EQUATIONS, INEQUALITIES

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EQUATIONS, INEQUALITIES ABSOLUTE VALUE

• CHAPTER 2
• DCT1043

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CONTENT

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

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2.1 Linear Equations
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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

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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

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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.

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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

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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)

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Solve a Linear Equation

• Solve the following equations

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Solve equations that lead to linear equations

• Solve the following equations

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An equation with no solution

• Solve the following equations

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Translating Written/Verbal Information into a
Mathematical Model
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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?

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2.2 Quadratic Expression Equations
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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

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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

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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

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Solve a Quadratic Equation by the Square Root
Method

• Solve the following equations

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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

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Solve a Quadratic Equation by the Method of
Completing the Square

• Solve the following equations by using completing
  the square method

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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

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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)

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Examples

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

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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.

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2.3 Inequalities
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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

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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
  

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Solving Linear Inequalities

• Solve the following inequality and graph the
  solution set

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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

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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?

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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

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Examples

• Solve the following inequality and graph the
  solution set

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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

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Examples

• Solve the following inequality and graph the
  solution set

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2.4 Absolute Value
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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

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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
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Properties of Absolute Value

• For any real number a and b

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Properties of Absolute Value

• Equations involving absolute value

• Inequalities involving absolute value

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Solve equations involving absolute value

• Solve the following equation

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Solve inequalities involving absolute value

• Solve the following inequalities. Graph the
  solution set

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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

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Thank You