EQUATIONS, INEQUALITIES - PowerPoint PPT Presentation

1 / 43
About This Presentation
Title:

EQUATIONS, INEQUALITIES

Description:

A statements in which 2 expressions (sides) at least one containing the variable ... clear the equation of fractions by multiplying both sides by the least ... – PowerPoint PPT presentation

Number of Views:411
Avg rating:3.0/5.0
Slides: 44
Provided by: notesU
Category:

less

Transcript and Presenter's Notes

Title: EQUATIONS, INEQUALITIES


1
EQUATIONS, INEQUALITIES ABSOLUTE VALUE
  • CHAPTER 2
  • DCT1043

2
CONTENT
  • 2.1 Linear Equation
  • 2.2 Quadratic Expression and Equations
  • 2.3 Inequalities
  • 2.4 Absolute value

3
2.1 Linear Equations
4
Objectives
  • At the end of this topic, you should be able to
  • Define linear equations
  • Solve a linear equation
  • Solve equations that lead to linear equations
  • Solve applied problems involving linear equations

5
Equation in one variable
  • A statements in which 2 expressions (sides) at
    least one containing the variable are equal
  • It may be TRUE or FALSE depending on the value of
    the variable.
  • The admissible values of the variable (those in
    the domain of the variable), if any, that result
    in a TRUE statement are called solutions or root.
  • To solve an equation means to find all the
    solutions of the equation

6
Equation in one variable, cont
  • An equation will have only one solution or more
    than one solution or no real solutions or no
    solution
  • Solution set the set of solutions of an
    equation, a
  • Identity An equation that is satisfied for
    every value of the variable for which both sides
    are defined
  • Equivalent equations Two or more equations that
    have the same solution set.

7
Linear Equations
  • A Linear Equation in one variable is equivalent
    to an equation of the form
  • where a and b are real numbers and
  • The linear equation has the single solution given
    by the formula
  • Simplify the given equations first, to solve a
    linear equations

8
Steps for Solving a Linear Equation
  • STEP 1 If necessary, clear the equation of
    fractions by multiplying both sides by the least
    common multiple (LCM) of the denominators of all
    the fractions.
  • STEP 2 Remove all parentheses and simplify
  • STEP 3 Collect all terms containing the variable
    on one side and all remaining terms on the other
    side.
  • STEP 4 Check your solution (s)

9
Solve a Linear Equation
  • Solve the following equations

10
Solve equations that lead to linear equations
  • Solve the following equations

11
An equation with no solution
  • Solve the following equations

12
Translating Written/Verbal Information into a
Mathematical Model
13
Solve applied problems involving linear equations
  • Example 1
  • A total of RM18000 is invested, some in stocks
    and some in bonds. If the amount invested in
    bonds is half that invested in stocks, how much
    is invested in each category?
  • Example 2
  • Amy grossed RM435 one week by working 52 hours.
    Her employer pays time-and-half for all hours
    worked in excess of 40 hours. With this
    information, can you determine Amys regular
    hourly wage?

14
2.2 Quadratic Expression Equations
15
Objectives
  • At the end of this topic you should be able to
  • Define quadratic expressions and equations
  • Solve quadratic equations by factorization,
    square root method, completing the squares and
    quadratic formula
  • Recognize the types of roots of a quadratic
    equation based on the value of discriminant
  • Solve applied problems involving quadratic
    equations

16
Quadratic Equations
  • A Quadratic Equation in is an equation equivalent
    to one of the form
  • where a, b and c are real numbers and
  • A Quadratic Equation in the form
  • is said to be in standard form
  • 4 ways to solve quadratic equations
  • a. Factoring b. Square
    root method
  • c. Completing the square c. Quadratic Formula

17
Solve a Quadratic Equation by Factoring
  • Solve the following equations
  • Repeated Solution / root of multiplicity 2 /
    double root
  • When the left side, factors into 2 linear
    equations with same solution

18
Solve a Quadratic Equation by the Square Root
Method
  • Solve the following equations

19
Solve a Quadratic Equation by the method of
Completing the Square
  • Adjust the left side of a quadratic equation, so
    that it becomes a perfect square (the square of
    first degree polynomial).
  • STEP

20
Solve a Quadratic Equation by the Method of
Completing the Square
  • Solve the following equations by using completing
    the square method

21
Solve a Quadratic Equation by the Quadratic
Formula
  • Use the method of completing the square to obtain
    a general formula for solving the quadratic
    equation
  • Solve the following equations

22
Discriminant of a Quadratic Equation
  • For a Quadratic Equation
  • If
  • there are two unequal real solutions
  • If
  • there is a repeated solution, a root of
    multiplicity 2
  • If
  • there is no real solution (complex roots)

23
Examples
  • Find a real solutions, if any, of the following
    equations

24
Application of Quadratic Equations
  • Example 1
  • The quadratic function
  • models the percentage of the U.S. population
    f (x), that was foreign-born x years after 1930.
    According to this model, in which year will 15
    of the U.S. population be foreign-born?
  • Example 2
  • The height of projectile at any given time is
    given by the equation
    , where h is the elapsed time in seconds, and
    v is the initial velocity in feet per second. The
    constant k represents the initial height of the
    object above ground level, as when a person
    releases an object 5 ft above ground in a
    throwing motion. If the person were on a hill 60
    ft, k would be 65 ft. A person standing on a hill
    60 ft high, throws a ball upward with initial
    velocity of 102 ft/sec. How many seconds until
    the ball hits the ground at the bottom of the
    hill.

25
2.3 Inequalities
26
Objectives
  • At the end of this topic you should be able to
  • Relate the properties of inequalities
  • Define and Solve linear inequalities
  • Define Solve quadratic inequalities
  • Understand and solve rational inequalities
    involving linear and quadratic expression

27
Properties of Inequalities
  • If a lt b and b lt c then a lt c
  • If a lt b and c is any number, then a c lt b c
  • If a lt b and c is any number, then a c lt b c
  • If a gt 0 and b gt 0 then a b gt 0
  • If a gt 0 and b gt 0 then ab gt 0
  • If a lt b then b a gt 0
  • If a gt b and a lt b
  • If a lt b and a gt b
  • If a lt b and c gt 0 then ac lt bc
  • If a lt b and c lt 0 then ac gt bc

  • reciprocal property
  • reciprocal
    property

28
Solving Linear Inequalities
  • Solve the following inequality and graph the
    solution set

29
Solves problems involving linear inequalities
  • At least, minimum of, no less than
  • At most, maximum of, no more than
  • Is greater than, more than
  • Is less than, smaller than

30
Examples
  • Sashas grade in her math course is calculated by
    the average of four tests. To receive an A for
    this course, she needs an average at least 89.5.
    If her current test scores are 84, 92, and 94,
    what range of scores can she make on the last
    test to receive an A for the course?
  • A painter charges RM80 plus RM1.50 per square
    foot. If a family is willing to spend no more
    than RM500, then what is the range of square
    footage they can afford?

31
Solving Quadratic inequalities
  • Step 1 - solve the related quadratic equation
  • Step 2 plot the solution on a number line
  • Step 3 Choose a test number from each interval
    substitute the number into the inequality
  • If the test number makes the inequality true
  • All numbers in that interval will solve the
    inequality
  • If the test number makes the inequality false
  • No numbers in that interval will solve the
    inequality
  • Step 4 State the solution set of the inequality
    ( It is a union of all intervals that solves the
    inequality)
  • If the inequality symbols are or ,
    then the values from Step 2 are included.
  • If the symbols are gt or lt, they are not solutions

32
Examples
  • Solve the following inequality and graph the
    solution set

33
Solving rational inequality
  • STEP 1 Solve the related equation
  • STEP 2 Find all values that make any denominator
    equal to 0
  • STEP 3 Plot the number found in Step 1 and 2 on
    a number line
  • STEP 4 Choose a test number from each interval
    and determine whether it solves the inequality.
  • STEP 5 The solution set is the union of all
    regions whose test number solves the inequality.
    If the inequality symbol is or , includes
    the values found in step 1
  • STEP 6 The solution set never includes the
    values found in Step 2 because they make the
    denominator equal to 0

34
Examples
  • Solve the following inequality and graph the
    solution set

35
2.4 Absolute Value
36
Objectives
  • At the end of this topic you should be able to
  • Define absolute value
  • Understand, state and use the properties of
    absolute value
  • Solve problems on equations and inequalities
    involving absolute value

37
What is Absolute Value
  • The absolute value can be define as
  • The absolute value represents the distance of a
    point on the number line from the origin

a
- a
38
Properties of Absolute Value
  • For any real number a and b

39
Properties of Absolute Value
  • Equations involving absolute value
  • Inequalities involving absolute value

40
Solve equations involving absolute value
  • Solve the following equation

41
Solve inequalities involving absolute value
  • Solve the following inequalities. Graph the
    solution set

42
Application of Absolute value
  • The inequality
  • describes the percentage of children in the
    population who think that being grounded is a bad
    thing about being kid. Solve the inequality and
    interpret the solution

43
Thank You
Write a Comment
User Comments (0)
About PowerShow.com