Download Mod 3 Ch 6-7

Mth 95 Notes
Module 3
Review – Factor completely.
6 x2  7 x  3
Spring 2014
5x2  7 x  6
6.5 - Solving Equations containing Rational Expressions
When equations to be solved contain fractions, usually the first steps is to remove the
denominators by multiplying the entire equation by the LCD(least common
denominator). Here is an example where the denominators are integers:
x 1
1
1 
x 1
Since the LCD is 6,    2  x   6
  2 x
6 3
2
2 
6 3
x
1
1
6 6  26 x6
6
3
2
x  2  12  3x
4x  10
x  2.5
Since we will have x’s in the denominators of our rational equations, we need to watch
out for extraneous solutions (an answer given by a symbolic solution that is not in the
domain of the function). This means you need to check your answers in the original
equation each time.
Solve and check your result.
x 12
 1
LCD
2 x
Domain
Check
1 1 4
 
3x x 15
LCD
Domain
Check
x
4

 2
x4 x4
Chapters 5 and 6
LCD
Domain
1
Mth 95 Notes
Module 3
Spring 2014
x 2  10
7x

x 5
x 5
LCD
Restriction(s)
15
9y  7

 9
y
y 2
LCD
Restriction(s)
32
4
2


x  25 x  5 x  5
LCD
Restriction(s)
2
5
4

 2
z 1
2z  2
z 1
LCD
Restriction(s)
2
Chapters 5 and 6
2
Mth 95 Notes
Module 3
Review. Simplify.
4 x2  4 x2  5x  6

3x  3
6
Spring 2014
5
5

x2 x2
State the domain of the rational function.
2
f  x 
3x  1
Solve and check your result.
x
11
1


2
x 3
x 9
x 3
LCD
Restrictions on the domain
Check
6.7 – Variation and Problem Solving
Direct Variation: Two quantities vary directly (directly proportional) if as one gets
bigger so does the other.
As t, time, increases so does D, distance. D = 20t can be read: At 20 mph
“distance varies directly with time” or distance is directly proportional to time.”
Inverse Variation: Two quantities vary inversely (inversely proportional to) if as one
gets bigger the other gets smaller.
30
If you need to travel 30 miles, as the speed increases the time decreases. t 
r
can be read: For the distance 30 miles, “time varies inversely with speed” or
“time is inversely proportional to speed.”
Joint Variation: A quantity varies as the product of one or more variables. (direct
variation)
Chapters 5 and 6
3
Mth 95 Notes
Module 3
Spring 2014
In each of the following formulas k is a nonzero number called the constant of
variation or constant of proportionality.
k
Direct: y = kx
Inverse: y 
Joint: z  kxy
x
Write an equation to describe each variation. Use k for the constant of variation.
y is directly proportional to x
a varies inversely as b
y varies jointly as x and z
y is inversely proportional to x2
y varies directly as x and inversely as p2
Write a sentence to describe each of the following equations as variations:
p=kq
y=kqt
z = k a4
k
j
m
Steps in Solving Variation Problems
1. Write an equation relating the unknowns, including k, using the formula below.
2. Substitute values of the variables and solve for k.
3. Rewrite equation using the variables and the value of k.
4. Use the equation to answer the question.
y varies directly with x.
y is 42 when x is 6.
y varies inversely with x.
y is 21 when x is 7.
z varies jointly with x and y.
z is 6 when x is 3 and y is 8.
Find y when x is 10.
Find y when x is 10.
Find z when x is 5 and y is 7.
Chapters 5 and 6
4
Mth 95 Notes
Module 3
Spring 2014
The cost of tuition is directly proportional to the number of credits taken. If 6 credits cost
$435, find the constant of variation and tell what it represents. Then find the cost of 11
credits.
The use of a particular toll bridge is inversely proportional to the toll. If the toll is $0.75, then
6000 vehicles are using the bridge. How many vehicles use the bridge if the toll is $0.50?
Hooke's Law states that the distance a spring stretches is directly proportional to the
weight attached to the spring. A 30-lb weight attached to a spring stretches the spring
4.5 inches. Find the distance a 45-lb weight attached to the spring stretches it.
The weight that can be supported by a 2-inch by 4-inch piece of pine (called a 2-by-4) is
inversely proportional to its length. A 10-foot 2-by-4 can support 500 lbs. What weight
can be supported by a 5-foot 2-by-4?
Chapters 5 and 6
5
Mth 95 Notes
Module 3
Spring 2014
7.7 - Complex Numbers
The Imaginary Unit is denoted i and represents the number whose square is -1.
i  1 and
i 2  1
The standard form of a Complex Number is written a + bi,
where a is the real part and bi is the imaginary part.
Every real number is a complex number because it can be written a + 0i.
For example, 8 = 8 + 0i.
An imaginary number is a complex number, a + bi, with b  0.
Complex Numbers
a + bi, where a and b are real numbers
Real Numbers
a + bi, with b = 0
Imaginary Numbers
a + bi, with b  0
4  2i, 6  2i, 3i, 2i 5
Rational Numbers
Irrational Numbers
-3, 2/3, 0, 0.4 , etc.
3,  , 3 12,etc.
Simplifying Square Roots of Negative Numbers
25  25  1  5i
81
Note: When a radical is multiplied by i, write i in front of the radical.
12  12  1  4  3  i 
6
3 8
5 50
Multiplying and Dividing Square Roots of Negative Numbers
First write each non-real number in terms of the imaginary unit.
Then multiply or divide.
a ) 2  7
b ) 25  1
c ) 27  3
d)
8
2
Chapters 5 and 6
6
Mth 95 Notes
Module 3
Spring 2014
Adding and Subtracting Complex Numbers - Write your answers in the form a + bi.
 8  2i    3  4i 
 8  3   2i  4i 
5  i  4  4i
 2  i    4  3i 
4i   2  i 
(2  5i )  ( 4  3i )
 5  i    4  4i 
 4  2i    6  i 
5i   3  2i 
 3  4i   5  2i 
12  3i    7  4i 
Multiply and simplify. - Write your answers in the form a + bi.
3  2  4i  
4i  7  3i  
2i  3  5i  
2i 3i 
3i 4i 
 2  3i  4i  
Multiplying Complex Numbers- Write your answers in the form a + bi.
F
O
I
L
Simplify.
 4  2i 3  5i   4 3  4 5i    2i 3   2i 5i 
Since i 2 


 12  20i  6i  10i 2
2
1  1 ,
 12 14i 10  1
 12 14i 10  22 14i
3  4i  2  3i 
Chapters 5 and 6
 4  5i  4  5i 
7
Mth 95 Notes
Module 3
Spring 2014
 6  2i  2
1 6i  4  3i 
 7  3i  2  5i 
(3  7i )2
 2  5i  4  6i 
(5  2i )(5  2i )
The complex conjugate of a + bi is a – b
The product of complex conjugates is always a real number.
Write the complex conjugate of each complex number
3 + 4i
Powers of i
i0  1
4 – 5i
i 4   i 2  i 2    1 1  1
i5   i 4  i  1i  i
i1  i
i 2  1
i3   i 2  i  1i  i
3 – 2i
4+i
i 6   i 4  i 2   1 1  1
i 7   i 4  i3   1 i   i
i8   i 4  i 4   11  1
i9   i8   i   1 i   i
i10
i11
Evaluating powers of i
Since i 0  i 4  i8  i12  i16  i 20 ...  1 , we can rewrite i 30  i 28  i 2  1  1  1
i 27 
i 36 
i 41 
Review – Perform the indicated operation, or simplify as needed. Simplify answers
completely and put them in a + bi form.
 6  3i    4  2i 
 3  5i 
2
Chapters 5 and 6
3 45
7  7
 7  2i  2  i 
8