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.