The Method to Solve Inequalities

The process of solving inequalities is similar to the process of solving equations. The linear equations can be easily solved with the help of the principle of transposing. In transposing only the terms that are unknown are kept on one side and the rest of the terms are brought to the other side. This will help in finding the unknown terms. The terms in an equation and inequalities consists of both constants and variables. The value of the constants does not change throughout the course of the equation or the inequalities but the value of the variables can take different values. The value of the variables can be found out if they are not given.

There are different methods to find the value of the unknown variable. In case of linear equations it is quite easy. But as the complexity of the equation increases it becomes more tedious to find the value of the unknown variable. Once the value of the unknown variable is found it should be checked whether the value obtained satisfies the given equation or the inequalities. If the equation or the inequalities is not satisfied the value of the unknown variable obtained is wrong. One must attempt the problem again and find the new value of the unknown variable.

The question how to solve inequalities is answered from the fact that they can be solved in the same manner the linear equations can be solved. This is because the basic difference between an equation and inequalities is nothing but the symbol used in them. The ‘equal to’ is replaced by the ‘greater than’ or ‘lesser than’ symbol. From this the method to solve inequalities can be learnt. The degree of equations plays a very important role in determining the solution of the equation. The same is true with inequalities as well.

The graphical method of finding the solution can be very helpful in this case. This is very good method of finding the solution. The solution obtained can be checked for its feasibility from the graph itself. From the graph first the feasible region and the rejection region are found out. The solution lies in the feasible region and not in the rejection region. So, once this is known the solution is obtained from the feasible and then a check is done for its feasibility. Every solution that is obtained must be checked for its feasibility.

Understanding square

The word ‘square’ in mathematics may refer to a square in geometry or the same in algebra. Both meanings are similar in some way and also different in some ways.

For the level of a 4th grader, the word usually refers to a plane shape of four sides as shown in the picture below. This can be called the square in geometry.

Shown above is one such. The properties of the same are as follows:
1. All sides are of equal length.
2. All angles are of equal measures.
3. It is also called a regular four sided polygon.

Some examples of a sq. can be stated as follows:
Any face of a cube is also same. A die used for playing board games is same shape. The base of Egypitian pyramids and the base of the Eiffel tower is also having the same shape.
Perimeter of a square: The perimeter refers to the sum of all the four sides of such figure. If the length of each of the sides of the same is ‘a’ units, then the sum of lengths of all the four sides would be = a + a + a + a = 4a. Thus perimeter of a sq. = 4a units.

Area of a sq. = If each of the sides of the same are ‘a’ units, then the area enclosed by the sq. polygon can be given by the formula:
A = a * a = a^2 sq. units.

Example: Find the area and the perimeter of the following:
Solution: Perimeter = 4a = 4*5 = 20 cm
Area = 5*5 = 5^2 = 25 sq cm.
Sq. value of a number:
In math, another concept for the same is regarding the square of a number. In general, such number refers to the area of a sq. figure such that the said number is the length of each of the side. Therefore if we say that we need the sq. of a number x, then the answer would be equal to the area of the same such that each of the sides of the sq. figure is x. So that would be x * x = x^2.

Finding Square of a number:

For smaller numbers, to find the sq. value is relatively easy. For example sq. value of 2 would be  = 2*2 = 4.
Sq. value of 7 is 7*7 = 49 and so on. For larger numbers, it can be found using various methods. Let us try to understand this better using an example.

Example: Find the sq. of the number 13.
Solution: We know that its nothing but 13 * 13
This could be rewritten as:
13^2 = 13*13
= (10+3) * (10+3)
= 10*10 + 10*3 + 10*3 + 3*3 (FOILing the terms)
= 100 + 30 + 30 + 9
= 100 + 60 + 9
= 169 <- answer="" p="">Alternatively, instead of FOILing the terms, we can also use the identity:
(a+b)^2 = a^2 + 2ab + b^2

What is Math Inequalitie

In algebra when we come across a mathematical sentence something like 3x+4y=0 it is called an equation which has an ‘=’ sign. It is not possible always to equate two values; sometimes the values are bigger or smaller relative to each other.
So there is another type of sentence which is used to show the relative size of two values which is called an inequality. The inequality math sentences use one of the following symbols, ‘>’ greater than, ‘<’ less than, ‘=’ less than or equal to and ‘=’ greater than or equal to. Inequality Examples are x+3<9 5-x="" x="-5;">9 etc.

The key words that are seen in the word problems and their meanings are, ‘At least’ means greater than or equal to; ‘Not more than’ means less than or equal to; ‘More than’ means greater than. Some more Examples of Inequalities are as given below:
The sum of x and 4 is greater than -5
Sum of x and 4 is ‘x+4’, the inequality used here is ‘>’, the resultant is -5
Finally the inequality would be, x+4 > -5
Subtracting 4 on both sides gives, x+4-4> -5 -4; x > -9 would be the final inequality. Here it means x can take all the values that are greater than -9. One point to remember here is -9 is not included as there is no equal to sign in the inequality.

The addition, multiplication and absolute principles of inequalities are as follows:

• If a>b then a+c>b+c
• If a>b and c is positive, then ac>bc and if c is negative, then ac
• If  X is any expression and c any positive integer such that |X|
• If X is any expression and c any positive integer such that |X|>c, this would be same as X>c, X>-c

Inequalities Word Problems
A taxi charges a flat rate of $1.85 in addition to $0.65 per mile. John has not more than $10 to spend on the ride. Write an inequality representing John’s situation and calculate the number of miles John can travel without exceeding his limit
Solution: Here the key words are ‘not more than’ so the inequality would be ‘=’
The variable here is number of miles= m
So, the inequality would be $0.65m + $1.85 = $10
Subtracting 1.85 on both sides, 0.65m+1.85-1.85 = 10-1.85
  0.65m = 8.15
Dividing on both sides with 0.65, 0.65m/0.65=8.15/0.65
m = 12.54
John can travel 12 miles without exceeding his limit

Progression definition

Progression definition – it means progress, growth, gain, move forward or instantaneous changes in forward direction. Electromagnetic waves travel forward. It means they are progressing. Plant is growing, it means it is progressing.
Patient health is improving or progressing. The student is progressing; it means that the student is passing his examinations from lower class to higher class. It is the progress in positive or negative direction. Speed may be increased or decreased. It may have acceleration or retardation both.

In mathematics there are three types of progressions.
Arithmetic progression(A.P) :- this take place according to a certain formula, for example
a, a+d, a+2d, a+3d, …………………a+(n-1)d
where a=initial or first term,
d= common difference between the two terms.
n= number of terms

Example1: 1,3,5,7………………..nth term
Sequence of terms is arranged in such a manner so that the difference of any two successive terms of the sequence remains constant

Geometric progression (G.P) :-The rule or formula for G.P is as under.
a,ar,ar²,ar³,…………………..ar^(n-1)
where a=initial or first term,
r= common ratio between the two terms.
n= number of terms
 Sequence of terms is arranged in such a manner so that the ratio of any two successive terms of the sequence remains constant
Harmonic Progression:- a H.P. is the reciprocals of an arithmetic progression. The form of HP is
1/a1,1/a2……………………
a1, a2, a3… is called an HP if  1/a1,1/a2..is an HP.
If a, b, c are in HP, then b is the harmonic mean between a and c.
In this case, b = 2ac/(a+c)
Examples are
12, 6, 4, 3,  , 2, … ,
10, 30, -30, -10, -6, -  , … ,

Progression In other fields:
Astrological:- It is used in Horoscope astrology for  forecast of future happenings
Horizontal:- the gradual movement from left to right or right to left of  a line .
Age:- the process of modification of a photograph of a person show  the  aging effect.
Tax: is a tax in which the amount of tax id directly proportional to the amount of income tax rate is increased as the income amount is increased.
Semantic:- It is the evolution of used word
Color:- the whole ranges of color for which the values changes smoothly through hues, and saturation, along with the illumination, or the combination of all these three factors.
Educational:- It is the progress of the  individual from lower classes to higher class.
Age:- the process of modification of a photograph of a person show  the  aging effect.
Tracking purpose:-It is used in video games to check the performance of the individuals.

Empty set

An empty set, as obviously indicates, is a set which is empty. In other words in an empty-set there are no elements. Sometimes it is also referred as null set. An empty set symbol can be any of the followings.      { } , (/),Ø. Of all these, the last symbol is very prominent.

It may be noted that it is not same as the Greek letter F. There is an interesting point in referring about empty-set. A set when empty is in a way common to all the sets that we think of. Therefore a set with no elements is better referred as ‘the empty set’, the word ‘the’ is more apt than the word ‘an’.

The null set, apart from the fact of having no elements has other properties too. The empty set can be a subset of any set. This may be a bit difficult to understand but a Venn diagram will be helpful to understand clearly. The union of the null set with any set A is the set A itself. It is obvious because we add only a 0 to the number of elements of A.

The intersection of the null set with any set A is again the null set. It is very clear that the intersection set cannot have any elements and it has to be empty. There can be only one subset of the null set which is the null set itself.

Now let us discuss about power set of the empty set. Before that let us see what a power set is. A power set of a set A is the set of all possible subsets of set A, including the null set. For example, if A is {a, b, c}, then all the possible subsets are { }, {a},{b},{c},{a, b},{b, c},{c, a},{a, b, c}. Therefore, the power set AP(A) is defined as {{ }, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}. Thus we find there are 8 subsets for a set A having three elements.

That is, 23 subsets for 3 elements and in set language the power set P(A) in this case can be expressed as P(A) = 23.  Extending this concept in general for a set A having ‘n’ number of elements, P(Ø) = 2n. Therefore, in case of a null set the power set of the null set is P(A) = 20 = 1. That is the null set itself. In other words, P(Ø) = { }.

Learning About the Field of Vectors

The quantities in mathematics can have different features. The features can be magnitude and sometimes direction as well. All the quantities do not have the same feature. This difference in features can be the difference between the quantities. The quantities that have only magnitude are known to be scalar quantities and the quantities that have both the magnitude and a direction can be called vectors or carriers.

This is the basic difference between the two types of quantities. This difference must be understood to understand the difference between the two types of quantities. This difference can be very helpful in understanding the difference between the quantities. The vector space over a field can be understood only when the concept of carriers is clear and understood. It is part of a mathematical in which the carriers which can be multiplied with a scalar.

The operation of addition is also possible within this space. The scalars can be any numbers. Sometimes they can also be real numbers or can also be complex numbers. The complex numbers have an imaginary part attached to them. The flux of vector field is nothing but a flow property of carriers. This property is concerned with a unit area.

The flow is per unit area. This concept is very important in the field of physics also. Both mathematics and physics are related to each other. So, this concept appears in both physics and mathematics. The knowledge of physics will help in understanding of the concept. The mathematical plot vector field will explain the concept.

The pictorial representation of the concepts will help in the understanding of the concepts. So, the pictorial representations are always helpful.

The flow of a vector field is related to the flux of the field. These both concepts are interrelated and the understanding of one concept will help in the understanding of the other concept. The plot can be shown in the Cartesian plane using the ‘x’ and ‘y’ axes.

The vector field graph shows the orientation of the various carriers in the Cartesian plane. The carriers are shown aligned in the different directions. So, the orientation of the carriers can be easily understood with the help of this graph.

This graph shows the different directions associated with the carriers. The carriers are shown oriented in different directions and this can be understood from the graph. So, the graph can be very useful.

Introduction to Linear Algebra

Linear Algebra is one of the branches of Mathematics.  These helps with the study of vectors, vector spaces, linear transformations, and systems of linear equations. Vector spaces are found in modern math thus, it is widely used in both abstract algebra and functional analysis.
It is also found in solving problems in analytic geometry.
 Its application is found extensively  the natural sciences and the social sciences.  This forms an intro to Linear Algebra.

 The mathematical approach of solving for variables in equations is Algebra.  
It does the manipulation of these variables using mathematical structures.
It is like taking algebra to another level by manipulating these variables using mathematical structures. This also helps to write the entire system of equations in the form of matrices. When expressed in the form of matrices this to study in depth of the individual parts of matrix which is known as vertex. A vertex is simply a representation of a coordinate.

Linear algebra is a branch of mathematics.  The creation of analytic geometry led to  the system of linear equations  acquire a new significance.
Linear-algebra is important for both pure mathematics and applied mathematics.    
Linear form of algebra when combined with calculus facilitates the solution of linear systems of differential equations.

Analytic geometry, engineering, physics, natural sciences, computer science, and the social sciences   use the techniques of Linear-algebra.
Linear mathematical models are sometimes used to approximate Non-linear mathematical models because linear algebra is such a well-developed theory.
The real world applications need Linear-Algebra.  Algebra is all about linear system of equations and their solutions Linear-Algebra usually consists of the linear set of equations as well as their transformations on it.  Linear Equations  consists of  topics  like  Linear Equations , Matrices, Determinants, Complex numbers, Second degree equations, Eigen values ,Linear Transformations.

Linear algebra help to mathematics: It is a useful branch of mathematics.   Below listed is some of the applications of Linear form of Algebra.  Constructing curves, Least square approximation, traffic flow, Electrical circuits, Determinants, Graph theory, Cryptography.
It   is used to draw graphs. Most equations of linear algebra will   be a straight line. For example draw a graph for a vehicle travelling at a constant speed at various time intervals.
The graph helps in determining the    unknown variable that is the distance by plotting in on the graph.  This can also be used for   a multitude of different functions that is a ready tool for  lots of different real life functions.