S2TN Properties and Solving Inequalities

Solving Inequalities                                                            S2 – N7           (S1 – N7 – O)

Recap: 

Symbols

                        >    greater than

                        <    less than

                        ≥    greater than or equal to

                        ≤    less than or equal to

                        =    equal

                        ≠  not equal to

Tip to remember

                                                   4  >  3

        greater = "open mouth"       >      “point” = less than

  

Similarly,

                                                   3  <  4

                  less than = “point”     <       greater = "open mouth"            

Example

    

            Solve 7 < 4x – 3 < 13

 

Step1: Remove number from the expression with x [ 4x – 3]

          [+ 3 to all the expression so that the inequalities remain the same]

 

            7 + 3 < 4x – 3 + 3 < 13 + 3

                10 < 4x < 16

 

Step2: Remove the numerical coefficient of x

                        10 < 4x < 16

                         4     4       4

2 ½ < x < 4

Multiplication property of Inequalities

     x +ve number

               if a < b and c > 0, then ac < bc    eg: 2 < 3 , x 3 => 6 < 9

               If a > b and c > 0, then ac > bc    eg: 3 > 2,  x 3 => 9 > 6

            

      x  –ve number

            If a < b and c < 0, then ac > bc       eg: 2 < 3 , x -3 => -6 > -9 

  => the sign changed from < to >

          

  If a > b and c < 0, then ac < bc       eg: 3 > 2 , x -3 => -6 < -9

  => the sign changed from > to <

   

             [[ x –ve    : change sign from < to > or > to <       ]]

Example

Find the value of x  

a.           5x ≥ 15                                   

               x ≥ 5            ( Divide both side by +5)                    

b.         -2x > 6

              -x < -3               (Divide both side by 2)

             -1 x -x > -3 x -1  (We need to solve for x, and not -x)

               x < 3                 (multiply by -1 => change sign )

S2 - G1 :Special Quadrilaterals and Regular Polygons

Interior and exterior angles of polygon

    

       

Sum of interior angles  = (n-2) x 180    (n=number of interior angles)

         

 Sum of exterior angles of any polygon = 360^o

  A Regular Polygon has equal sides and equal angles.

  Each external angle of any regular n-sided polygon = 360/n

  Each Exterior angle + interior angle = 180^o

Common Polygon Name

            No. of sides              Name of Polygon       Sum of Interior Angles

                         3                        Triangle                    180^o

                         4                        Quadrilateral             360^o

                         5                        Pentagon                  540^o

                         6                        Hexagon                   720^o

                         7                        Heptagon                  900^o

                         8                        Octagon                  1080^o

                         9                        Nonagon                 1260^o 

                        10                       Decagon       1440^o

                        12                       Dodecagon       1880^o

S2/3-S1: Data Analysis

 DATA ANALYSIS 

Average = mean = Add all numbers/ how many numbers there are

Mode = the number/things that appears the most time.

Median = middle number in a ‘sorted list of values’

Example: 

Number list [ 2,3,9,3,6,2,3,4]

Average/Mean = 2+3+9+3+6+2+3+4 / 8 = 4

Arranged Number list [2,2,3,3,3,4,6,9]

Mode = 3 (appear the most number of times)

Median = (3 + 3) / 2 = 3 (for even list of number0

If arranged number list is [2,2,3,3,4,5,6,9,9]

        median = 4 [(Total number in list – 1) / 2]

From Frequency Table

            

Mean = (0x3) + (1x6) + (2x0) + (3x2) + (4x1) / (3+6+0+2+1)

Mode = 2 (has the highest frequency)

Median:

Total Frequency = 3+6+2+1 = 12

For even frequency, position of median number is 12/2 = 6 => Position is 6 and 7.

Median = 1

For odd frequency, position of median number is (total frequency -1)/2

Probability of Single Event

What is Probability?

• the likelihood that an event will happen from all the possible outcome.

   

What is an event?

An “event” can be one or more outcomes. 

A dice has 6 faces - 1, 2, 3, 4, 5 and 6. 

What a dice is tossed, one of six events can happen:- the number is 1, 2, 3, 4, 5, or 6

Probability

Probability = Positive events / Total number of possible outcomes

Example

When a dice is tossed, what is the probability of having the number 2?

Dice Number                 1        2        3        4        5        6

Number(Event)              1        1        1        1         1       1

Probability                   1/6    1/6     1/6    1/6     1/6      1/6 

=>

Total number of events = 6     [Step1: Total possible outcome]

No. 2 happens = 1                  [Step2: Count of positive event ]

Probability of number is 2 = 1/6    [Step3:Compute probability]

Also, probability is

     ~ expressed as a value between 0 and 1. 

     ~ an event that will not happen at all (0%), probability = 0

     ~ an event that will definitely happen(100%), probability  = 1

     ~ Sum of Probability of all possible events = 1

Example

There are 2 red balls and 1 blue ball in a basket. What is the probability of picking up a red ball from the basket?

Colour                R        R        B

Event                  1        1        1   (total = 3)

Probability        1/3     1/3     1/3

  Event of Red = 2, Probability = 1/3+ 1/3 = 2/3 or 2/3

=>

Total events = 3         [Step1: red ball, red ball, blue ball]

Picking a red ball = 2 [Step2: positive event = 2]

Probability = 2/3         [Step3: Compute probability]

Probability with a Frequency Distribution Table

Example

The following is the data of people going to Johore Bahru

  Transport Frequency

             Walk 40

             Train          24

             Bus            56

What is the probability of a person walking to Johore Bahru?

   Number of people walking = 40   [Step1: positive event]

   Total number of people = 40 + 24 + 56 [Step 2: Total]

                                         = 120

   Probability = 40/120 = 1/3            [Step 3: Compute]

   

N63-G23 Graphs of Power Functions

Graph of power functions y = ax^n (n = -2, -1, 0, 1, 2, 3)

Graph of y = ka^x(a is positive integer)

N52-G23 Expansion of Linear Expressions

Expansion of Linear Expressions

Expansion usually involves removing the bracket.

=>   a (b + c) = ab + ac

Expand 3( p + 2q)

    3(p + 2q) 

    = 3 x p + 6 x q                 (Step : Use a ( b + c) = ab + ac)

    = 3p + 6q

Example

(i) Expand        4 (a+ b) 

                        = 4 x a + 4 x b           (Step : Use a ( b + c) = ab + ac)

                        = 4a + 4b

            

(ii) Expand        4 + a

                           5          

                        =  1 x (4 + a).       (Step : Use a ( b + c) = ab + ac)

                            5                                

                        =  4 + 1a 

                            5    5

                        =  4 + a

                            5    5

Examples

Expand and simplify

6(a + b) - 3a(2 - 5a)

        = 6a + 6b - 6a +15a² Step 1: Open/remove bracket

  = 6a + 6b - 6a  + 15a² Step 2 : Group and simplify

= 15a² +6b

Example

(1) Simplify (6y²– 4y – 3) – 2(2y²– 3y + 2)

           (6y²– 4y – 3) – 2(2y²– 3y + 2)      Step 1: Open bracket

         = 6y²– 4y – 3 – 4y²+ 6y – 4                          ( -2 x -3y = 6x [- x - = +] )

         = 6y²– 4y²– 4y + 6y – 3 – 4           Step 2: Group variables (y²with y², y, numbers)

         = 2y²+ 2y – 7

(2) Simplify (2x² + 2) + (3x² + x + 1)

2x² + 2 + 3x² + x + 1 (Step 1 : Open bracket)

        = 2x² + 3x² + x + 1 + 2                     (Step2: Group x², x , numbers)

        = 5x² + x + 3

(3) Simplify and factorise (2x²+ 3x - 7) + (3x²– 4x + 3) 

              2x²– 3x + 4 + 3x²– x  - 3             (Step 1 : Open bracket)

          =  2x² + 3x²+ 3x – 4x - 7 + 3          (Step2: Group x², x , numbers)

          =  5x²– x - 4 (Step3 : Factorise)

          =  (5x - 4)(x - 1)

Expansion of 2 Linear Expressions

                                    

                        ( a + b) ( c + d) =   ac + ad     +   bc + bd                         

                                                         /\                        /\

                                             [      a x (c + d)  +    b x (c + d)    ]

 It is also called the "rainbow" arrow.

Example

Expand (y + 2)(y – 3)

Practice

1.   Expand the followings:

a.  5a(3 + b)

b.  2(9a - 2b)

c.  4(1 + 2a)

d.  7(3b - 4)

2.  Expand the followings

a.  (2a + b)(a + b)

b.  (3a - b)(2a + b)

c.  (4a - 2b)(a - b)

d.  (2a + 7b)(2a - 3b