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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