Understanding the Concept of Inequalities

In mathematics there is system of linear equations. The equations are formed when there is an ‘equal to’ symbol present in the given expression. If the ‘equal to’ symbol is replaced by a ‘greater than’ or ‘lesser than’ symbol the equation becomes an inequality. When a set of equations are given it forms a system of equations. The equations can be linear in nature and also quadratic in nature. Equations of higher degrees are also present. The higher the degree of the equation the more complex the equation becomes. It is difficult and also time consuming to solve the equations of higher degrees. But the linear equations can be easily solved. The simple concept of transposing is used in solving them. There are both variables and constants present in an equation. The value of constants does not change throughout the equation but the value of the variables changes. The variables can assume various values.

Just like the system of equations there is also the system of inequalities in mathematics. The method of solving them is similar to the system of equations. But the ‘greater than’ or ‘lesser than’ inequality must be kept in mind before solving them, otherwise the whole answer can go wrong. The inequality can also be treated as a constraint. The final solution must satisfy this inequality or constraint. If the solution does not do so then it is not the correct answer. The solution must be found out again and must be checked for its feasibility. The systems of inequalities can be easily solved just like the system of equations.

The art of solving systems of inequalities can be easily learnt if one is thorough with the concept of solving the system of linear equations. The difference between the two is only in the symbols used. The graphical method can also be used to solve the same. The graphical method is a pictorial method of finding the solution to this method. The equation or the inequality given in the question is plotted on the graph and the solution is found accordingly. It is an easy method of understanding the concept. Once the solution is obtained through this method it is very easy to check its feasibility. The feasibility of the solution can be understood by looking at the graph itself. Once the solutions that are obtained are found to be feasible they must be accepted as the solutions.

Graph Logarithmic Functions

A logarithmic function relates the output corresponding to the logarithms of the variables. In many applications in real life, it is easier to study the output to the logarithmic values of the variable rather than the values of the variables. This is because; in certain cases the output increases very rapidly with even nominal increases of values of the variables.

Since the logarithm picks up only the increase in the exponent size, the increase rate is controlled to smaller figures. For example if a function is described as f(x) = x, for every1 increase of the variable, the function rises by 1. On the other hand if we define a function g(x) = log [(f(x)] = log (x), then the function rises by 1 only for every 1 increase of exponent of 10.(Here as per normal conventions we referred ‘log’ to mean logarithm to base 10).

In other words, a logarithmic-function has a low increase for large inputs. But however much we explain, the concept can be more clearly understood with a logarithmic function graph. Always a visual presentation is much more effective than verbal algebraic explanation. Thus graphing logarithmic functions have taken a very important place in algebra especially in the study of functions.
Because of the large variations of scales of the input and output in such functions and also because of the domain restrictions one must adopt the correct technique to graph logarithmic functions.

The following may be used as some tips to do the graphing for functions of logarithm.
1) First find the domain of the given functions. Remember that logarithms are defined only for positive expressions. So accordingly determine the domain and start graphing only from the minimum domain value. This will give an idea about the scaling the variable axis.
2) According to the domain of the function, determine the range. You can determine a practical range so that graph is properly sized. This will help in selecting proper scale for the output or function axis.
3) Next find the x-intercept and y-intercept. In some cases one or even both may not exist at least in the selected domain and range of the graph. Plot these points on the set of coordinate axes.
4) Make a table of values for a few points (practically it will be only 2 or 3) selecting compatible values of the variables. For example, if f(x) = log (x), make the table for values of powers of 10. Plot the table values on the same set of axes.
5) Draw a smooth curve connecting all points that are plotted. The ends of the curve should have arrow heads in the appropriate directions to indicate that the function is continuous.

Quadratic functions

Let us first see how quadratic functions are defined. A function is a relation between two or more variables which is one to one and onto. If the independent variable is single with degree 2 and the dependent variable is of degree 1, then the function is said to be quadratic. That is, the output of the function is related to the square of the input.

The general form of a quadratic function is y = f(x) = ax2 + bx + c, where a, b and c are all constants. Obviously, the constant a cannot be 0, in which case the degree of the variable will be no longer 2 and hence the function will not qualify as quadratic function.

Quadratic-functions are the most prominent functions in algebra as many practical situations can be cited as quadratic functions examples. Thus the knowledge of solving quadratic functions is extremely important. The solving can mean in two types of solving. One is to find y for a given value of x. But this is very simple task to do and has no great practical implications.

On the other hand, for a given value of y, to solve forx is extremely important. This is what generally referred as solving quadratic functions. The very first exercise is to solve for solve for x,when y = 0. The values of x under such conditions are called as the zeroes of the function or the x-intercepts of the function. In case of linear functions y-intercepts are more prominent but in case of functions of quadratic form, the x-intercepts are more prominent.

In certain cases, you may find the solutions to be imaginary and at a later stage we will explain the implication of such cases.

A quadratic function can be transformed to a quadratic equation for a given value of y. That is, the constant term is adjusted accordingly and is equated to 0. There are many methods of solving for the variable. The easiest method, if possible, is to factor the trinomial as the product to two binomials and apply the zero product property. In cases where factorization is not possible, one can use the quadratic formula and find the solutions.

We cannot predict for sure, the shape of a graph for a given function except in two cases. In case of a linear function, the graph is a straight line. The shapes of graphs of quadratic functions are invariably vertical parabolas. The points of intersections of the parabolas with x-axis are the zeroes of the function. If there are no such intercepts, the zeroes of the functions are imaginary.

The Art of Simplifying Rational Expressions

The concept of simplifying rational expressions can be understood only after the studying the basics. Once the basics are clear, it becomes very easy to solve these. The concept of solving algebraic equations can also be learnt from these basics.

Once this is understood simplifying algebraic expressions can be very simple. Basic algebraic expressions will contain the arithmetic operations of addition, subtraction, multiplication and division. They also contain constants and variables. The value of constants does not change throughout and the value of the variables can change. This is the basic difference between the two.

To simplify rational expressions one must understand what the definition of these expressions is. These can be written with the fractions. The denominator in the fraction must not be zero; otherwise it is difficult to define the fraction.

The degree of the expression plays a very important role in deciding the method to solve the expression. The question how to simplify rational expressions can be understood only if the basics are understood. It is important to learn the concept of polynomials to understand this concept. Ratio of the polynomials can be treated as an expression representing the concept. The concept of ratio is very clear and simple to understand.

There can be various terms in the polynomial. These terms can be rational or need not be rational in nature. But this will be true only for the real numbers. The complex numbers do not come into the picture here. The complex numbers consist of a real part and an imaginary part in them.

So, this cannot become a part of the polynomials that appear in the ratio. The polynomial that appears in the denominator must not be reducible to zero otherwise the whole expression is valid.This concept is also useful in geometry. The expression can be represented graphically. Simplifying these fractions can be learnt from the simplification of simpler fractions. The process in simplification is very simple.

The common terms that are present in both the numerator and the denominator are cancelled, so that the common terms do not appear twice. This is the basic idea of simplification. The same process is followed in simplifying these expressions as well. The polynomials must be reduced into simpler terms. If there are any common terms in the numerator and the denominator they must be cancelled. This makes the term simpler and easy to understand.

Mixture Problems as Words

We are solving the equations in mathematics always having direct problems but when we are solving the problems in the word or description format then it is known to be as mixture problems. Formation of the question or getting the problems in the form of word problems is the approach to check your analytical skills.

It is very useful in the future profits as if you go further in the studies then you have to face the Operational Research type of subject or the optimization techniques used in the process of various functional aspect of the organization.

When the problem comes in the word format then you have a better experience to do that it means that you have to do the lot of problems before attempt the problems given in your papers. As it is must to know the formats in the word problems. Algebra mixture problems are creating a mix of two or more than two things to determine something related to quantity of the mixture problems.

It is the conversion of qualitative analysis into quantitative analysis. Say for example of mixture problems in algebra like if the trains are moving in a same direction and both the speeds of the train is 7km/s then tell us the relative speed of both the trains?

This is an example of the problems which is word problem. Solving this kind of the equation you have to focus on what the question want and then proceed further to reach to the final equation.

The other part of solving the equation is form the tabular form as when the polynomial type of the equation is given and then solves it as per the logical reasoning. Mixture problems in algebra is to mix up the qualitative type of problems in a quantify manner to check the data interpretation of yours while solving this kind of problems.

As if you understand these kinds of problems then you are able to solve all the problems in the mathematics. Operation research is a big scope for good data interpreter and it works in your competitive advantage also. So try to better solve more and more number of problems based on word problems to increase your IQ level.

Few tips we want to share is use go through the problem and in rough do the analysis part what is given in the problem and then to start the solution. Procedure wise solve the question to reach the final goal what you want to achieve and make it as a sequential wise approach.

Concept of Simplification of Radicals

Various concepts are used in mathematics. Radicals are also one among them. They can also be represented with the help of a symbol. Basically they are used to represent the root of a number. The number can from the various number systems in mathematics. There are various number systems in mathematics.

The natural number system, whole number system, integer number system and real number system are some of them. There can also be irrational numbers and also complex numbers. The complex numbers have a real part and an imaginary part attached to them.

Do they are said to be complex numbers. The operations that are performed on real numbers like addition and subtraction can also be performed on the complex numbers as well. But to perform these operations one must be clear with the concept of the complex numbers.

The next question that arises is how do you simplify radicals in mathematics. The radicals have to be simplified in order to arrive at the final answer. The process to simplify radicals is quite simple. The root that has been applied to a number can be a square root, cube root and so on.

Sometimes fourth root can also be used. The number inside the square root symbol must be present two times then it can be taken out of the square root symbol. In case of cube root symbol it should be three times and in the case of the fourth root it should four times and so on.

In simplifying radicals practice is required; otherwise it becomes very tough to solve the problems. In mathematics practice is very important otherwise it will be very difficult to arrive at the final answer. Many mistakes can also be committed en-route to the answer. This can be avoided by proper practice.

There can questions to simplify radicals with variables like the simplification of normal radicals. The process is quite similar to the process of simplification of the radicals of the normal nature. Only proper practice can help one to reach at the right answer. There can be signs that can be used before the radical symbol.

The signs can be positive or negative in nature. Depending on the sign preceding the radical symbol the answer also changes. The positive denotes a positive number and a negative sign denotes a negative number. The magnitude also plays a very important role.

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