inequalities

Inequality is a collection of operators which is used to represent the inequality of algebraic equations. An inequality is a statement of algebraic expression to calculate the value of unknown variables. In general aspect we can say that inequality is used to calculate the algebraic expression that is not same in both sides of equal sign. The term inequality can be applied to any type of statement by using the various types of symbol like ‘>’ (greater then), ‘<’ (less then), ‘<=’ (less then equal to), ‘>=’ (greater then equal to) and so on. The concept of inequality helps the students of Grade VII to understand basic concepts of mathematics.
Here we show you the fundamental properties of inequalities to understand the concept of inequalities:
a)      x, y and z are the real numbers if x ≤ y then x + z ≤ y + z.
b)      x, y and z are the positive real numbers if x ≤ y then xz ≤ yz.
A solution of an inequality is a number which is substituted for the variable makes the inequality a true statement. In the mathematics there are various properties defined for inequality to solve equations. In the next session we are going to discuss Multistep problems.
a) Transitive property: if a > b and b > c then a > c.
b) Addition property: if a > b then a + c > b + c.
c) Multiplication property: if a > b then ab > ac.
d) Subtraction property: if a > b then a – c > b – c.
The above given properties of inequality helps the students to Graphing inequalities into the graph. Inequalities can be performed by solving the inequalities into the algebraic expressions. There are some rules given below:
a)      Adding and subtracting the same number on both sides.
b)      After performing the above rule interchange the sides and changing the orientation of the given inequality symbols.
c)      If needed, then perform the multiplication and division of same positive or negative number on both sides of equal sign then changing the orientation of the inequality symbol.

In the next session we are going to discuss Multistep problems. 

Simplifying numerical expressions

This unit is for Grade VII. We will learn how to solve the expressions. We know that the expressions are the numerals joined together with the help of different operators. When more than one operator appears in the expression and some fixed laws are not designed for which operation is to be performed first, we will get variety of outcomes for all the persons who operate the calculations. So we will set the pattern by which hierarchies of the operators are decided and we all get the symmetry of the output for any expression whichever is solved.  Simplifying numerical expressions is the planned and the systematic methods of calculating the mathematical expressions. The order of solving the equations is as follows: BODMAS, where we have,
B – Bracket or braces,
O- Of operation,
D- Division,
M- Multiplication,
A-Addition,
S- Subtraction,
It means that when we solve the mathematical expressions, we will first open the braces and so whichever expression appears in the braces will be solved first. Next step of solution includes the calculating an “of” operator.  After solving the operator of, we will take up all the division calculations in the expression. After completing division, comes multiplication of the terms which are joined by multiplication “*” operator. After multiplying, we will solve the operation of Addition of the terms in the expression and in the end is left the operator of subtraction, if any exist in the main given expression. Thus proceeding in the systematical way and following the steps of the calculation as explained above all the expressions will follow a set pattern and hence the results we get will be universally same, who so ever does it and whenever they are solved. (know more about cbse text books, here)

Thus we say that the above method of solving the equations will help us to get symmetry of outputs
In the next session we will discuss about inequalities. 

Linear non-linear functions

Linear functions can be defined as the functions that are denoted by the straight line or their equation is presented in the form of y = m x + c .Sometimes Linear functions are also describe as the equation that has variable with no power means highest power of the variable is one .This topic helps to understand basic concepts of grade VII.
When we talk about the non linear function then it is denoted as the functions whose graphs are not a straight line. These non linear functions are also known by different names as quadratic equation or cubic equations etc. means when the power of the variable in the given equation is greater than one then these types of equations are known as non linear equation. As if we define non linear equations as p a² + q a + r = b is an example of quadratic equation that is defined within the non linear equation or a cubic equation that is also the non linear equation is defined as,
b = p a³ + q a² + r a + s but we should keep in mind that value of ‘a’ will never be zero.
If we want to know that which equation is linear or which one is non linear then, check the exponent value on the variable ‘x’, if value of exponent is not greater than one then the expressions is defined as the linear function otherwise expression is non linear function . (know more about cbse question papers, here)
If the expression in the form of f (a) = p a + q then these types of functions are surely linear functions.
The other way to find the linear or non linear function is vertical line set.
In the next session we are going to discuss Simplifying numerical expressions. 

Operations on monomials

This Unit is for  Grade VII, here we are going to learn about  monomials operations. A monomial is an algebraic expression which has only one term.  Now we will learn about Operations on monomials. All mathematical operators namely addition, subtraction, multiplication and division can be performed on monomials. Before we discuss the methods of operation we must know the terms like terms and unlike terms. The terms with same variables are called like terms as 2xy and 5yx are like terms, on another hand 3xz and 3x are unlike terms.ely, without considering if they are like terms or not
When we perform addition and subtraction of the two monomials, then we add the terms which are like and the unlike terms are simply written in original form  with their operators. If the two monomials are multiplied or divided, we perform the operation of multiplication with constants and the variables separately
 Addition and subtraction of monomials is possible with like terms only, if the terms are unlike, then  only the terms are represented as follows:
Add 3xy and 4xz will give 3xy + 4xz Ans
On another hand if we have  to add 4x , 3y  and 8x, we get
=4x + 3y  + 8x
= 4x + 8x + 3y = 12x + 3y Ans.
Now we take the problem of subtraction: Find the difference between 6x and 2x will give:
= 6x – 2x = 4x Ans
If the problem is as follows  Subtract 4x from 6y, we get
6y – 4x  Ans as the two monomial terms are not like. (know more about cbse board papers, here)

 In case of multiplication, we can multiply the terms without considering if they are like or unlike. In multiplication, the numerals of the two monomials are multiplied and the powers of the like variables are added up. Similarly when we divide one monomial with another, in that too we need not to have like terms. We will simply divide the numeral with the numeral and the powers of the same variables are subtracted. Let us make it more clear with the following examples:
 Multiply 4x²y and 3y, we can write the above expression as = 4x²y * 3y =4 * 3 * x²y * y
= 12 x²y².
Divide  12x²y and 3y, we can write the above expression as = 12x²y / 3y = (12/3)* x² * y^1-1.
 = 4 x² * y⁰ , but we know that any number raise to the power 0 is 1, we get  4 x^{2    Ans}


In The Next Session We Are Going To Discuss Linear non-linear functions. 

Geometry

In this unit we are going to study Geometry of Grade VII. In earlier classes we have studied about lines and angles.
If the two lines are drawn in such a way that they do not meet at any point are called parallel lines. Also we conclude that the perpendicular distance between the two lines is equal at all the points. Now if ‘l’ and ‘m’ are any two lines such that l || m. Here if a line ‘n’ is drawn such that it intersect both the lines at  a point, then line ‘n’ is called the transversal at parallel lines ‘l’ and ‘m’.  Now we observe the following situations to occur if two parallel lines are intersected by a transversal:
a)      The pair of corresponding angles is equal.
b)       The pair of interior alternate angles is equal.
c)      The pair of exterior alternate angles is also equal.
d)      The pair of angles on the same sides of the transversal is supplementary. It means that the sum of angles on the same side of the transversal is 180 degrees. (know more about cbse books, here)
 Thus we conclude that if any two lines are given and we need to check that they are parallel or not, it means that we need to check any one of the above given conditions are satisfied, then the lines are parallel. So we will try to check that which of the above condition we can show are equal. More over we come across the problems in which we are given that the pair of lines is parallel and there exist a transversal passing through both the lines. Further any one angle among all the angles is given and we need to find the value of all the other angles so formed. We can find all the angles by applying the property of corresponding angles are equal and the interior alternate angles are equal, when the lines are parallel.
In the next session we will discuss about Operations on monomials. 

Multistep problems

In this unit we are going to learn how to solve multistep problems. This unit is designed for Grade VII.  When we have certain equations, which are having one variable, we proceed in the way that in every step, we move towards separating all constant values from the variables. This can be done in single steps, when the equations are small. But in the bigger equations which include multi operators existing in the equation, can be solved step by step. In the initial step, we will first shift all the variables to one side of the equation. While shifting we must remember that the positive term changes to a negative term and the negative term changes to a positive term. Here we can do the same process in another way, if the term which is negative on the right side of the equation has to be shifted to the right side of the equation, and then we simply add positive value of the same term on the both side of the equation. Thus the positive and the negative term of the right side of the equation will be cancelled, on another hand the positive of the same value will be added to another side of the equation.

We can proceed in the same method for subtraction too. Thus any negative value from one side of the equation is to be removed, for this we will add the same value on the both sides of the equation. This becomes possible as the addition and the subtraction are inverse of each other.

 Now we look at multiplication and division sign which appears in the equation. To separate the variable, we need to see that the constant value with the variable consists of the multiplication operator or the division operator. If the operator is of multiplication, divide both sides of the equation by the same number and if the operator is of division, multiply both sides of the equation by that number. It will give the resultant value of the variable.

In the next session we are going to discuss Geometry. 

 

Arithmetic sequences

Arithmetic sequences can be defined as a sequence of a number. It means that sequence of number specify the difference between the consecutive number that is constant. Suppose there is an arithmetic sequence 2, 5, 8, 11, 13, 16…….with the common difference of 3. In mathematics, sometime arithmetic sequence is known as arithmetic progressions. The calculation of arithmetic sequence is very easy to understand. This topic helps to understand basic concepts of grade VII. If the initial number of an arithmetic sequence is ‘x₁’ and the difference of the successive members is ‘df’, then the ‘x_n’ term of the sequence is given by:
                         x_n = x₁ + (n – 1) df,
And in the mathematical terms, it can be defined as,
                         x_n = x_m + (n – m) df.
A finite portion of an arithmetic sequence is called as a finite sequence and sometime it is known as arithmetic progression. The total of different arithmetic progression is known as arithmetic series. The number added or subtracted at each stage of an arithmetic sequence is known as the common difference “df”. In simple language, an arithmetic sequence can be defined as a set of number that follows a particular pattern. The term and pattern in the number sequence depends on the behavior of the common difference means on ’df’. Shown below are some of the instances, (know more about ICSE Board Syllabus, here)
a) If the common difference is positive in the sequence then the terms of the sequence grow towards the positive infinity.
b) If the common difference is negative in the sequence then the terms of the sequence grow towards the negative infinity.
The total of arithmetic sequence can be defined as an arithmetic series.
Sequence of ‘n’ numbers = x₁ + (x₁ + df) + (x₁ + 2 * df) + ……… + (x₁ + (n – 1) df),

In the next session we will discuss about Multistep problems. 

equations

This unit is for the students of Grade VII. In this unit we will learn about equations. We must remember that equations with variables, are inter related terms. We form the equations to express the mathematical statements in form of the expressions. The mathematical expressions joined with the numerical operators are called equations, when there exist two sides of the equation LHS and the RHS. Also by word equation we mean whatever placed on the left side of the equation is equal to the expression placed on the right side of the equation. By word variable, we mean the unknown value which may change every time. When we solve an equation with variable ‘x’ and another equation with same variable, then value of ‘x’ may change from equation to equation, which may satisfy the equation.
The basic purpose of solving any equation is to find the value of the value of the variable which is unknown in the given equation. When we say that we need to find the value of the unknown variable in the equation, we mean that the value we calculate must satisfy the equation. Thus we say that the  when we put the value of the unknown variable in the equation,  the value we get after solving both the sides of the equation must be equal.  (know more about cbse board books, here)
There are different ways to solve the equations and finding the values for the unknown variables in the equation. To find the value, we may adopt hit and trial method, where we put the values of the variables and then check it that particular value satisfies the given equation or not. The value that satisfies the given equation actually is the required solution to the given equation.
Another method to solve the equation is by shifting the variable to one side and all the constants to another side of the equation and  In the next session we will discuss about  Arithmetic sequences.