INEQUALITIES

INEQUALITIES

INTRODUCTION

An inequality says that two values are not equal. 

a ≠ b says that a is not equal to b

There are other special symbols that show in what way things are not equal.

a < b says that a is less than b

a > b says that a is greater than b

(those two are known as strict inequality) 

a ≤ b means that a is less than or equal to b

a ≥ b means that a is greater than or equal to b.

SOLVING INEQUALITIES

Sometimes we need to solve Inequalities like these:

Symbol	Words	Example
>	greater than	x + 3 > 2
<	less than	7x < 28
≥	greater than or equal to	5 ≥ x - 1
≤	less than or equal to	2y + 1 ≤ 7

SOLVING

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

Something like:		x < 5
or:		y ≥ 11

We call that "solved".

EXAMPLES SOLVING INEQUALITIES

Example 1:
Alex and Billy have a race, and Billy wins!
What do we know?
We don't know how fast they ran, but we do know that Billy was faster than Alex:

Billy was faster than Alex

We can write that down like this:

b > a

(Where "b" means how fast Billy was, ">" means "greater than", and "a" means how fast Alex was).
We call things like that inequalities (because they are not "equal").

Example 2:
Alex plays in the under 15's soccer. How old is Alex?
We don't know exactly how old Alex is, because it doesn't say "equals"
But we do know "less than 15", so we can write:

Age < 15

The small end points to "Age" because the age is smaller than 15.

Example 3 is equal to:
You must be 13 or older to watch a movie.
The "inequality" is between your age and the age of 13.
Your age must be "greater than or equal to 13", which is written:

Age ≥ 13

Exercise :

Solve x + 5 < 12

Below is the video of Inequalities:

ref:

http://www.mathsisfun.com/definitions/inequality.html

http://www.mathsisfun.com/algebra/inequality-solving.html

https://youtu.be/lo1kvxu-Dc8

SET & VENN DIAGRAM

SET & VENN DIAGRAM

SETS

A set is a collection of things.

For example, the items you wear is a set: these would include shoes, socks, hat, shirt, pants, and so on.

You write sets inside curly brackets like this:

{socks, shoes, pants, watches, shirts, ...}

You can also have sets of numbers:

• Set of whole numbers: {0, 1, 2, 3, ...}
• Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

EXAMPLE:

Ten Best Friends

You could have a set made up of your ten best friends:

• {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}

Each friend is an "element" (or "member") of the set (it is normal to use lowercase letters for them.)

Now let's say that alex, casey, drew and hunter play Soccer:

Soccer = {alex, casey, drew, hunter}

(The Set "Soccer" is made up of the elements alex, casey, drew and hunter).

And casey, drew and jade play Tennis:

Tennis = {casey, drew, jade}

You could put their names in two separate circles:

UNION

You can now list your friends that play Soccer OR Tennis.

This is called a "Union" of sets and has the special symbol ∪:

EXAMPLE:

Soccer ∪ Tennis = {alex, casey, drew, hunter, jade}

Not everyone is in that set ... only your friends that play Soccer or Tennis (or both).

We can also put it in a "Venn Diagram":

Venn Diagram: Union of 2 Sets

A Venn Diagram is clever because it shows lots of information:

• Do you see that alex, casey, drew and hunter are in the "Soccer" set?
• And that casey, drew and jade are in the "Tennis" set?
• And here is the clever thing: casey and drew are in BOTH sets!

INTERSECTION

"Intersection" is when you have to be in BOTH sets.

In our case that means they play both Soccer AND Tennis ... which is casey and drew.

The special symbol for Intersection is an upside down "U" like this: ∩

And this is how we write it down:

EXAMPLE:

Soccer ∩ Tennis = {casey, drew}

In a Venn Diagram:

Venn Diagram: Intersection of 2 Sets

DIFFERENCE

You can also "subtract" one set from another.

For example, taking Soccer and subtracting Tennis means people that play Soccer but NOT Tennis ... which is alex and hunter.

And this is how we write it down:

EXAMPLE:

Soccer − Tennis = {alex, hunter}

In a Venn Diagram:

Venn Diagram: Difference of 2 Sets

Summary So Far

∪ is Union: is in either set
∩ is Intersection: must be in both sets
− is Difference: in one set but not the other

THREE SETS

You can also use Venn Diagrams for 3 sets.

Let us say the third set is "Volleyball", which drew, glen and jade play:

Volleyball = {drew, glen, jade}

But let's be more "mathematical" and use a Capital Letter for each set:

• S means the set of Soccer players
• T means the set of Tennis players
• V means the set of Volleyball players

The Venn Diagram is now like this:

Union of 3 Sets: S ∪ T ∪ V

You can see (for example) that:

• drew plays Soccer, Tennis and Volleyball
• jade plays Tennis and Volleyball
• alex and hunter play Soccer, but don't play Tennis or Volleyball
• no-one plays only Tennis

We can now have some fun with Unions and Intersections ...

This is just the set S

S = {alex, casey, drew, hunter}

This is the Union of Sets T and V

T ∪ V = {casey, drew, jade, glen}

This is the Intersection of Sets S and V

S ∩ V = {drew}

And how about this ...

• take the previous set S ∩ V
• then subtract T:

This is the Intersection of Sets S and V minus Set T

(S ∩ V) − T = {}

Hey, there is nothing there!

That is OK, it is just the "Empty Set". It is still a set, so we use the curly brackets with nothing inside: {}

The Empty Set has no elements: {}

UNIVERSAL SET

The Universal Set is the set that contains everything. Well, not exactlyeverything. Everything that we are interested in now.

Sadly, the symbol is the letter "U" ... which is easy to confuse with the ∪ for Union. You just have to be careful, OK?

In our case the Universal Set is our Ten Best Friends.

U = {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}

We can show the Universal Set in a Venn Diagram by putting a box around the whole thing:

Now you can see ALL your ten best friends, neatly sorted into what sport they play (or not!).

And then we can do interesting things like take the whole set and subtract the ones who play Soccer:

We write it this way:

U − S = {blair, erin, francis, glen, ira, jade}

Which says "The Universal Set minus the Soccer Set is the Set {blair, erin, francis, glen, ira, jade}"

In other words "everyone who does not play Soccer".

COMPLEMENT

And there is a special way of saying "everything that is not", and it is called "complement".

We show it by writing a little "C" like this:

S^c

Which means "everything that is NOT in S", like this:

S^c = {blair, erin, francis, glen, ira, jade}
(just like the U − C example from above)

SUMMARY

∪ is Union: is in either set
∩ is Intersection: must be in both sets
− is Difference: in one set but not the other
A^c is the Complement of A: everything that is not in A
Empty Set: the set with no elements. Shown by {}
Universal Set: all things we are interested in

ref: https://www.youtube.com/watch?v=rRk0d6P4oUI

PROBABILITY

PROBABILITY



How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin  When a coin is tossed, there are two possible outcomes: heads (H) or tails (T) We say that the probability of the coin landing H is ½. And the probability of the coin landing T is ½.

Throwing Dice  When a single die is thrown, there are six possible outcomes:1, 2, 3, 4, 5, 6. The probability of any one of them is 1/6.

Probability 

In general:

Probability of an event happening = Number of ways it can happenTotal number of outcomes

EXAMPLES:

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 16

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 45 = 0.8

Probability Line

We can show probability on a Probability Line:

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads.

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Example 1:

The chances of rolling a "4" with a dice:

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 16

Example 2:
There are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?


Number of ways it can happen: 4 (there are 4 blues)
Total number of outcomes: 5 (there are 5 marbles in total)
So the probability = 45 = 0.8

ref: https://www.youtube.com/watch?v=uzkc-qNVoOk

MEASURES OF DISPERSION - Part 2

3.  Variance:  To find the variance:
     • subtract the mean, , from each of the values in the data set, .
     • square the result
     • add all of these squares
     • and divide by the number of values in the data set.

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4.  Standard Deviation:  Standard deviation is the square root of the variance.  The formulas are:

Mean absolute deviation, variance and standard deviation are ways to describe the difference between the mean and the values in the data set without worrying about the signs of these differences. These values are usually computed using a calculator.

ref : http://www.regentsprep.org/regents/math/algtrig/ats1/dispersion.htm