How to solve linear inequalities

In the previous post we have discussed about Absolute Value Inequalities and In today's session we are going to discuss about How to solve linear inequalities. If graph of an equation is a straight line then equation is called as linear equation. For example: q = mp + c; here ‘m’ shows the slope of line and ‘c’ shows Y- intercept where line crosses q – axis (here ‘p’ is along to the horizontal axis and ‘q’ is along to vertical axis. If (<, >) these symbols are there in a linear expression then it comprise inequality in it. Now we will understand the concept of linear inequalities. It is fully depends on symbol that present in inequality. If less than sign present in linear expression then we found inequality under the line. If grater than sign is present in linear expression then we found inequalities top the line.
Let’s understand the concept of solving systems of linear inequalities. Let's we have a linear inequality 4a + b < 15, then we can calculate this linear inequality as mention below:
In the above given linear inequality is there so after calculating, coordinates we found is under the line. Here set different values for one variable to get other coordinates. So it can be written as:
=> 4a + b < 15, to find coordinates of linear inequalities replace inequality symbol by equal sign.
=> b = 15 – 4a.
On putting value of a = 1 we get:
=> b = 15 – 4 (1),
=> b = 11.
On putting value of a = 3 we get:
=> b = 15 – 4 (3), on further solving we get:
=> b = 15 – 12,
=> b = 3.
In this way we can find different values. So here we get (1, 11), (3, 3). If we plot the graph we get inequalities below the line.
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Absolute Value Inequalities

In mathematics, Absolute value is a measure of how energies a number is from 0. For example: ‘15’ is 15 point far from zero and -58 is 58 point far from zero term. Absolute value of number 0 is 0 and absolute value of 150 is 150. The entire values are the examples of absolute value function. In the case of absolute value negative numbers are not taken. Let's we have given any negative number then it is very essential to avoid negative sign from number. So we can have only positive values and zero. Absolute value function is denoted by the symbol '|'. This symbol is also said to be bar. If we plot any negative number among this symbol then we found positive number outside this symbol. Now we will discuss process of calculating Absolute Value Inequalities.
If any of the given symbol (<,> <, >) are there in any expression then we can say that equation have inequality in it. Let's us discuss how to calculate absolute value inequalities. Let's we have | 2a + 3 | < 8, absolute value inequality.
Solution: Given inequality | 2a + 3 | < 8, then first we will calculate linear inequality. So it can be written as:
=> - 8 < 2a + 3 < 8, it means 2a + 3 is larger than -8 and shorter than 8. Then subtract 3 from inequality. On subtracting value 3 we get:
=> - 8 – 3 < 2a + 3 – 3 < 8 – 3, on moving ahead we get:
=> -11 < 2a < 5.
Then divide entire inequality by 2, on dividing we get:
=> - 11 / 2 < a < 5 / 2,
On solving | 2a + 3 | < 8, we get –11 / 2 < a < 5 / 2. In this way we can solve any inequalities values.
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Solving Inequalities

Inequalities are the expressions that shows without using any equal sign means it will show as an expression that have signs of less than or greater than. Some times it is possible that these expression also have the equal sign but along with the less than or greater than sign that is also describe as a less than equal to or greater than equal to sign. Solving Inequalities having the same pattern like equations that is describe as an expression with the equal sign. One thing keep in the mind at the time of solving the inequality is that whenever change the side of the variable or values sign will change as > greater sign will change into the < and < less than sign will change into the > greater than sign.
There are some rules that does not effects the change of the sign as follows:
Whenever any number will be subtracted from any one of side then the same process is also done from other side.
When multiply both side of the inequality with positive number will also not effect the sign of inequality.
But there are also some ways that will change the sign as if multiply with the negative number or when we slide the number from one side to another side. we can simply define it by an example: 3 n < 7.
Now in the above inequality we have to put the values in place of n that satisfy inequality. there is no need to change in the sign.
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Solving Inequalities

If we have two values and both given values are not equal then we say that there is an inequality in both the values. Some of the conditions of an inequality are:

Suppose we have two variables ‘u’ and ‘v’. If u ≠ v; which represents u is not equal to v, and if u < v then we say that u is less than v, and if u > v then we say that u is greater then v. when u <= v, it means u is less than equal to v. When u >= v then u is greater than equal to v.

Now we will understand how to Solving Inequalities using multiplication and division? To find inequalities by multiplication and division we need to follow some steps which are shown below:

Step1: To solve inequality first of all we have to take an equation.

Step2: Remember that the inequality equation should only be defined in above symbols.

Step3: If equal sign is present in between the equation then the given equation never contain any inequality. Now we will understand it with the help of small example: (know more about Solving Inequalities, here)

Lets we have –a / 5 > 21; then solving inequalities by multiplication or division method. To solve inequality we need to follow all the above mention steps:

To find inequality first we have to take inequality, and if it has < or > signs then it is an inequality. So, the given equality can be written as:

= –a / 5 > 21,

To solve the inequality we have to multiply -5 on both sides of equation. On multiplying -5 we get:

= – a / 5 > 2,

= (-5) * -a / 5 > 21 * -5,

= 5a / 5 > -105, On further solving inequality we get:

If we divide 5 from 5 then we get 1. Now we have equation like this:

= a > -105, So it can be written as:

= a < -105. In this way we can solve inequality.

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solving systems of inequalities by graphing

In the previous post we have discussed about analyzing equations and inequalities and In today's session we are going to discuss about solving systems of inequalities by graphing. If variables are not equals to each other then we can say it is an inequality. Now we will discuss some of the conditions for an inequality which are: let we have two variables ‘p’ and ‘q’ then:
Condition1: If p ≠ q; that represents ‘p’ is not equals to ‘q’;
Condition 2: If p < q then it represents ‘p’ is less than ‘q’;
Condition3: If p > q then it represents ‘p’ is greater than ‘q’.
If the given conditions are present then we say that inequality is present. Now we will discuss how to solving systems of inequalities by graphing. We need to follow some steps to plot the graph. We know that there are many methods for solving systems of inequalities but solving graphically is the one of the best method. (know more about Inequality, here)
Step 1: To plot graph first we have an inequality equation. Let we have an equation x + y ≤10.
Step 2: Then put the different values of x – coordinate to get the other value of y – coordinate. So we can write the above equation as:
Y ≤- x + 10,
Step 3: If we put the value of x coordinate is ‘1’ then we get the value of y – coordinate is 9.
If we put the value of ‘x’ is 0 then we get the value of ‘y’ is 10. If we put value of ‘x’ is ‘4’ then we get the value of ‘y’ is ‘6’ and we put the value of ‘x’ coordinate is ‘6’ then we get the value of ‘y’ is ‘4’. So we get the ‘x’ and ‘y’ coordinates as: (1, 9), (0, 10), (4, 6), (6, 4). Now we can easily plot the graph of the given inequality. The graph is shown below.



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analyzing equations and inequalities

Before analyzing equations and inequalities first of all it is necessary to know about the equation and inequality. As we know that if equal sign is present in an expression then it is said to be equation. If less than, greater than, less than equal to and greater than equal to operator is present in an expression then it is said to be an inequality. We have to focus on some points to analyzing equation and inequality. Now first of all we will see the properties of equation or (Real number). (know more about Inequality, here)

	Addition	Multiplication
Commutative	P + q = q + p	pq = qp
Associative	(p + q) + r = p + (q + r)	(pq) r = p (qr)
Identity	P + 0 = p = 0 + p	p (1) = p = 1 (p)
Inverse	P + (-p) = 0 = (-p) + p	If p ≠0 then p (1/p) = 1 = 1/p(p)

If we talk about the inequality property then we can write the properties as: For any two real number p and q, one of the given statements is true. p < q, p = q, p > q

Addition and subtraction properties for inequality	For any real numbers p, q and r: 1.      If p > q then p + r > q + r and p – r > q – r 2.      If p < q then p + r < q + r and p – r < q – r
Multiplication and Division property of inequality	For any real number p, q and r: 1.      If  r is positive and p < q then pr < qr and p/r < q/r 2.      If  r is positive and p > q then pr > qr and p/r > q/r 3.      If  r is negative and p < q then pr > qr and p/r > q/r 4.      If  r is positive and p > q then pr < qr and p/r < q/r

If we apply these properties then we can easily analysis the equation and inequality. VSEPR Theory is based on chemistry. The icse guess papers 2013 are useful for the preparation of exam.

How to Solve Inequalities

In the previous post we have discussed about How to Draw Bar Graph and In today's session we are going to discuss about How to Solve Inequalities, Algebra can be consider as a important part of mathematics that perform the task of converting the real world problem in the form of mathematical equation or find the value of a variable in mathematical equation. In mathematics, an algebraic equation represents the relationship between the both side of equal sign in algebraic equation. It means anything happen with number on one side also reflect the changes on other side.
Algebraic equation basically deals with solving an equation by finding the value of variable. It also perform task of solving an inequalities of algebraic equation. Here in this section we are going to discuss about the topic how to solve inequalities.
In mathematics, the concept of inequalities can be represented by some of the symbol that are given below:
a ) >= greater then equal to
b )> greater then
c ) <= less then equal to
d ) < = less then
as we all are very well aware that algebraic expression is a combination of number and variables to represent the real world problem in a mathematical form. When such kind of symbol are used with algebraic expression then a question arises in our mind that how to solve inequalities.
The solution of solving inequalities from an equation is already defined by mathematics. To solve the inequalities from an algebraic expression can be done by two ways.
A ) first is “ By adding or subtracting a value from both side”: It means to say that inequalities can be solve by adding or subtracting the same value on both side. Like there is a equation a + 5 < 10 then it can be solve as a + 5 – 5 < 10 – 5 = a < 5. The obtained result shows that the value of x must be less then 5 to set the equation true.
B ) By dividing and multiplying the value. The concept of 

Measures of Central Tendency can be performed by using mean, median or mode of mathematics that helps in extracting the exact values. Icse stands for Indian certificate for secondary education that provide icse board sample papers for class 12 to the students for guiding the students in the examination by providing a list of questions. 

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.
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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. 

Solving inequalities by addition and subtractions

Previously we have discussed about first order differential equation and In this session we are going to study about the inequalities and how to solve them that comes under cbse 12th syllabus. Let us define an equation. An equation is an expression that contains combination of numbers and variables and shows relationship between them. After that these expressions are written in both sides of equality sign. In grade VI we study about the Algebraic Equations and solve the problem of inequalities.
Inequalities in algebra means two variables are not equal that is x≠y. There are various types of symbol to show the inequalities between the two variables like >(greater than),<(less than),>=(greater than or equal to),<=(less than or equal to) etc. Here, we will be solving inequalities by addition and subtractions. We will add and subtract the numbers into the expression.(Know more about inequalities in broad manner, here,)
Example of equation is: 4y + 5 = 9
Let us see some examples of solving inequalities:
Solving inequalities by addition:
Example1: Solve x-7 >=15
Solution: Now we solve this by adding digit 7 to both sides of inequality
                  x-7 +7  >=15+7
                  We know that -7 +7 = 0 and 15 + 7 =22
                  Thus x >=22
Example2: Solve a – 13 >= 28
Solution: We solve this by adding digit 13 to both sides of inequality
                     a – 13 >= 28
                     a – 13 +13 >= 28+13
                      a >=41
 Solving inequalities by subtractions:
Example3: Solve inequality by suitable method for:
                      X + 5 >=10
Solution: We subtract 5 from both sides of inequality
                     X + 5 – 5 >= 10 - 5
                            X > = 5
Example 4: Remove the inequality by adding or subtracting the numbers for given equation
                         a + 1 >=7
Solution: Here we solve the equation by subtracting 1 from both sides of inequality
                        a + 1-1 >= 7 - 1
                         a >= 6

In the next topic we are going to discuss Solving Multi Step Inequalities and if anyone want to know about Simplifying Rational Expressions then they can refer to Internet and text books for understanding it more precisely.

Solving Inequalities with Rational Numbers

Previously we have discussed about how to solve inequalities with fractions and Today I am going to discuss about solving inequalities by rational numbers which comes under cbse syllabus 12th. Before I start telling you the actual procedure to solve these inequalities by rational numbers, we should know about the inequalities and rational numbers. Inequality means unequal. If two quantities or parts or anything which are not equal then this leads to inequality. Inequality may arise if two quantities have a relation of greater than (>) or less than (<) or greater than equal to (>=) or less than equal to (<=).
Now we come to rational numbers. Rational numbers are basically the fraction of two numbers and have a form of a/b where a and b are two integers and b is not equal to 0.
Solving inequalities with rational numbers is a very interesting and easy task and can be practiced using Inequalities Worksheet. To solve this we must follow a simple procedure that is as follows:
1.      First of all to solve the equation we must get the variable alone on left side of the equation, so that we can find its value.
2.      Now to get the variable we must use an inverse operation. This inverse operation will undo whatever had been done to the variable.
3.      Here inverse operations are: addition and subtraction or multiplication and division.
4.      To maintain the equality we should do the same operation on both the sides.
Now to get this whole procedure let us take an example:
Here the question is to determine the value of y in the given equation:  3/2 y = 5/4
2/3 * (3/2) y = 2/3 * (5/4) [Multiply both the sides by 2/3]
y = 10/12
Now again solving this we will get y = 5/6.
So by using this procedure we can use rational numbers to solve inequalities. In the next session we will discuss Solving inequalities by addition and subtractions and Read more maths topics of different grades such as Multiplying Rational Expressions in the upcoming sessions here.   

Math Blog on Solve two step linear equations

Hi friends!
Linear equation is an equation that has many variables. A pair of equations with same variables is said to form a system of simultaneous linear equations. Linear equation can have two or more variables. In this section, we shall be discussing two step of linear equations in solving simple problems from different areas.
If a and b are two real numbers such that b is not 0 and a is a variable, then we have learnt that an equation of the form ax+b=0 is called linear equation of single (one) variable, where a and b are real numbers and x is a variable.
In linear equation of single (one) variables, we can have solution within two steps. There are many types of properties:-addition, subtraction, multiplication and division.
We take some addition and subtraction examples to solve the linear equation of single variable.
Example 1:- Solve equation a-b=c with addition properties of linear equation of single variable. Solve this equation for a, where b=2 and c=3.
Solution:-
Step 1:- a-2=3 (we can add 2 both side of equation. a-2+2=3+2 .now L.H.S part is a and R.H.S part is 5.
step 2:-a=5.
thus, b=2 and c=3 satisfy both the equation of the given system.
Hence, b=2 and c=3 is a solution of the given system.
Example 2:- Solve equation a+b=c with subtraction properties of linear equation of single (one) variables. Solve this equation for a. where b=2and c=5.
Solution:-
Step 1:- a+2=5 (we can subtract 2 both side of equation. a-2+2=5-2 .now L.H.S part is a and R.H.S part is 3.
Step 2:-a=3
Thus, b=2 and c=5 satisfy both the equations of the given system.
Hence, b=2 and c=5 is a solution of the given system.
In the next topic we are going to discuss Solving inequalities by addition and subtractions of linear equations.

Math Blogs on Solving two step linear equations

Today I am going to explain how we can solve two step linear equations. Before this we should know about the linear equations. A linear equation is one of the important parts of the algebra. It is an algebraic equation that consists of either the product of a constant and a variable or a constant.  Whenever we plot them on a graph we always get a linear line. It has a form y = m.x + c, where m is the slope of the line and c is a constant and x, y are the variables of the linear equation.
Now we come on the topic that how to solve two step linear equations. In this the main task is applied on the variables, i.e. to get the variable alone on either left or right side of the equal sign. For this we make the equation balanced by making same changes in both sides of the given linear equation. We must keep the equation balanced so that we get the right solution.
To understand this concept more let us take an example two step linear equations:
We have linear equation 5x + 2 = 57. Then to solve this we need to undo the multiplication of five and the sum of two in the equation. If we first divide the whole equation with 5 then we will get fractions that are not desirable so we should avoid it and instead of this we prefer to do addition or subtraction.
5x + 2 = 57
       -2 = -2
5x = 55
So x = 55/5 so x = 11
From the above example we get the method of two step linear equation. (To get help on cbse books click here)
Similarly one more example: 3x + 5 = x – 3 so here we will subtract x from both the sides as
3x + 5 = x – 3
-x       = -x
2x + 5 =-3 then here we will subtract -5 from both the sides
2x + 5 =-3
      -5 = -5
2x = 2 here x = 2/2 so x = 1.
So today we learnt the two step linear equations. and In the next session we will discuss about Solving Inequalities with Rational Numbers. 

Math Blog on Solving inequalities by addition and subtraction

In the mathematical aspect solving inequalities by addition and multiplication is a simpler task, when you are little bit aware of equations and their functionality. Equations are nothing but an expression in both sides of equal sign, which are written in math format and check the equality & inequality from the expression. Now focusing on solving inequalities by addition and subtractions is just placing the values in both sides of equal sign.  (To get help on central board of secondary education click here)

                      Example: A + 5 = 14
This is the example of equation which shows the mathematical expressions in the both sides of equal sign. Here we have to calculate the value of A by adding or subtracting the numbers in both sides of equal sign. Now we are focusing on solving the inequalities of equation by addition and subtraction. Solving inequalities by addition and subtraction some examples are given below:
 Example:  a – 6 = 18            
                Now if we follow the addition rule (according to addition rule if x=y then x+a = x+y) i.e. add some number in both sides of equal sign to remove the inequality from the equation. Then in next step equation will be
                a – 6 + 6 = 18 + 6
                    a = 24
Now if we want to find out that obtained solution is correct or not then we can do that by putting the obtained answer into the initial equation. It means put a = 24 in the initial equation.
                Initial equation is   a – 6 = 18
                                              24 – 6 = 18
                                                  18 = 18
We can see that both sides of equation have same value then we can say that our obtained result is correct. The above example shows the way of solving inequality by addition. Now we see solving inequalities by subtractions.
Example 2:      solve z + 10 = 17
                  Now we solve this inequality by subtracting the values from both sides of equal sign.
Then equation will be         z + 10 – 10 = 17 – 10
                                                        z = 7
  In the next topic we are going to discuss Solving Inequalities by Multiplying and Dividing and In the next session we will discuss about Math Blogs on Solving two step linear equations. 

Two Step Linear Inequality

Earlier we have discussed about verifying trigonometric identities and now we are going to start Two step Inequalities or two step equations which falls under gujarat secondary education board,They are the inequalities or equations which cannot be solved in single step operation. It involves the series of steps one after another to get the solution for the given variable in the equation. While we are solving inequality, we must remember that if any negative number is multiplied or the inequality is divided by any negative number, then the sign of inequality changes.
Now here are some equations and inequalities which can be solved by two steps:
2x + 4 = 10
In this equation, first we subtract 4 from both the sides,
We get the following form by solving equations :
   2x + 4 - 4 = 10 - 4
 or,  2x = 6
 This solution we get by the first step, but still the value of x is not obtained. So to obtain the value we proceed to second step of solution:

In second step we divide both sides of the equation by 2,
we get 2x / 2 = 6 /2
or x = 3 is the solution to the given equation.

Let us take another example:(Know more about Linear Inequality in broad manner here,)
35 = 5x - 10
Here in first step of solution, we add 10 to both sides of the given equation.
we get,
35 + 10 = 5x -10 + 10
or, 45 = 5x
Now in second step we divide both sides of the equation by 5
we get 45 / 5 = 5x/ 5
            9 = x is the solution of the given
Now let us take an example of Solving Two step Inequalities,
        3x -5 >= 16
For Solving Inequalities
   Add 5 to both sides we get
     3x - 5 + 5 > = 16 + 5
 or 3x  >= 21
Now we divide both sides of the inequality by 3,
we get
 3x /3 > = 21 / 3
 x > = 7 Ans.

This is all about two step equations and Inequalities. In the next article we are going to discuss about solving two step linear inequality and if anyone wants to know about Math Blog on Estimating Quotients then they can refer Internet.

Solving Two Step Linear inequality

Earlier we have discussed about law of total probability and Now new topic, As we know that Any equation and inequality requires certain steps to reach to the desired solution of the variable and usually it comes under every education board. As we all know that Solving equations which cannot be solved in single steps operation requires multiple steps. It involves the series of steps one after another to get the solution for the given variable in the equation.
While solving two step inequalities, we must always remember that if any negative number is multiplied or the inequality is divided by any negative number, then the sign of inequality changes.
Now here are some equations and inequalities which can be solved by two steps:
We first start with solving two step equations:(Know more about inequality in a broad manner, here,)
  3x + 4 = 7
 In this equation, first we subtract 4 from both the sides,
 We get
   3x + 4 - 4 = 7 - 4
 Or, 3x = 3
 This is the form of the equation we get by the first step, but still we need to follow certain steps to attain the value of x
In second step we divide both sides of the equation by 3
We get 3x / 3 = 3 / 3
Or x = 1 is the solution to the given equation.

Let us take another example:
  30 = 2z - 20
Here in first step of solution, we add 20 to both sides of the given equation.
We get,
 30 + 20 = 2z -20 + 20
Or, 50 = 2z
We will proceed to second step to find the value of z
 So, we divide both sides of the equation by 2
We get 50 / 2 = 2z / 2

              z= 25 is the required solution.

 Children, here we have another example of Solving two step inequality:
        5x -6 =< 16
   We proceed for first step of solution by adding 6 on both sides
     5x - 6 + 6 < = 16 + 6
 Or   5x <= 22
Now in second step we divide both sides of the inequality by 5
We get
 5x /5 < = 22 / 5
 x < = 22/7 Ans.
This is all about the Two Step Linear inequality and if anyone want to know about Solving Multi Step Inequalities then they can refer to Internet and text books for understanding it more precisely. Read more maths topics of different grades such as Probability Distribution in the next session here.