It is the existing inequality between two algebraic expressions, connected through the signs: greater than>, less than <, less than or equal to ≤, as well as greater than or equal to ≥, in which one or more unknown values ​​called unknowns appear, in addition to certain known data.


The existing inequality between the two algebraic expressions is only verified, or rather, it is only true for certain values ​​of the unknown.

The solution of a formulated inequality means to determine, through certain procedures, the value that satisfies it.

If we formulate the following algebraic inequality, we will be able to notice in it the elements indicated above. Let’s see:

9x – 12 <24

As can be seen in the example, there are two members in the inequality. The member on the left and the member on the right are present. In this case the inequality is connected through the century less than. The quotient 9 and the numbers 12 and 24 are the known facts.

Classification of inequalities

There are different types of inequalities. These can be classified according to the number of unknowns and according to their degree. To know the degree of an inequality, it is enough to identify the largest of them. Thus, we have the following types:

  • Of an unknown
  • Of two unknowns
  • Of three unknowns
  • Of n unknowns
  • First grade
  • Second grade
  • Third grade
  • Fourth grade
  • Inequalities of degree N

Operating with inequalities

Before solving an example of inequalities, it is convenient to indicate the following properties:

  • When a value that you are adding passes to the other side of the inequality, a minus sign is put on it.
  • If a value that is subtracting passes to the other side of the inequality, a plus sign is put on it.
  • When a value you are dividing passes to the other side of the inequality, it will multiply everything on the other side.
  • If a value is multiplying passes to the other side of the inequality, then it will pass dividing everything on the other side.

It is indifferent, to go from left to right or from right to left of the inequality. The important thing is not to forget the sign changes. Also, it does not matter which way we solve the unknowns.

Worked example of inequality

To see in depth the process of solving an inequality, we are going to propose the following:

15x + 18 <12x -24

To solve this inequality we must solve for the unknown. To do this, first, we proceed to group like terms. Basically, this part consists of passing all the unknowns to the left side and all the constants to the right side. So we have.

15x – 12x <-24 - 18

Adding and subtracting these like terms. Have.

3x <- 42

Finally, we now proceed to take off the unknown and determine its value.

x <- 42/3

x <- 14

In this way, all values ​​less than -14 correctly satisfy the formulated inequality.

Inequality systems

When two or more inequalities are formulated together, we then speak of systems of inequalities. An example of the formulation of an inequality system is the following:

18x + 22 <12x - 14 (1)

9x> 6 (2)

In this system, the two inequalities must be fulfilled for the system to have a solution. That is, the solution is the values ​​of ‘x’ that allow inequality (1) and (2) to be fulfilled at the same time.

Worked Example of Inequation System

The process of solving an inequality system does not turn out to be complicated, since for its resolution it is enough to solve each of the formulated inequalities separately.

To see this resolution process, let’s take the following inequality system as a reference:

18x + 22 <12x - 14

9x> -6

We solve the first inequality of the system, through the procedure seen in the resolution of inequalities.

18x – 12x <-22 -14

6x <-36

x <-36/6

x <- 9

Now we solve the second inequality of the system.

9x <-9

X <-9/9

X <-1

It should be noted that not all systems of inequalities have a solution.