How to solve inequalities? How to solve fractional and square inequalities?
The concept of mathematical inequality arose inof antiquity. This happened when a primitive man had a need to compare and multiply the number and magnitude of counts and actions with different objects. Since ancient times, Archimedes, Euclid, and other celebrated figures of science have used inequalities in their reasoning: mathematicians, astronomers, designers and philosophers.
But they, as a rule, applied in their worksverbal terminology. For the first time, modern signs for the notions of "more" and "less", as they are known to every schoolboy today, have been invented and applied in practice in England. The mathematician Thomas Garriott provided such a service to descendants. And it happened about four centuries ago.
There are many types of inequalities. Among them, simple, containing one, two or more variables, square, fractional, complex relationships and even represented by a system of expressions. And to understand how to solve inequalities, it is best on different examples.
Do not miss the train
To begin with, imagine that a villagerThe area rushes to the railway station, which is located at a distance of 20 km from his village. To not be late for the train departing at 11 o'clock, he must leave the house in time. In which hour is it necessary to do this if the speed of its movement is 5 km / h? The solution of this practical task is to fulfill the conditions of expression: 5 (11 - X) ≥ 20, where X is the time of departure.
This is understandable, because the distance that is neededto overcome the peasant to the station is equal to the speed of movement, multiplied by the number of hours on the road. A man can come earlier, but he can not be late. Knowing how to solve inequalities, and applying your skills in practice, we eventually get X ≤ 7, which is the answer. This means that the peasant should go to the railway station at seven in the morning or a little earlier.
Number gaps on the coordinate line
Now find out how to display the describedrelations on the coordinate line. The inequality obtained above is not strict. It means that the variable can take values less than 7, and can be equal to this number. We give other examples. To do this, carefully consider the four figures below.
On the first one you can see the graphicgap image [-7; 7]. It consists of a set of numbers placed on the coordinate line and located between -7 and 7, including borders. In this case, the points on the graph are represented in the form of filled circles, and the gap is recorded using square brackets.
The second picture is a graphicrepresentation of strict inequality. In this case, the boundary numbers -7 and 7, shown by punctured (not shaded) points, are not included in the specified set. And the record of the gap itself is made in parentheses as follows: (-7; 7).
That is, figuring out how to solve inequalitiesof this type, and having received a similar answer, you canconclude that it consists of numbers between the considered boundaries, except for -7 and 7. The following two cases must be evaluated in a similar way. In the third figure, images of the intervals (-∞; -7] U [7; + ∞) are given, and on the fourth figure, images (-∞; -7) U (7; + ∞) are given.
Two expressions in one
You can often find the following entry: 7 <2X - 3 <12. How to solve double inequalities? This means that two conditions are immediately superimposed on the expression. And each of them must be taken into account in order to obtain the correct answer for the variable X. Taking this into account, we obtain from the relations 2X-3> 7 and 2X-3 <11 the following:
5 <X <7. The final answer is written thus: (5; 7). This means that the variable takes a set of values enclosed in the gap between numbers 5 and 7, excluding boundaries.
Similar properties with equation
Equation is an expression,united by the sign =, which means that both its parts (left and right) are identical in magnitude. Therefore, often these relationships are associated with the image of old scales, having bowls installed and fastened by means of a lever. This device is always in balance if both ends are weight-balanced. In this case, the position does not change if the left and right parts are supplemented or lose loads of the same mass.
In the mathematical equation to both partsequality, so that it is not violated, you can also add the same number. In this case, it can be positive or negative. How to solve inequalities in this case, and can you do the same with them? Previous examples have shown that yes.
The difference from equation
Both parts of the expression, connected by <or> signs,can be multiplied and divided by any positive number. In this case, the truth of the relation is not violated. But how to solve the inequality with fractions with negative and integer multipliers, before which there is a minus sign? Here the situation is completely different.
Let's analyze this with the example: -3X <12. To select a variable on the left side, you have to divide each of them by -3. In this case, the inequality sign reverses. We get: X> -4, which is the answer to the problem.
Method of intervals
An inequality is said to be quadratic if it contains a variable raised to the second power. An example of such a relation is the following expression: X2- 2X + 3> 0. How to solve quadratic inequalities? The most convenient method is the interval method. To implement this, the left side of the ratio should be factorized. It turns out: (X - 3) (X + 1). Then it is recommended to find the zeros of the function and arrange the resulting points in the correct order on the coordinate line.
Next, you need to distribute the signs of the resultingintervals, substituting in the expression any of the numbers belonging to a given interval. In simple cases, it is usually enough to understand at least one of them, and the rest - to arrange by the rule of alternation. In conclusion, it only remains to select the appropriate intervals to get the final solution.
Square inequalities here obey the lawthe correspondence of negative areas to minuses, and positive ones to pluses. That is, if the expression is greater than zero, then we must take the numerical gaps marked with the + sign. In the opposite case, the solution will be the sections marked with the -. Thus, the solution of our inequality can be written as follows: (-∞; -1) U (3; + ∞).
Other examples of the application of the interval method
The described method gives an answer to the otheran important question: how to solve fractional inequalities, if in this case the same method of intervals is quite applicable? Let us consider in more detail how this can be done, using the example of the relation presented below.
Here the zeros of the function are the points -9 and 4. To find the solution, we must apply them to the coordinate line and determine the signs of the intervals, selecting those that are marked with the plus sign. It should be noted that only the number 4 will be filled.
The other point will be deleted, since -9 does not enterin the range of values that are acceptable. After all, the denominator is zero, which is impossible in mathematics. How to solve fractional inequalities? In this case, the final answer is the union of gaps: (-∞; -9) U [4; + ∞).
Parabolas on the graph
To find out all about inequalities often help notOnly the figures on the coordinate line, but also the images in the Cartesian plane. The graph of a quadratic dependence is known to be a parabola. Even a schematic drawing of this type is able to provide almost complete answers to the questions posed. We consider some of the types of parabolas giving ideas about the solution of quadratic inequalities.
Here, first of all, we will clarify some truths for ourselves. Any expression of this type is reduced to the form: ax2+ вх + с = 0. In this case, if the coefficient a turns out to be positive, then the parabola should be drawn with the branches up, in the opposite case - down. And the roots of the equation are the points where the graph of the function intersects the axis OX.
Know the above statements are very important forunderstanding of square inequalities and answers to questions related to them. Having drawn the parabola scheme on the Cartesian plane, it is necessary to find out at which point the function (that is, the values of the coordinates of the points along the OY axis) takes the + and - indices. Moreover, if the inequality contains the sign>, then its solution will be the set of values accepted by the variable X for positive Y.
In the case of the sign <in the response, the indicatorsfor X with negative Y. It happens that the parabola does not intersect the axis OX at all. This occurs in cases when A <0. Then, if the graph is located in the upper half-plane, the answer (-∞; + ∞) for the square inequality with the sign> turns out to be. And for <the solution there is an empty set. With the lower half-plane, this is the case with accuracy and vice versa.
About the benefits of graphics
Images on the Cartesian plane significantlyfacilitate the problem for systems of equations. Figures clearly show solutions that are points of intersection of the lines applied. It remains only to calculate their coordinates and record the answer.
The same is true for inequalities. For example, the solution y ≤ 6 - x (as is clear from the figure) is the straight line y = 6 - x, as well as the half-plane located below this boundary. For an exact answer, we can take any point on the graph (for example (1; 3) and substitute its coordinates in the inequality.) We get: 3 ≤ 6 - 1, that is, the correct ratio.
Inequality y ≥ x2is described by a region on the Cartesian plane located in the bowl of the parabola, including its boundaries. And at the intersection of these sectors, we can find a solution of the relation written in the form: x2≤ y ≤ 6 - x. It will be limited from below by the line of the parabola and cut off from above by a straight line. To be sure, we will make a check again, substituting the coordinates of any point that belongs to this region.
Take (1; 4). We get: 1 ≤ 4 ≤ 6 - 1, that is again the correct ratio. Here again it makes sense to note that inequalities have many similarities with equations, although they are endowed with significant differences.