1. 2.1 Equations Solving Equations with the Variable on Both Sides
Objectives:
to solve equations with the variable on both sides.
to solve equations containing grouping symbols.
Solve linear equations in one variable.
Apply these skills to solve practical problems.
Justify steps used in solving equations.

2. To solve these equations,
Use the addition or subtraction property to move all variables to one side of the equal sign.

3. Let’s see a few examples:
1) 6x - 3 = 2x + 13
-2x x
4x - 3 = 13
4x = 16
x = 4
Be sure to check your answer!
6(4) - 3 =? 2(4) + 13 =?
21 = 21

4. Let’s try another! Check: 2) 3n + 1 = 7n - 5 -3n -3n
3(1.5) + 1 =? 7(1.5) - 5 =?
5.5 = 5.5
2) 3n + 1 = 7n - 5
-3n n
1 = 4n - 5
6 = 4n
Reduce! 3 = n 2

5. Here’s a tricky one! 3) 5 + 2(y + 4) = 5(y - 3) + 10 Check:
Distribute first.
5 + 2y + 8 = 5y
Next, combine like terms.
2y + 13 = 5y - 5
Now solve. (Subtract 2y.)
13 = 3y (Add 5.)
18 = 3y (Divide by 3.)
6 = y
Check:
5 + 2(6 + 4) =? 5(6 - 3) + 10
5 + 2(10) =? 5(3) + 10 =?
25 = 25

6. Let’s try one with fractions!4)
3 - 2x = 4x - 6
3 = 6x - 6
9 = 6x so x = 3/2
Steps:
Multiply each term
by the least common
denominator (8) to
eliminate fractions.
Solve for x.
Add 2x.
Add 6.
Divide by 6.

7. Two special cases: 6(4 + y) - 3 = 4(y - 3) + 2y
21 = -12 Never true!
21 ≠ -12 NO SOLUTION!
3(a + 1) - 5 = 3a - 2
3a = 3a - 2
3a - 2 = 3a - 2
-3a a
-2 = -2 Always true!
We write IDENTITY.

8. 2.1 Answers (4-64x 4) (4-24 graphing) (28-44 solve)
4. 0
8. 5
20. .5
28. Null Set /6
36. Null Set
40. All Reals, x ≠ ±5/2
44. Null Set /4
56. Choose any a and b such that
b = -5/3a
60. r=C/2
64. I = V/r

9. Understanding Algebra Word Problems
2.2 Applied Problems
Focus here is on problems using one variable. Many of the same types of problems can be solved using two variables. However, the approaches are the same.

10. Word Problem Types
Distance Problems
Mixture Problems
Work Problems

11. Distance = (rate)(time)
Distance Problems
Distance = (rate)(time)
Basic formula for all distance problems is

12. Types Traveling in the Same Direction Traveling in Opposite Directions
Overtaking
Separating
Traveling in Opposite Directions
Traveling Toward Each Other
Traveling Away From Each Other
Going and Returning

13. Overtaking Problem
A fishing boat leaves Tampa Bay at 4:00 a.m. and travels at 12 knots. At 5:00 a.m. a second boat leaves the same dock for the same destination and travels at 14 knots. How long will it take the second boat to catch the first?
Let t be travel time of first boat
Then t – 1 is the travel time of second boat
14(t – 1) = 12t
One starts at 4:00 a.m., one starts at 5:00 a.m.

14. Separating in Same Direction
Key: Travel time same for both
Faster Vehicle = rft
Slower Vehicle = rst
Solution: rft - rst = Constant
Constant is desired distance between

15. Separating Problem
At the auto race one car travels 190 mph while another travels 195 mph. How long will it take the faster car to gain two laps on the slower car if the speedway track is 2.5 miles long?
Let t be the racing time
195t – 190t = (2)(2.5)

16. Coming Together Problem
A freight train leaves Centralia for Chicago at the same time a passenger train leaves Chicago for Centralia. The freight train moves at a speed of 45 mph, and the passenger train travels at a speed of 64 mph. If Chicago and Centralia are 218 miles apart, how long will it take the two trains to meet?
Let t be time to meet
45t + 64t = 218
In this case, the travel time is the same for both trains. It need not be for the approach to work

17. Going and Returning Problem
Brandon and Shanda walk to Grandma’s house at a rate of 4 mph. They ride their bicycles back home at a rate of 8 mph over the same route that they walked. It takes one hour longer to walk than to ride. How long did it take them to walk to Grandma’s?
Let h be time to walk
Then h – 1 is time to ride
4h = 8(h – 1)
What is the hidden assumption in the problem? Bicycles already at grandma’s house!
I like to use a variable that hints at the quantity for which I am solving

18. Mixture Problems

19. Coin Problem Let n be number of nickels n + 4 = quarters 2n = dimes
A coin bank contains four more quarters than nickels, twice as many dimes as nickels, and five more than three times as many pennies as nickels. If the bank contains $22.25, how many of each coin are in it?
Let n be number of nickels
n + 4 = quarters
2n = dimes
3n + 5 = pennies
Best to multiply by 100 first to eliminate decimals.
(3n + 5)(0.01) + n(0.05) + 2n(0.10) + (n + 4)(0.25) = 22.25

20. Interest Problem
An investor has $500 more invested at 7% than he does at 5%. If his annual interest is $515, how much does he have invested at each rate?
Let p be amount at 5%
Then p is amount at 7%
p(0.05)(1) + (p + 500)(0.07)(1) = 515
Best to multiply by 100 first to clear decimals.

21. Solutions Problem
How many gallons of cream that is 30% butterfat must be mixed with milk that is 3% butterfat to make 45 gallons that are 12% butterfat?
Let c be gallons of cream
Then 45 – c is gallons of milk
0.30c (45 – c) = 0.12(45)
Concentrations must be changed from percents to decimals.

22. work done = (rate of work)(time spent)
Work Problems
work done = (rate of work)(time spent)

23. Work Problem
Ron, Mike, and Tim are going to paint a house together. Ron can paint one side of the house in 4 hours. To paint an equal area, Mike takes only 3 hours and Tim 2 hours. If the men work together, how long will it take them to paint one side of the house?
Let t be time needed to paint the side.
(1/4)t + (1/3)t + (1/2)t = 1

24. 2-2 Word Problems Solutions
3. Gross Pay – deductions = Net (take home) pay ; X - .40x = 492
6. 40($10) + x($15) = 595; 13 hrs. overtime
9. x(2) + (600-x)(5) = 2400
12. x(1) + (15-x)10 = 15(2); 13.3 of 1% & 1.6 of 10%
15. (a) 1.5t + 2t = 224; (b) 64(1.5) = 96 m; 64(2) = 128 miles
18.1st – 1 + 4t miles; 2nd – 6t; r*t = d; 4(t+.25) + 6t = 2;
6 min or 1:21
21.Time (to target) + Time (from target) = Time (total); x/ x/1100 = 1.5
24.1st story = (8*30) = 240; 2nd story = (30*3) + ½(30)(h-3); 15h + 45 = 240; 13ft.
27. V = 2/3л*r3 + лr2h = 11250л
30. 1/8 + 1/5 = 1/x; 40/13 hrs.
33. GPA = total weighted honor pts. / total credit hours; 3.2 = (4.8(2.75) + x(4.0)) / (48+x)

25. 2.3 Quadratic Equations,

26. Solving a Quadratic Equation
by factorization
by graphical method
by taking square roots
by quadratic formula
by using completing square

27. By factorization
roots (solutions)

28. By graphical method y
roots x O

29. By taking square roots

30. Solving a Quadratic Equation by the quadratic Formula

31. By quadratic formula

32. a = 1
b = -7
c = 10

33. What are we going to do if we have non-zero values for a, b and c but can't factor the left hand side?
This will not factor so we will complete the square and apply the square root method.
First get the constant term on the other side by subtracting 3 from both sides. 9 9
Let's add 9. Right now we'll see that it works and then we'll look at how to find it.
We are now going to add a number to the left side so it will factor into a perfect square. This means that it will factor into two identical factors. If we add a number to one side of the equation, we need to add it to the other to keep the equation true.

34.  Now we'll get rid of the square by square rooting both sides.
Now factor the left hand side.
This can be written as:
Now we'll get rid of the square by square rooting both sides.
two identical factors
Remember you need both the positive and negative root! 
Subtract 3 from both sides to get x alone.
These are the answers in exact form. We can put them in a calculator to get two approximate answers.

35. Let's solve another one by completing the square.
To complete the square we want the coefficient of the x2 term to be 1. 2 2 2 2
Divide everything by 2 16 16
Since it doesn't factor get the constant on the other side ready to complete the square.
So what do we add to both sides?
the middle term's coefficient divided by 2 and squared
Factor the left hand side
Square root both sides (remember )
Add 4 to both sides to get x alone

37. In general, a quadratic equation may have :
(1) two distinct (unequal) real roots
(2) one double (repeated) real root
(3) no real roots

38. Two distinct (unequal) real roots
x-intercepts

39. One double (repeated) real roots
x-intercept

40. No real roots
no x-intercept

41. Nature of Roots

42. △ = b2 - 4ac
Since the expression b2 - 4ac can be used to determine the nature of the roots of a quadratic equation in the form ax2 – bx + c = 0, it is called the discriminant of the quadratic equation.

43. Two distinct (unequal) real roots
△ = b2 - 4ac > 0
x-intercepts

44. One double (repeated) real roots
△ = b2 - 4ac = 0
x-intercept

45. No real roots
△ = b2 - 4ac < 0
no x-intercept

46. Homework pp. 90-91 (25-35 all, 35 graphing, 47, 55)
b) 9
c) ±10
d) ±9
28.
30.
32. -3, 2/5
55. a) b)

47. The Discoverers of Imaginary Numbers
Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. However, Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers were regarded by some as fictitious or useless, much as zero and the negative numbers.

48. Imaginary Number Timeline

49. 2.4 Complex Numbers OBJECTIVES
Use the imaginary unit i to write complex numbers
Add, subtract, and multiply complex numbers
Use quadratic formula to find complex solutions of quadratic equations

50. Consider the quadratic equation x2 + 1 = 0.
What is the discriminant ?
a = 1 , b = 0 , c = 1 therefore the discriminant is
02 – 4 (1)(1) = – 4
If the discriminant is negative, then the quadratic equation has no real solution. (p. 114)
Solving for x , gives x2 = – 1
We make the following definition:

51. Note that squaring both sides yields
Real numbers and imaginary numbers are subsets of the set of complex numbers.
Complex Numbers
Imaginary
Numbers
Real Numbers

52. Definition of a Complex Number
If a and b are real numbers, the number a + bi is a complex number written in standard form.
If b = 0, the number a + bi = a is a real number.
If , the number a + bi is called an imaginary number.
A number of the form bi, where , is called a pure imaginary number.
Write the complex number in standard form

53. Equality of Complex Numbers
Two complex numbers a + bi and c + di, are equal to each other if and only if a = c and b = d
Find real numbers a and b such that the equation ( a + 6 ) + 2bi = 6 –5i .
a + 6 = 6 2b = – 5
a = 0 b = –5/2

54. Addition and Subtraction of Complex Numbers, p. 127
If a + bi and c +di are two complex numbers written in standard form, their sum and difference are defined as follows.
Sum:
Difference:

55. Perform the subtraction and write the answer in standard form.
( 3 + 2i ) – ( i )
(3 – 6) + (2 – 13)i
–3 – 11i 4

56. Properties for Complex Numbers p.126
Associative Properties of Addition and Multiplication
Commutative Properties of Addition and Multiplication
Distributive Property of Multiplication
Multiplying complex numbers is similar to multiplying polynomials and combining like terms.
Perform the operation and write the result in standard form. ( 6 – 2i )( 2 – 3i )
F O I L
12 – 18i – 4i + 6i2
12 – 22i + 6 ( -1 )
6 – 22i

57. Consider ( 3 + 2i )( 3 – 2i )
9 – 6i + 6i – 4i2
9 – 4( -1 )
9 + 4 13
This is a real number. The product of two complex numbers can be a real number.

58. Complex Conjugates and Division p. 129
Complex conjugates-a pair of complex numbers of the form a + bi and a – bi where a and b are real numbers.
( a + bi )( a – bi )
a 2 – abi + abi – b 2 i 2
a 2 – b 2( -1 )
a 2 + b 2
The product of a complex conjugate pair is a positive real number.

59. To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator.

60. Perform the operation and write the result in standard form. (Try p

61. Principle Square Root of a Negative Number,
If a is a positive number, the principle square root of the negative number –a is defined as

62. Use the Quadratic Formula to solve the quadratic equation.
9x2 – 6x + 37 = 0
a = 9 , b = - 6 , c = 37
What is the discriminant?
( - 6 ) 2 – 4 ( 9 )( 37 )
36 – 1332
-1296
Therefore, the equation has no real solution.

63. 9x2 – 6x + 37 = 0
a = 9 , b = - 6 , c = 37

64. 2.4 Answers: (3-42 x 3)
2.5 Answers (3-39 x 3) i i
18. -1
24. i
36. x=10; y = 3
42.
6. 2
, 0
18. 1
24. 9
30. 4
36. ±1, ±2

65. 2.5 Other Types of Equations Radical Equations; Absolute Value Equations;

69. Thus 85 is NOT a solution.
The solution set is {5}.

70. The solution set is {7,-6}.

71. A constant function is a function of the form
f(x)=b y b x

72. Identity function is a function of a form:
f(x)=x
(1,1)
(0,0)

73. The square function

74. Cube Function
(1,1)
(-1,-1)

75. Square Root Function

76. Rational Function

77. 2.5 Answers (3-39 x 3)
6. 2
, 0
18. 1
24. 9
30. 4
36. ±1, ±2

78. Solving Absolute Value Equations & Inequalities

79. 
<
>    
Linear Inequalities
> 
>
<
<   

80. Remember---these mean the same thing---just two different notations.
There are two kinds of notation for graphs of inequalities: open circle or filled in circle notation and interval notation brackets. You should be familiar with both.
Remember---these mean the same thing---just two different notations. [ 6 4 8 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
circle filled in
squared end bracket
Both of these number lines show the inequality above. They are just using two different notations. Because the inequality is "greater than or equal to" the solution can equal the endpoint. That is why the circle is filled in. With interval notation brackets, a square bracket means it can equal the endpoint.

81. Remember---these mean the same thing---just two different notations.
Let's look at the two different notations with a different inequality sign. ) 6 4 8 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
circle not filled in
rounded end bracket
Since this says "less than" we make the arrow go the other way. Since it doesn't say "or equal to" the solution cannot equal the endpoint. That is why the circle is not filled in. With interval notation brackets, a rounded bracket means it cannot equal the endpoint.

82. ( ] Compound Inequalities
Let's consider a "double inequality" (having two inequality signs). ( ] 6 4 8 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
I think of these as the "inbetweeners" x is inbetween the two numbers. This is an "and" inequality which means both parts must be true. It says that x is greater than –2 and x is less than or equal to 3.

83. Compound Inequalities
Now let's look at another form of a "double inequality" (having two inequality signs). ) [ 6 4 8 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
Instead of "and", these are "or" problems. One part or the other part must be true (but not necessarily both). Either x is less than –2 or x is greater than or equal to 3. In this case both parts cannot be true at the same time since a number can't be less than –2 and also greater than 3.

84. Just like graphically there are two different notations, when you write your answers you can use inequality notation or interval notation. Again you should be familiar with both. [ -7 -6 -5 -4 -3 -2 -1 1 5 7 3 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 2 4 6 8
Inequality notation for graphs shown above.
Interval notation for graphs shown above.

85. This means x is unbounded above
Let's have a look at the interval notation.
This means x is unbounded above
unbounded [ 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
For interval notation you list the smallest x can be, a comma, and then the largest x can be so solutions are anything that falls between the smallest and largest.
The bracket before the –1 is square because this is greater than "or equal to" (solution can equal the endpoint).
The bracket after the infinity sign is rounded because the interval goes on forever (unbounded) and since infinity is not a number, it doesn't equal the endpoint (there is no endpoint).

86. Let's try another one.
Squared bracket means can equal 4
Rounded bracket means cannot equal -2
The brackets used in the interval notation above are the same ones used when you graph this. ( ] 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
This means everything between –2 and 4 but not including -2

87. ) Let's look at another one
This means the largest x can be is 4 but can't equal 4
This means x is unbounded below ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
Notice how the bracket notation for graphing corresponds to the brackets in interval notation.
Remember that square is "or equal to" and round is up to but not equal. By the infinity sign it will always be round because it can't equal infinity (that is not a number).

88. ) [ Now let's look at an "or" compound inequality2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) [
When the solution consists of more than one interval, we join them with a union sign.
There are two intervals to list when you list in interval notation.

89. Properties of Inequalities
Essentially, all of the properties that you learned to solve linear equations apply to solving linear inequalities with the exception that if you multiply or divide by a negative you must reverse the inequality sign.
So to solve an inequality just do the same steps as with an equality to get the variable alone but if in the process you multiply or divide by a negative let it ring an alarm in your brain that says "Oh yeah, I have to turn the sign the other way to keep it true".

90. Ring the alarm! We divided by a negative!
Example:
- 4x x
Ring the alarm! We divided by a negative!
We turned the sign!

91. 2.6 Inequalities Involving Absolute Value

92. Absolute Value (of x) Symbol |x|
The distance x is from 0 on the number line.
Always positive
Ex: |-3|=3

93. Ex: x = 5
What are the possible values of x?
x = or x = -5

94. To solve an absolute value equation:
ax+b = c, where c>0
To solve, set up 2 new equations, then solve each equation.
ax+b = c or ax+b = -c
** make sure the absolute value is by itself before you split to solve.

95. * Plug in answers to check your solutions!
Ex: Solve 6x-3 = 15
6x-3 = or 6x-3 = -15
6x = or 6x = -12
x = 3 or x = -2
* Plug in answers to check your solutions!

96. Get the abs. value part by itself first!
Ex: Solve 2x = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
2x = 4 or 2x = -18
x = 2 or x = -9
Check the solutions.

97. Solving Absolute Value Inequalities
ax+b < c, where c>0
Becomes an “and” problem
Changes to: –c<ax+b<c
ax+b > c, where c>0
Becomes an “or” problem
Changes to: ax+b>c or ax+b<-c

98. Ex: Solve & graph.
Becomes an “and” problem

99. Solve & graph. Get absolute value by itself first.
Becomes an “or” problem

100. Answers to 2.6 pp. 117-118 (3-33 x3, 41) (-3, ∞) (-5, -3] x > -3
( −8 5 , ∞)
30. (-2, −1 3 ]

101. Inequalities with Quadratic Functions
2.7 More on Inequalities
Inequalities with Quadratic Functions

102. Quadratic inequalities
…means “for what values of x is this quadratic above the x axis”
ax2+bx+c>0
e.g. x2+ x - 20 >0
…means “for what values of x is this quadratic below the x axis”
ax2+bx+c<0
e.g. x2+ x - 20 < 0

103. Quadratic inequalities (2)
e.g. x2+ x - 20 >0
Factorises to (x-4)(x+5) >0
Numbers that multiply together to give more than 0?
A) both greater than zero
B) both less than zero
So, (x-4)>0
and (x+5)>0
x > 4 and x > -5
Only possible if x > 4
(then it must be >-5)
So, (x-4)<0
and (x+5)<0
x < 4 and x < -5
Only possible if x < -5
(then it must be <4)
Either:
x > 4 or
x < -5

104. Quadratic inequalities (3)
e.g. x2 + x - 20 <0
Factorises to (x-4)(x+5) < 0
Numbers that multiply together to give less than 0?
A) first >0, second <0
B) first <0, second >0
So, (x-4)>0
and (x+5)<0
x > 4 and x < -5
IMPOSSIBLE
So, (x-4)<0
and (x+5)>0
x < 4 and x > -5
DO-ABLE
x < 4
and
x > -5
-5 < x < 4

105. Quadratic inequalities (4) - may be easier just looking at a graph
e.g. (x-4)(x+5) >0
…means “for what values of x is this quadratic above the x axis”
When x>4
and x<-5
Crosses at
x=4
and x=-5
When x<4
and x>-5
e.g. (x-4)(x+5) < 0
…means “for what values of x is this quadratic below the x axis”

106. Try this one For what values of x is x2 - 3x - 18 > 0
[ Factorises to (x+3)(x-6) >0 ]
Numbers that multiply together to give more than 0
A) both greater than zero
B) both less than zero
So, (x+3)>0
and (x-6)>0
x > -3 and x > 6
Only possible if x > 6
So, (x+3)<0
and (x-6)<0
x < -3 and x < 6
Only possible if x < -3
Either:
x > 6 or
x < -3

107. Quadratic with linear Solve: x2 – 8x + 16 > 2x +7 Estimate ?
y = 2x + 10
y = x2 – 8x +16

108. Quadratic with linear (2)
Solve: x2 – 8x + 16 > 2x +7
Algebraically:
1. Rearrange first
2. Solve like the others
x2 – 8x + 16 > 2x +7
x2 – 10x + 16 > 7
(-2x)
(-7)
x2 – 10x + 9 > 0
Like the ones we did
(x-9)(x-1) > 0
x>9 or x<1

109. Try this one Solve: x2 + x + 4 > 4x +14 Algebraically:
First: try a sketch
Algebraically:
1. Rearrange first
2. Solve like the others
x2 + x + 4 > 4x +14
x2 – 3x + 4 > 14
(-4x)
(-14)
x2 – 3x - 10 > 0
(x+2)(x-5) > 0
x<-2 or x>5

110. Answers to 2.7 pp. 125-126 (3-30 x 3, 43, 44) 6. (-∞, −3]∪[−1, ∞)
6. (-∞, −3]∪[−1, ∞)
(- ∞, -3]∪[4,∞)
(- ∞, −3 4 )∪( 3 4 , ∞)
24. (-3, 3)∪(3,∞)
30. (-4, 0) ∪(4,∞)
≤𝑡 ≤12

111. Answers to Ch. 2 Review Packet
x = (0, 14]
y = ( −8 7 , )
9 mi./hr x < 2 or x > 5
4 11 mi./hr.
h = 5.32 ft.
x = -14
width = 6 ft.
21i – 20 74
10+24𝑖 13 i
2𝑖 −247
13. x = 2 ± 5i
x = ± 3, x = ± 3i
x = 6, x = -12
( −11 7 , −4 7 ]