Concept mathematical inequality arose in ancient times. This happened when primitive man there was a need for counting and operations with various items compare their number and size. Since ancient times, Archimedes, Euclid and other famous scientists: mathematicians, astronomers, designers and philosophers used inequalities in their reasoning.
But they, as a rule, used verbal terminology in their works. For the first time modern signs to denote the concepts of “more” and “less” in the form in which every schoolchild knows them today, they were invented and put into practice in England. The mathematician Thomas Harriot provided such a service to his descendants. And this happened about four centuries ago.
There are many types of inequalities known. Among them are simple ones, containing one, two or more variables, quadratic, fractional, complex ratios, and even those represented by a system of expressions. The best way to understand how to solve inequalities is to use various examples.
Don't miss the train
To begin with, let’s imagine that a resident rural areas hurries to railway station, which is located at a distance of 20 km from his village. In order not to miss the train leaving at 11 o'clock, he must leave the house on time. At what time should this be done if its speed is 5 km/h? The solution to this practical problem comes down to fulfilling the conditions of the expression: 5 (11 - X) ≥ 20, where X is the departure time.
This is understandable, because the distance that a villager needs to cover to the station is equal to the speed of movement multiplied by the number of hours on the road. Come formerly man maybe, but there’s no way he can be late. Knowing how to solve inequalities and applying your skills in practice, you will end up with X ≤ 7, which is the answer. This means that the villager should go to the railway station at seven in the morning or a little earlier.
Numerical intervals on a coordinate line
Now let's find out how to map the described relations onto the The above inequality is not strict. It means that the variable can take values less than 7, or it can be equal to this number. Let's give other examples. To do this, carefully consider the four figures presented below.
On the first one you can see graphic image gap [-7; 7]. It consists of a set of numbers placed on a coordinate line and located between -7 and 7, including the boundaries. In this case, the points on the graph are depicted as filled circles, and the interval is recorded using
The second drawing is graphical representation strict inequality. In this case, the borderline numbers -7 and 7, shown by punctured (not filled in) dots, are not included in the specified set. And the recording of the interval itself is made in parentheses as follows: (-7; 7).
That is, having figured out how to solve inequalities of this type and received a similar answer, we can conclude that it consists of numbers that are between the boundaries in question, except -7 and 7. The next two cases must be evaluated in a similar way. The third figure shows images of the intervals (-∞; -7] U (0) (0) )
