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