Equations and Inequalities

1
Chapter 1

• Equations and Inequalities

2
Aim 1.1 How do we graph and interpret
information?

• Analytic Geometry this new branch of geometry
  founded by Rene Descartes brings Algebra and
  Geometry together.
• Key Terms
• Rectangular Coordinate System or Cartesian
  Coordinate System
• X-axis,
• Y-axis
• Quadrants

3
Graphs of Equations

X Y 4 x2 (X, Y)
3 -5 4 - 32 (3, -5)

Solution of an equation that satisfies the
equation.
4
Using a Graphing Calculator

• Using your calculator-
• Y 4 x2
• Understanding the Viewing Rectangle
• -10, 10, 1

Minimum X-value, Maximum X- value and X scale
5
Intercepts

• X-intercept- is the point on the graph where it
  intersects with the X- axis. Ex. (X, 0)
• Y-intercepts is the point on the graph where it
  intersects with the Y-axis. Ex, (0, Y)
• Look at Text- Identifying Intercepts

6
Interpreting Information from Graphs

• Line Graph used to illustrate trends or data over
  time.
• Look at Text

7
Summary Answer in complete sentences.

• What is the rectangular coordinate system?
• Explain why (5, -2) and (-2, 5) do not represent
  the same point.
• What does -20, 2, 1 by -4, 5, 0.5 viewing
  rectangle mean?
• Determine whether the following is true or false.
  Explain.
• If the product of a points coordinates is
  positive, the point must be in quadrant I.

8
Warm-up 8/23/2011

• Take out your homework and unit plan.-
• Copy the questions and Answer.
• Explain how to graph an equation in the
  rectangular coordinate system.
• Explain how to graph the point (7, -8) on the
  rectangular coordinate system.
• Sketch a function that models the following
  situation.
• As the blizzard got worse, the snow fell harder
  and harder.
• (Label the x-axis Time and Y-axis snowfall)

9
Aim 1.2 How do we solve equations?

• What is a linear equation?
• A linear equation has one variable and can be
  written in the form of y mx b, where a and b
  are real numbers and a ? 0.
• How do we solve? 4x 12 0

10
Practice

• 4x 5 29
• 6x 3 63

11
How do we solve other types linear equations?

• Solve 2 (x 3) 17 13
• 4( 2x 1) 29 3 (2x 5)

12
Practice

• 5x (2x 10 ) 35
• 3x 5 2x 13
• 13 x 14 12 x 5

13
How do we solve an equation w/ a fraction?

• Steps
• Multiply the entire equation by a multiple of the
  denominator. For this example, it would be
• This would get rid of all denominators.
• Then combine like terms and isolate the variable.

14
Guided Practice
15
What is a Rational Equation?

• It is an equation containing one or more rational
  expressions.
• Solving

• Hint Multiply the entire equation by the LCD or
  multiple of x, 5, and 2x.

16
Solving a Rational Equation
17
Solving a Rational Equation
18
Categorizing the Different Equations

• Key Terms
• Identity Equation is an equation that is true for
  all values of x. Ex. x 3 x 2 1
• Conditional Equation is true for a particular
  value of x. Ex. 2x 5 10 x 2.5
• Inconsistent Equation is an equation that is not
  true for any value of x. Ex. x x 7

19
Summary Answer in complete sentences.

• What is a linear equation in one variable? Give
  an example. What are some other types of linear
  equations?
• 2. Does the following make sense? Explain your
  reasoning.
• Although I can solve 3x 1/5 ¼ by first
  subtracting 1/5 on both sides, I find it easier
  to multiply by 20, the least common denominator,
  on both sides.
• 3. Is the equation (2x- 3)2 25 equivalent to 2x
  3 5? Explain.

20
Warm-up 8/25/2011

• Take out your homework and unit plan.-
• Copy the questions and Answer.
• Solve for X.
• 7x 5 72
• 6x 3 60
• 13x 14 12 x 5

21
Warm-up 8/25/2011

• Take out your homework and unit plan.
• Copy the questions and Answer.
• Solve for X.

22
Aim 1.3How do we use linear equations to model
situations?

• Steps to solving Word Problems
• 1. Read the problem. Twice.
• 2. Define x
• 3. Write your equation.
• 4. Solve.
• 5. Check your solution. Does it make sense?

23
Using the 5-step strategy and find the number.

• When two times a number is decreased by 3, the
  result is 11. What is the number?
• Let x number
• Equation 2x 3 11 Solve for x.

24

• When a number is decreased by 30 of itself, the
  result is 20. What is the number?
• Let x the number
• Equation x __ ___
• Solve and Check your answer.

25
Solving a Formula for a Variable

• 2L 2W P Solve for l

• Your goal is to isolate L.
• Subtract 2W on both sides.
• Then divide by 2 on both sides to isolate L.
• Whats your final answer?
• Note 2W is like one term because you are
  multiplying W by 2.

26
Check for Understanding

• Solve the formula for W.
• 2L 2W P

27
Solving a Formula for a Variable that Occurs Twice

• Solve for P.
• A P P r t

• Factor out the P from P P rt.
• Then divide by the expression in parenthesis to
  isolate P.

28
Check for Understanding

• Solve the formula for C.
• P C MC

29
Summary Answer in complete sentences.

• Explain how to solve for P below and then solve.
  
• T D pm
• What does it mean to solve for a formula?
• Write an original word problem that can solved
  using a linear equation. Then write out all the
  steps for the solution.

30
Warm-up 8/26/2011

• Take out your homework and Unit Plan.
• What is the difference between an identity,
  conditional and inconsistent equation?
• Solve for r.

• Agenda
• Go over hw.
• Quiz
• Review

31
Warm-up 8-29-2011

• Take out your homework and Unit Plan.
• Solve.

• Reminder- Quiz Friday- P.6B through- 1.5
• Note We are skipping 1.4 at this time.

32
Warm-up 8/30/2011

• Take out your homework and Unit Plan.
• Solve for x.

• Reminder Quiz Friday P.6B -1.5

33
Warm-up 8-31 or 9-1

• Take out your homework and unit plan.
• First, write the value (s) that make the
  denominator (s) zero. Then solve the equation.
• What type of equation is this?

34
Aim 1.5 How do we solve quadratic equations?

• Definition of a Quadratic Function
• A quadratic equation is an equation that can be
  written in the general form of ax2 bx c 0,
• where a, b, and c are real numbers and a?0.
• i.e. Quadratics are second degree polynomial
  functions.

35
Solving Equations by the Square Root Property

• Notice in the examples to the left that there is
  no b term.
• Steps
• 1. Isolate the x2 term.
• 2. The find the square root of both sides.
• 3. Simplify.

36
Check for Understanding

• Solve by the Square Root Property.

37
Zero-Product Principle

• If the product of two algebraic expressions is
  zero, then at least one of the factors must equal
  to zero.
• If AB 0 then A 0 or B 0
• Example x2 7x 10
  0
• Factor. (X
  5)(x2) 0
• Set each factor 0 X 5 0 or x2
  0
• Solve. x -5 or x
  -2

38
Solving Quadratic Equations by Factoring

1. Is there a GCF, that can factored out?
2. Subtract 4 on both sides and set equation 0.
   Now factor.

39
Solving Quadratics by Completing the Square

• Completing the Square is a strategy to solve
  quadratics when
• The Trinomial can not be factored
• Zero Product property can not be used
• Completing the square allows us to convert the
  equation so that it can be solved using the
  square root property.

40

• Completing the Square
• If x bx is a binomial, then by adding ,
  which is the square of half the coefficient of
  x,a perfect square trinomial will result. That
  is,

41
Completing the Square

• What term should be added to each binomial so
  that it becomes a perfect square trinomial? Write
  and factor the trinomial.
• x2 8 x
• Solution

42
Completing the Square

• What term should be added to each binomial so
  that it becomes a perfect square trinomial? Write
  and factor the trinomial.

43
Completing the Square

• What term should be added to each binomial so
  that it becomes a perfect square trinomial? Write
  and factor the trinomial.

44
Solving Quadratics using Completing the Square

• Steps
• 1. Subtract 4 on both sides.
• 2. Take the b term and divide by 2 and square it.
• 3. Now add it to both sides of the equation.
• 4. Now you can express the left hand side as a
  square.
• 5. Apply the square root property and solve for x.

45
Solving Quadratics using Completing the Square

• Guided Practice

46
Solving Quadratics using Completing the Square

• Steps
• Divide the entire equation by 9, so that a 1.
• Add -4/9 to both sides.
• Complete the square.
• Then solve for x.

47
Solving Quadratics using Completing the Square

• Guided Practice

48
What is the Quadratic Formula?

• Quadratic formula If ax2 bx c 0 and a?0

49
Using the Quadratic Formula

• Solve x2 6 5x

• Steps
• Subtract 5x on both sides, so the equation 0.
• Identify the values for a, b, and c.
• a 1, b -5, c 6
• 3. Then substitute into the

50
What is the discriminant?

• Property of the Discriminant
• For the equation ax2 bx c 0, where a ? 0,
  you can use the value of the discriminant to
  determine that number of solutions.
• If b2 4ac gt 0, there are two solutions.
• If b2 4ac 0, there is one solution.
• If b2 4ac lt0, there are no solutions

51
Using the Discriminant

• For each equation, compute the discriminant. Then
  determine the number and type of solutions.

52
SummaryAnswer the following in complete
sentences.

• What is a quadratic equation?
• What are at least 3 different ways of solving a
  quadratic?
• When is using the square root property helpful?
• (Think of at least 2 ways.)
• 4. Solve using any method. Explain the method you
  choose and why.

53
Warm-up 9-2-2011

• Take out your homework and Unit Plan.
• Find the x intercepts of the equation. (i.e.
  Solve.)
• BE READY, YOU WILL BE CALLED TO THE BOARD IN 10
  MINUTES.

54
Warm-up 9-6-011

• Take out your homework and Unit Plan.
• I will collect Chapter Review with work on TEST
  DAY!!!
• Find the x-intercepts.

BE READY, YOU WILL BE CALLED TO THE BOARD IN 10
MINUTES.
55
Warm-up 9-9-2011

• For the following equation, compute the
  discriminant.
• Then determine the number of solutions and type
  of solutions

BE READY, YOU WILL BE CALLED TO THE BOARD IN 8
MINUTES.
56
Aim1.5B How do we solve problems modeled by
quadratic equations?

• In a 25-inch television set, the length of the
  screens diagonal is 25 inches. If the screens
  height is 15 inches, what is its width?

• You need to use the Pythagorean Theorem
• a2 b2 c2
• Sketch a figure.
• Substitute into the equation what you know and
  Solve.
• Write your final answer in a sentence.

57
Problem

• What is the width of a 15-inch television set
  whose height is 9 inches?

58
Summary Answer in complete sentences.

• If you are given a quadratic equation, how do you
  determine which method to use to solve it?
• Describe the relationship between the solutions
  of
• ax2 bx c 0 and the graph of y ax2 bx
  c.
• Write a quadratic equation in general form whose
  solution set is

59
Warm-up 9-12-2011

• Take out your homework.- Do not copy problem.
  Show all work and solve.
• Define your variables, write your equation, solve
  and express final answer as a sentence.
• After a 20 reduction, you purchase a television
  for 336. What was the televisions price before
  the reduction?
• Including 5 sales tax, an inn charges 252 per
  night. Find the inns nightly cost before the tax
  is added.
• Each side of a square is lengthened by 2 inches.
  The area of this new, larger square is 36 sq.
  inches. Find the length of a side of the original
  square.

60
Aim 1.6 How do we solve polynomial equations
by factoring?

• A polynomial equation is the result of setting
  two polynomials equal to each other.
• The equation is in general form if one side is 0.
  and the polynomial on the other side is in
  descending powers of the variable.
• The degree of a polynomial is the same as the
  degree of any term in the equation. Here are some
  examples of polynomial equation
• 3x 5 14 Degree of 1 also linear
• 2x27x 4 Degree of 2 also quadratic
• x3 x2 4x 4 Degree of 3 also cubic

61
Solving a Polynomial by Factoring

• Steps
• 1.Move all terms to one side and set equation
  0.
• Subtract 27x2 on both sides.
• 2. Factor the GCF.
• 3. Set each factor to 0
• 4. Solve.

62
Guided Practice

• Solve by factoring
• 4x412x2

63
Solving a Polynomial Equation

• x3 x2 4x 4

• Steps
• Move all terms to one side and set equation 0.
• Factor using the grouping strategy.
• Set each factor to zero and solve.

64
Guided Practice

• Solve by factoring
• 2x3 3x2 8x 12

65
Practice

• Solve each polynomial equation by factoring and
  then use zero-product principle.
• 5x4- 20x2 0
• 4x3 -12x2 9x 27
• x 1 9x3 9x2
• 9y3 8 4y 18y2

66
How do we solve radical equations?

• A radical equation is an equation in which the
  variable occurs in a square root, cube root or
  any higher root. An example is

Squaring both sides. Eliminates radical sign.
67
Note

• We solve radical equations with nth roots by
  raising both sides of the equation to the nth
  power. Unfortunately, if n is even, all the
  solutions of the equation raised to the even
  power may not be solutions of the original
  equation.
• For example x 4
• If we square both sides, we obtain x2 16
• x v16 4 This equation has two new
  solutions, -4 and 4. By contrast only 4 is a
  solution to the original equation.

68

• When raising both sides of an equation to an even
  power, always check proposed solutions in the
  original equation.

69
Solving Radical Equations Containing nth Roots

1. If necessary, arrange terms so that one with the
   radical is isolated on one side of the equation.
2. Raise both sides of the equation to the nth power
   to eliminate the nth root
3. Solve the resulting equation. If there is still a
   radical repeat step 1 and 2.
4. Check all proposed solutions.

70

• Extraneous solutions or extraneous roots are
  solutions that do not satisfy the original
  equation.

71
Solving a Radical Equation

• Steps
• Isolate radical on one side by subtracting 2 on
  both sides.
• Raise both sides to the nth power. Because n, the
  index is 2, we square both sides.
• Solve the resulting equation.

72
Guided Practice

• Solve

73
How do we solving an equation that has two
radicals?

• Steps
• Isolate a radical on one side.
• Square both sides.
• Simplify.
• Note the resulting equation still has a radical
  sign so, repeat steps 1 and 2.
• Solve resulting equation.

74
Guided Practice
75
Practice

• Solve each radical equation. Check all proposed
  solutions.

76
Summary Answer in complete sentences.

• Without actually solving the equation, give a
  general description of how to solve x3 - 5x2 x
  5 0.
• In solving why
  is it a good idea to isolate a radical term? What
  if we dont do this and simply square each side?
  Describe what happens.
• What is an extraneous solution to a radical
  equation?

77
Warm-up 9-13-11

• Take out your homework and Unit Plan.
• Solve each polynomial equation by factoring. Then
  use the zero product principle.
• 3x3 2x2 12x 8
• 2x4 16x
• 2x3 x2 18 x 9 0
• BE READY, YOU WILL BE CALLED TO THE BOARD IN 6
  MINUTES.
• Reminder Quiz Warm-up due Friday!

78
Warm-up 9-14-2011

• Take out your homework and Unit Plan.
• Solve each radical equation. Check all proposed
  solutions.

79
Aim 1.6B How do we solve equations with
rational exponents?

• We know that expressions with rational
  expressions represent radicals

80
SOLVING RATIONAL EQUATIONS OF THE FORM

• Assume that m and n are positive integers, m/n is
  in lowest terms, and k is a real number.
• Isolate the expression with the rational
  exponent.
• Raise both sides of the equation to the n/m
  power.

If m is even If m is odd

81
NOTE

• It is incorrect to insert symbol when the
  numerator of the exponent is odd. An odd index
  has only ONE root.
• 3. Check all proposed solutions in the original
  equation to find out if they are actual solutions
  or extraneous solutions.

82
Solving Equations Involving Rational Exponents

• Solve

• Steps
• Goal is to isolate the expression with the
  rational exponent.
• Undo the addition or subtraction.
• Undo the Multiplication or division.
• Raise both sides by the reciprocal of the
  exponent.
• Note exponent is odd so we do not add .
• Check proposed solution.

83
Guided Practice

• Solve

84
Practice
85
How do we solve equations that are in quadratic
form?

• An equation that is quadratic in form is one that
  can be expressed using an appropriate
  substitution.

86
Solving an Equation in Quadratic Form

• Steps
• 1. Replace x2 with u.
• 2. Factor.
• 3. Apply the zero-product principle.
• 4. Solve for u.
• 5. Then replace u with x2 and now solve for x.

87
Guided Practice

• Solve

88
Solving an Equation in Quadratic form

• Steps
• Replace with u.
• Factor.
• Set each factor to 0.
• Solve for u.

89
How do we solve equations with Absolute Value?

• An absolute value is the distance a number is
  from zero.
• An absolute value equation
• What values of x would make the above equation
  true?

90
Solving Absolute Value Equations

• Solve

• Steps
• Rewrite equation without absolute value bars.
• Write one equation 11 and a second -11
• Solve each equation.
• Check all proposed solutions.

91
Solving Absolute Value Equations

• NOTE
• Before we can solve we need to isolate the
  absolute value expression.

92
Practice

• Solve

93
Summary Answer in complete sentences.

• Explain in words how to solve. Then show the
  work.
• How do we solve an absolute value equation?
  Include an example to support your answer.
• What do solving absolute value equations and
  radical equations have in common?
• What other type of equation did we learn to solve
  in this Aim 1B? Give an example.

94
Warm-up 9/ 16/2011

• Take out your homework.
• Make the appropriate substitution, then solve.

95
Warm-up 9-19-2011

• Solve each equation. Make the appropriate
  substitution.
• Be ready, in 8 minutes you will be called to the
  board.

96
Aim 1.7 What is interval notation and how do we
use it?

• Goals of this Aim are
• Use of Interval Notation
• Find Intersections and Unions of Intervals
• Solve Linear Inequalities

97
What is interval notation?

• Subsets of real numbers can be represented using
  interval notation.
• Examples
• Open Interval (a, b) represents the set between
  the real numbers a and b and not including a and
  b.
• (a, b) x /a lt x lt b
• b. Closed Interval a, b represents the set
  between the real numbers a and b including a and
  b.
• a, b x /a lt x lt b

98
More Examples of Infinite Notation

• The infinite interval represents the set
  of real numbers that are greater than a.
• The infinite interval represents the set
  of real numbers that are less than or equal to b.
• Remember
• Parentheses indicate endpoints not included in
  an interval. Square brackets indicate that
  endpoints that are included interval

99
Using Interval Notation

• Express each interval using set builder notation
  and graph.
• (-1, 4 x /-1lt x lt 4
• (2.5, 4
• (-4,

100
How do we find the intersection and union of
intervals?

• Use graphs to find each set

• Steps
• Graph each interval on a number line.
• 2 a. To find intersection, take the portion of
  the number line that the two graphs have in
  common.
• b. To find the union, take the portion of the
  number line representing the total collection of
  numbers in the two graphs.

101
Practice

• Use graphs to find each set

102
How do we solve Linear Inequalities in One
Variable?

• Solve and graph 3- 2x lt 11.
• Try 2-3x lt 5

103
Solving a Linear Inequality

• Solve and graph.
• -2x 4 gt x 5

• Solve and graph.
• 3x 1 gt 7x - 15

104
How do we recognize inequalities with unusual
solution sets?

• Examples x lt x 1
• The solution is all real numbers. Or using
  interval notation its___.
• If you attempt to solve an inequality that has no
  solution sets, you will eliminate the variable
  and obtain a false statement such as 0 gt 1.
• If you attempts to solve an inequality that is
  true for all real numbers, you will eliminate the
  variable and obtain a true statement such as 0 lt
  1.

105
Solving Linear Inequalities

• Solve each inequality.
• 2 (x 4)gt 2x 3

• x 7 lt x - 2

106
Summary Answer in complete sentences.

• When graphing the solutions of an inequality,
  what does a parenthesis signify? What does a
  bracket signify?
• Describe the ways that solving a linear
  inequality is similar to solving a linear
  equation.
• Describe ways that solving a linear inequality is
  different from solving a linear equation.

107
Warm-up 9-20-2011

• Take out your homework and Unit Plan.
• Solve
• Solve. Express answer in interval notation and
  graph.
• C. 18x 45 lt 12x - 8 d. -4 (x 2) gt 3x 20

108
Warm-up 9-21or 9-22

• Take out your homework and Unit Plan.
• Solve each linear inequality.
• Be READY you will be CALLED in 8 minutes to the
  board.

109
Aim 1.7B How do we solve other types of
inequalities?

• Two inequalities, such as
• -3 lt 2x 1 and 2x 1 lt 3
• That can written as a compound inequality.
• -3lt 2x 1 lt 3

110
How do we Solve a Compound Inequality?

• Solve and graph.
• -3 lt 2x 1 lt 3
• Now you try 1lt 2x 3 lt 11

• Goal is to isolate variable.
• Subtract from all parts.
• Simplify.
• Divide each part by 2.
• Simplify.

111
How do we Solve Inequalities with Absolute Value?

• Solving an Absolute Value Inequality
• If X is an algebraic expression and c is a
  positive number.
• The solutions of are the numbers that
  satisfy clt X lt c.
• The solutions of are the numbers that
  satisfy X lt -c or X gt c.
• These rules are valid if lt replaced by lt,
• And gt is replaced by gt.

112
Solving an Absolute Value Inequality

• Given the inequality
• Solve the compound inequality.
• The solution set written in interval notation is
• And the graph is

113
Solving an Absolute Value Inequality

• Solve and graph the solution on a number line

114
Solving an Absolute Value Inequality

• Steps
• Remember to isolate Absolute Value Expression
• Then before you can rewrite without bars

115
Solving an Absolute Value Inequality

• Solve and graph on a number line

116
Solving an Absolute Value Inequality

• Solve and graph the solution set on a number
  line

117

• Solve and graph on a number line

118
Applications

• Acme car rental agency charges 4 a day plus
  0.15 per mile. Interstate rental agency charges
  20 a day and 0.05 per mile. How many miles must
  be driven to make the daily cost of an Acme
  rental a better deal than an Interstate rental?

• First define your variable
• Let x
• Represent both quantities in terms of x.
• Which inequality symbol do we use and why?
• Write and solve the inequality.
• Check your proposed solution.

119
Practice

• A car can be rented from Basic Rental for 260
  per week with no extra charge for mileage.
  Continental charges 80 per week plus 25 cents
  for each mile driven to rent the same car. How
  many miles must be driven in a week to make this
  rental cost for Basic Rental a better deal than
  Continentals?

120
SummaryAnswer in complete sentences.

• Describe how to solve an absolute value
  inequality involving this symbol lt. Give an
  example.
• Describe the solution set of
• Whats wrong with this argument? Suppose x and y
  represent two real numbers, where x gt y

121
Warm-up 9-23-2011

• Take out your homework and Unit Plan.
• Solve each inequality.

122
Please Read DO NOW 9-26-2011

• Take out your Review, Work and Unit Plan.
• Turn in last weeks warm-up in the tray on table.
• Complete Todays Warm-up in your notebook.
• Reminder Crossword and Unit Plan due Tuesday,
  Sept. 27th
• Test Thursday