How to Draw Bar Graph

In the previous post we have discussed about slope worksheets and In today's session we are going to discuss about How to Draw Bar Graph. Statistics is the branch of the mathematics and in statistics we use graphs for calculating the data. In statistics there are different types of graphs such as histogram, line graph, frequency polygon and many more. Bar graph is one of them and also used for calculating data by graphically. They are rectangular in shape and following types of bar graph can be feasible. (know more about Bar chart, here)

I.        simple bar graph

II.        Double bar graph

III.        Divided bar graph

Bar graphs are used to display the categorical type data. This type of data have no order, with the help of bar graph we can represent the numerical data in the pictorial or graphical form. These bars or rectangular shape can be draw horizontally or vertically.
There are some steps to construct the bar graphs.
Step 1: - First of all change the given frequency distribution from inclusive form to exclusive, if it is already in exclusive form then there is no need to change the form.
Step 2: - Then taking suitable scale, make the class interval as base along the x axis.
Step 3: - After that make the respected frequencies as heights along y axis.
Step 4: - Repeat second and third step until all the frequency distribution is finish.
Step 5: - Now we will create the bars by taking the base and heights. We can color and label them also.
Suppose we have some names as a base and their age as heights. Than we can make the bar graph by taking names at x axis and age at y axis.



Combinations and permutations have the great importance in modern mathematics. We use combinations when order does matter and we use permutations when order does not matter.
Syllabus of CBSE board includes all of those units that is very essential for the growth of students.

slope worksheets

A slope defines the inclination of a line or the steepness of the line. In general slope can be defined as the proportionate ratio of rise when divided by the run among two points on the line. The slope worksheets helpful in estimating the points of a line, which in the plane consisting of the x and y axes which is represented by the letter 't' and is defined as the difference in y coordinate by the difference in x coordinate among the two distinct point on the line. We can also express it as
         âˆ†y      rise
t =  ------ = -------- .
         âˆ†x        run
here the Greek symbol âˆ† (pronounced as delta) is used to show the change or the difference.
Let us assume that ,If the two points are given (x1,y1) and (x2,y2) then the variation in x from one to the another will be considered as x2 - x1 ( taking as run) whereas the change in y will be y2 - y1 (considered as rise) now the formula generate will takes place which is given below:
                 y2 - y1
       t = ----------
               x2 - x1
In the case of vertical line this formula will not work.
To understand this in the more precise manner we will consider one illustration:
suppose a line runs through two points : s = (4,1) and t(13,8) now as stated in the above formula we will follow the following steps.
           âˆ†y        y2 - y1
   m = ------ = -----------
           âˆ†x        x2 - x1

            8 – 4
       = -------- (place the value in the formula)
           13 – 1

       = 4/12 (calculation will be conducted)
       = 1/3
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how to solve inequalities with fractions

Before discuss the inequality with fraction, it is necessary to know about the inequality and fractions. So first we see the equation of inequality look like, what are the conditions of inequality.
If we have two variable y and z, then there is many more condition for the inequality equations.
Suppose y is less than equal to z;
⇒y ≤ z;
Inequality present;
y is greater than equal to z then we can write:
⇒y ≥ z;
And if y is greater than z then we can write;
⇒y > z;
And if y is smaller than z then we can write;
⇒y < z;
These all are the properties of inequality. If in any equation these symbols are present then we can say there is an inequality in the expression otherwise the expression is simple equation.
And any number which is written in the form of o/p, that type of number is known as fraction. Now we will see how to solve inequalities with fractions? To solve the inequality with fraction, we change it into an equation without fraction. By this technique we know how to solve the equation. This technique is said to be clearing.
Before solving these types of equation you have to recall all the rules for adding, subtraction, multiplying and dividing fractions.
     In these types of equations, firstly we take the constant term to one side of the equation and variables are taken on another side. And solve the equation and after solving we get the value of unknown variables.
There are some steps which explain how to solve the inequality equations with fractions:
Step1: First we will Multiply both side of the equation by both LCM of the denominator.
Step 2: Then after solving every denominator will then cancel, after this we will get the equation without Fractions.
Step3: After that we find the L.C.M. and multiply the L.C.M. on both side of the equation.
We have to follow these steps to solve the equation.
 By using above steps we will see some of the examples which are given below
Example: -    P + P – 4 ≤ 9; solve the inequality equations with fractions?
                     5     6

Solution: - By using all the above steps we can easily solve this inequality.

                       P  + P – 4   ≤ 9;
                       5         6
 The LCM of 5 and 6 is 30. Therefore multiply every term on both sides by the value 30.

       6P + 5P –20 ≤ 9;
              30
⇒6P + 5P – 20 ≤ 30 * 9;
Now we add all the like term which are present in the equation:
⇒11P – 20 ≤ 270;
On further solving this equation we get
 ⇒11p ≤ 270 + 20;
 11P ≤ 290;
 P ≤ 290/ 11;
P ≤ 26.36;
After solving the inequality we get the value of P is greater then 26.36;
Example 2: -    P   - 4P   ≥ 1; solve the equations with fractions?
                        3       2       4
 Solution: -       P - 4P   ≥ 1;
                        3      2       4
Multiply both side of the equation by both LCM of the denominator.
 After solving every denominator will then cancel, after this we will get the equation without Fractions. (Know more about inequalities in broad manner, here,)
 The LCM of 3, 2 is 6. Therefore multiply every term on both sides by the value 6.

         P – 4P ≥ 1,
         3    2      4

         2P – 12P ≥ 1;
               6          4

           -10P ≥ 6 / 4;

           -40P ≥ 6;
            P ≥ 6 / -40;
           P ≥ -0.15;
After solving the inequality equation we get the value of P is less than 0.15;
Now we see Binomial Probability Formula:
= (n) x^p y ^{n - p}
   (p)
Where n is number of trials:
p is number of successors;
n - p is number of failures;
These all are used in board of secondary education ap ,
In the next session we will discuss about Compound Inequalities and if anyone want to know about Multiplying Rational Expressions then they can refer to Internet and text books for understanding it more precisely.

Inequalities

Previously we have discussed about variables and expressions worksheets and In today's session we are going to discuss about Inequalities which is a part of ap state intermediate board, In this part we Simplify expressions and statements. Some of them are equal to each other and some of them are not equal to each other. Inequalities contain those expressions and statements that are not equal to each other, they can be strictly greater than and only greater than, strictly less than and only less than and not equal too, but they cannot be equal. Some notations that are used in inequalities are:-
-The sign x < y denotes that x is less than y.
-The sign x > y denotes that x is greater than y.
-The sign x != y denotes that x is not equal to y. But in this notation we cannot say that which is greater than or less than to each other.
In all the above mentioned statements x is not equal to y, it means that they are strict inequalities. Let’s see more inequality statements that are not strict.
-The x ≤ y statement means that x is less than or equal to y.
-The x ≥ y statement means that x is less than or equal to y
We can use more inequalities statements that are much strict, they are:-
- The x << y means x is much less than y.
-The x >> y means x is much greater than y.
Now the question arises how to solve inequalities, to solve this we have to follow some procedure that is:-
-First of all same number should be add and subtract from both sides.
-Secondly shifting the sides and changing the adjustment of inequality sign.
-The same positive number should be multiply/divide from both the ends.
-The same negative number should be multiply/divide from both the ends and changing the adjustment of inequality sign.(Know more about Inequalities in broad manner, here,)
Remembering above rules helps to solve inequalities in an easy manner.
In the next session we are going to discuss Compound Inequalities
and if anyone want to know about Equations with no Solution then they can refer to Internet and text books for understanding it more precisely.

Compound Inequalities

Previously we have discussed about antiderivative of sin2x and In today's session we are going to discuss about Compound Inequalities which comes under board of intermediate education ap, Equations are a combination of two or more variables with the numbers. Solving Absolute Value Equations by using various types of properties and different type of operations is shown here. In this session, we will talk about inequalities, especially Compound Inequalities. An inequality is similar to an equation as we solve the inequality by adding or subtracting the variables from it. The difference is that we use comparison operators (>, <, >=, <=, ≠) rather than equality symbol (=). Here we are going to discuss about the concept of solving Compound Inequalities.
We can define Compound Inequalities as the combination of two or more inequalities bound with the ‘and’ or ‘or’ symbol.
Let’s show you the example of inequalities:
Suppose two inequalities are given:
 (a) 5y – 4 < 7
 (b) y  + 12 > 13
Then both the above inequalities can be represented as Compound Inequalities as:
             5y – 4 < 7 and y + 12 > 13
The above representations of inequalities are known as compound inequalities or generally known as conjunction of inequalities.(want to Learn more about Inequalities, click here),
 Example2: (a) 5y > 65
                    (b) m + 7 < 3
The above inequalities can be represented in below given format:
                5y > 65  or  m + 7 < 3
The above representations of inequalities are known as disjunction of inequalities. Now we show you how to solve the compound inequalities:
Example: Solve the given compound inequalities 5y – 4 < 7 and y + 12 > 13 ?
Solution: Here we solve the inequalities in different way:
       =>          5y – 4 < 7
Now we add 4 in both sides
ð 5y – 4 + 4 < 7 + 4
Here – 4 and + 4 cancelled to each other:
ð 5y < 11
   Divide both sides by 5
ð y < 2.2
Now solve second inequality
y + 12 > 13
Now we subtract 12 from both sides:
  y + 12 – 12 >  13 – 12
        y  > 1
In the next session we are going to discuss Inequalities and Read more maths topics of different grades such as Equations with no Solution in the upcoming sessions here.

Equations and Inequalities

Previously we have discussed about rational expression calculator and In today's session we are going to discuss about Equations and Inequalities which comes under cbse books for class 11,  Equations are defined as the combination of number and variables that have an equal sign and both sides of an equation must be equal to each other .To solve the equation or for finding the values of variable of a equation are same in meaning .It means that when find the values in form of real number and when it is substituted then it will provide the identity as an example a given equation
3 ( a + 4 ) = -4 ( 2 – 2 a )
By simplify it we get
3 a + 12 = - 8 - 8 a
- 5 a = - 20
a = 4 ( dividing the both side of equation by – 5 ) .
But when we talk about the inequalities, all the rules of equations will be applied except some of the rules of division or multiplication by a negative number .inequalities can be understood by an example :
3 < 4 is multiplied by - 5 then it gives
3 * - 5 > 4 * - 5
- 15 > - 20 means in solving the inequality or finding the values of the variable the solution belongs to an interval of real numbers . (Know more about  Inequalities in broad manner, here,)
Some example of inequalities that describe the rules are as follows :
example : An inequality - ( 3 + a ) < 2 ( 3 a + 2 )
By simplifying it we get
-3 – a < 6 a + 4
-a – 6 a < 4 + 3
- 7 a < 7
By dividing the ( - 7 ) , we get a > -1 . In form of interval notation it is written as ( - 1 , infinity ).
In the next session we are going to discuss Compound Inequalities and Read more maths topics of different grades such as Rationalizing the Denominator in the upcoming sessions here.

Solving Multi Step Inequalities

Previously we have discussed about statistics worksheets and In today's session we are going to discuss about Solving Multi step inequalities which is a part of cbse 11th syllabus and is same like Solving Multistep Equations, It includes one or more operations that can be solved by avoiding the operations in reverse order. It is just like solving the equations with one or more than one operations. An inequality or an equation with one or more operations has two steps to solve it that are as follows:
( a ) By taking the inverse of addition or subtraction .
( b ) By using the inverse of division or multiplication for simplifying it.
It should be remembered at the time of solving the inequalities that while multiplying or dividing with the negative numbers the symbol of inequalities is reversed.
We can take an example for describe the process of solving the inequalities:
Here we take an equation with two step inequalities that means it has two operations and has two step solutions : 3 p – 10 >= 14
Here in two step solutions, we start with the variable p and understand it step by step .(want to Learn more about Inequalities, click here),
In the above equation variable p is multiplied with number 3 as ' 3*p ' and then number 10 is subtracted from the term 3p as 3 p - 10 . After it getting the answer 14 that is written as
3 p -10 > 14 .So there are steps that follow is start with the variable p then multiply by 3 then subtract 10 and at last equal to 14 .
So , for solving an inequality goes from the backward side and using the reverse operations. Start with the last step that is result 14 .
Now , by follow the reverse process, 10 is added to the 14 means 14 + 10 that is the reverse process of the subtracting 10 .
Next , the inverse of multiplying by 3 is divide by the 3 that is p > = ( 14 + 10 ) / 3
p > = 24 / 3
p > = 8 .
At last 8 is the answer of given inequality that is greater than equal to 8 .
In the next session we are going to discuss Equations and Inequalities and if anyone want to know about Math Blog on Subtracting Rational Expressions then they can refer to Internet and text books for understanding it more precisely.