (m/n) exponent, you
must isolate that expression and then raise
BOTH sides to the (n/m) power.
• WHY?? When you raise one exponent to
another, you multiply the 2 exponents.
( a  b)
m
n
m
n
n
m
(( a  b) )  a  b
What if the equation involves an
expression inside absolute value
brackets?
• Recall what absolute value means: What is
within those brackets could be positive or
negative and still have the same overall value.
3 x  7  15
I )3 x  7  15
22
3 x  22, x 
3
II )  (3 x  7)  15
 3 x  7  15
8
 3 x  8, x 
3
Solve:
3x  7  2  1
1.
2.
3.
4.
No solution.
{7/3}
{10/3, 4/3}
{-7/3, 7/3}
1.7
• Linear Inequalities and Absolute Value
Inequalities
Objectives
•
•
•
•
Use interval notation.
Find intersections & unions of intervals.
Solve linear inequalities.
Recognize inequalities with no solution or
all numbers as solutions.
• Solve compound inequalities.
• Solve absolute value inequalities.
Linear inequalities
• For equalities, you are finding specific
values that will make your expression
EQUAL something. For inequalities, you
are looking for values that will make your
expression LESS THAN (or equal to), or
MORE THAN (or equal to) something.
• In general, your solution set will involve an
interval of values that will make the
equation true, not just specific points.
What if you have more than one
inequality?
• If two inequalities are joined by the word
“AND”, you are looking for values that will
make BOTH true at the same time. (the
INTERSECTION of the 2 sets)
• If two inequalities are joined by the word
“OR”, you are looking for values that will
make one inequality OR the other true (not
necessarily both), therefore it is the
UNION of the 2 sets.
What IS an absolute value
inequality?
• Recall that absolute value refers to the
expression inside the brackets being either
positive or negative, therefore the absolute value
inequality involves 2 separate inequalities
• IF absolute value expression is LESS THAN a
value, you’re looking for values that are WITHIN
that distance (intersection of the 2 inequalities)
• IF absolute value expression is MORE THAN a
value, you’re looking for values that are getting
further away in both directions (union of the 2
inequalities)
If the absolute value is greater than a
number, you’re considering getting further
away in both directions, therefore an OR.
(get further away left OR right)
• See next slide for example:
2x  4  3
2x  4  3
7
x
2
OR  ( 2 x  4)  3
 2x  8  3
5
x
2
5
7
x   x 
2
2
• If, however, the absolute value was LESS
than a number (think of this as a distance
problem), you’re getting closer to your
value and staying WITHIN a certain range.
Therefore, this is an intersection problem
(AND)
• Same problem as before, but solved as a
LESS than inequality. (next slide)
2x  4  3
2x  4  3
7
x
2
AND  (2 x  4)  3
 2x  8  3
5
x
2
5
7
x   x 
2
2
Don’t leave common sense at the
door!
• Remember to use logic!
• Can an absolute value ever be less than
or equal to a negative value?? NO!
(therefore if such an inequality were
presented, the solution would be the
empty set)
• Can an absolute value ever be more than
or equal to a negative value?? YES!
ALWAYS! (therefore if such an inequality
were given, the solution would be all reals)