How to solve linear inequalities

In the previous post we have discussed about Absolute Value Inequalities and In today's session we are going to discuss about How to solve linear inequalities. If graph of an equation is a straight line then equation is called as linear equation. For example: q = mp + c; here ‘m’ shows the slope of line and ‘c’ shows Y- intercept where line crosses q – axis (here ‘p’ is along to the horizontal axis and ‘q’ is along to vertical axis. If (<, >) these symbols are there in a linear expression then it comprise inequality in it. Now we will understand the concept of linear inequalities. It is fully depends on symbol that present in inequality. If less than sign present in linear expression then we found inequality under the line. If grater than sign is present in linear expression then we found inequalities top the line.
Let’s understand the concept of solving systems of linear inequalities. Let's we have a linear inequality 4a + b < 15, then we can calculate this linear inequality as mention below:
In the above given linear inequality is there so after calculating, coordinates we found is under the line. Here set different values for one variable to get other coordinates. So it can be written as:
=> 4a + b < 15, to find coordinates of linear inequalities replace inequality symbol by equal sign.
=> b = 15 – 4a.
On putting value of a = 1 we get:
=> b = 15 – 4 (1),
=> b = 11.
On putting value of a = 3 we get:
=> b = 15 – 4 (3), on further solving we get:
=> b = 15 – 12,
=> b = 3.
In this way we can find different values. So here we get (1, 11), (3, 3). If we plot the graph we get inequalities below the line.
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Absolute Value Inequalities

In mathematics, Absolute value is a measure of how energies a number is from 0. For example: ‘15’ is 15 point far from zero and -58 is 58 point far from zero term. Absolute value of number 0 is 0 and absolute value of 150 is 150. The entire values are the examples of absolute value function. In the case of absolute value negative numbers are not taken. Let's we have given any negative number then it is very essential to avoid negative sign from number. So we can have only positive values and zero. Absolute value function is denoted by the symbol '|'. This symbol is also said to be bar. If we plot any negative number among this symbol then we found positive number outside this symbol. Now we will discuss process of calculating Absolute Value Inequalities.
If any of the given symbol (<,> <, >) are there in any expression then we can say that equation have inequality in it. Let's us discuss how to calculate absolute value inequalities. Let's we have | 2a + 3 | < 8, absolute value inequality.
Solution: Given inequality | 2a + 3 | < 8, then first we will calculate linear inequality. So it can be written as:
=> - 8 < 2a + 3 < 8, it means 2a + 3 is larger than -8 and shorter than 8. Then subtract 3 from inequality. On subtracting value 3 we get:
=> - 8 – 3 < 2a + 3 – 3 < 8 – 3, on moving ahead we get:
=> -11 < 2a < 5.
Then divide entire inequality by 2, on dividing we get:
=> - 11 / 2 < a < 5 / 2,
On solving | 2a + 3 | < 8, we get –11 / 2 < a < 5 / 2. In this way we can solve any inequalities values.
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Solving Inequalities

Inequalities are the expressions that shows without using any equal sign means it will show as an expression that have signs of less than or greater than. Some times it is possible that these expression also have the equal sign but along with the less than or greater than sign that is also describe as a less than equal to or greater than equal to sign. Solving Inequalities having the same pattern like equations that is describe as an expression with the equal sign. One thing keep in the mind at the time of solving the inequality is that whenever change the side of the variable or values sign will change as > greater sign will change into the < and < less than sign will change into the > greater than sign.
There are some rules that does not effects the change of the sign as follows:
Whenever any number will be subtracted from any one of side then the same process is also done from other side.
When multiply both side of the inequality with positive number will also not effect the sign of inequality.
But there are also some ways that will change the sign as if multiply with the negative number or when we slide the number from one side to another side. we can simply define it by an example: 3 n < 7.
Now in the above inequality we have to put the values in place of n that satisfy inequality. there is no need to change in the sign.
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Solving Inequalities

If we have two values and both given values are not equal then we say that there is an inequality in both the values. Some of the conditions of an inequality are:

Suppose we have two variables ‘u’ and ‘v’. If u ≠ v; which represents u is not equal to v, and if u < v then we say that u is less than v, and if u > v then we say that u is greater then v. when u <= v, it means u is less than equal to v. When u >= v then u is greater than equal to v.

Now we will understand how to Solving Inequalities using multiplication and division? To find inequalities by multiplication and division we need to follow some steps which are shown below:

Step1: To solve inequality first of all we have to take an equation.

Step2: Remember that the inequality equation should only be defined in above symbols.

Step3: If equal sign is present in between the equation then the given equation never contain any inequality. Now we will understand it with the help of small example: (know more about Solving Inequalities, here)

Lets we have –a / 5 > 21; then solving inequalities by multiplication or division method. To solve inequality we need to follow all the above mention steps:

To find inequality first we have to take inequality, and if it has < or > signs then it is an inequality. So, the given equality can be written as:

= –a / 5 > 21,

To solve the inequality we have to multiply -5 on both sides of equation. On multiplying -5 we get:

= – a / 5 > 2,

= (-5) * -a / 5 > 21 * -5,

= 5a / 5 > -105, On further solving inequality we get:

If we divide 5 from 5 then we get 1. Now we have equation like this:

= a > -105, So it can be written as:

= a < -105. In this way we can solve inequality.

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solving systems of inequalities by graphing

In the previous post we have discussed about analyzing equations and inequalities and In today's session we are going to discuss about solving systems of inequalities by graphing. If variables are not equals to each other then we can say it is an inequality. Now we will discuss some of the conditions for an inequality which are: let we have two variables ‘p’ and ‘q’ then:
Condition1: If p ≠ q; that represents ‘p’ is not equals to ‘q’;
Condition 2: If p < q then it represents ‘p’ is less than ‘q’;
Condition3: If p > q then it represents ‘p’ is greater than ‘q’.
If the given conditions are present then we say that inequality is present. Now we will discuss how to solving systems of inequalities by graphing. We need to follow some steps to plot the graph. We know that there are many methods for solving systems of inequalities but solving graphically is the one of the best method. (know more about Inequality, here)
Step 1: To plot graph first we have an inequality equation. Let we have an equation x + y ≤10.
Step 2: Then put the different values of x – coordinate to get the other value of y – coordinate. So we can write the above equation as:
Y ≤- x + 10,
Step 3: If we put the value of x coordinate is ‘1’ then we get the value of y – coordinate is 9.
If we put the value of ‘x’ is 0 then we get the value of ‘y’ is 10. If we put value of ‘x’ is ‘4’ then we get the value of ‘y’ is ‘6’ and we put the value of ‘x’ coordinate is ‘6’ then we get the value of ‘y’ is ‘4’. So we get the ‘x’ and ‘y’ coordinates as: (1, 9), (0, 10), (4, 6), (6, 4). Now we can easily plot the graph of the given inequality. The graph is shown below.



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analyzing equations and inequalities

Before analyzing equations and inequalities first of all it is necessary to know about the equation and inequality. As we know that if equal sign is present in an expression then it is said to be equation. If less than, greater than, less than equal to and greater than equal to operator is present in an expression then it is said to be an inequality. We have to focus on some points to analyzing equation and inequality. Now first of all we will see the properties of equation or (Real number). (know more about Inequality, here)

	Addition	Multiplication
Commutative	P + q = q + p	pq = qp
Associative	(p + q) + r = p + (q + r)	(pq) r = p (qr)
Identity	P + 0 = p = 0 + p	p (1) = p = 1 (p)
Inverse	P + (-p) = 0 = (-p) + p	If p ≠0 then p (1/p) = 1 = 1/p(p)

If we talk about the inequality property then we can write the properties as: For any two real number p and q, one of the given statements is true. p < q, p = q, p > q

Addition and subtraction properties for inequality	For any real numbers p, q and r: 1.      If p > q then p + r > q + r and p – r > q – r 2.      If p < q then p + r < q + r and p – r < q – r
Multiplication and Division property of inequality	For any real number p, q and r: 1.      If  r is positive and p < q then pr < qr and p/r < q/r 2.      If  r is positive and p > q then pr > qr and p/r > q/r 3.      If  r is negative and p < q then pr > qr and p/r > q/r 4.      If  r is positive and p > q then pr < qr and p/r < q/r

If we apply these properties then we can easily analysis the equation and inequality. VSEPR Theory is based on chemistry. The icse guess papers 2013 are useful for the preparation of exam.

How to Solve Inequalities

In the previous post we have discussed about How to Draw Bar Graph and In today's session we are going to discuss about How to Solve Inequalities, Algebra can be consider as a important part of mathematics that perform the task of converting the real world problem in the form of mathematical equation or find the value of a variable in mathematical equation. In mathematics, an algebraic equation represents the relationship between the both side of equal sign in algebraic equation. It means anything happen with number on one side also reflect the changes on other side.
Algebraic equation basically deals with solving an equation by finding the value of variable. It also perform task of solving an inequalities of algebraic equation. Here in this section we are going to discuss about the topic how to solve inequalities.
In mathematics, the concept of inequalities can be represented by some of the symbol that are given below:
a ) >= greater then equal to
b )> greater then
c ) <= less then equal to
d ) < = less then
as we all are very well aware that algebraic expression is a combination of number and variables to represent the real world problem in a mathematical form. When such kind of symbol are used with algebraic expression then a question arises in our mind that how to solve inequalities.
The solution of solving inequalities from an equation is already defined by mathematics. To solve the inequalities from an algebraic expression can be done by two ways.
A ) first is “ By adding or subtracting a value from both side”: It means to say that inequalities can be solve by adding or subtracting the same value on both side. Like there is a equation a + 5 < 10 then it can be solve as a + 5 – 5 < 10 – 5 = a < 5. The obtained result shows that the value of x must be less then 5 to set the equation true.
B ) By dividing and multiplying the value. The concept of 

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