Solving systems of linear inequalities graphically. The system of inequalities is the solution. System of linear inequalities

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any collection of two or more linear inequalities containing the same unknown quantity is called

Here are examples of such systems:

The intersection interval of two rays is our solution. Therefore, the solution of this inequality is all X located between two and eight.

Answer: X

The application of this type of mapping of the solution of a system of inequalities is sometimes called roof method.

Definition: The intersection of two sets BUT and AT is called such a third set, which includes all the elements included in and in BUT and in AT. This is the meaning of the intersection of sets of arbitrary nature. We are now considering numerical sets in detail, therefore, when finding linear inequalities, such sets are rays - co-directed, counter-directed, and so on.

Let's find out on real examples finding linear systems of inequalities, how to determine the intersection of the sets of solutions to individual inequalities included in the system.

Compute system of inequalities:

Let us place two lines of force one below the other. On the top we put those values X, which fulfill the first inequality x>7 , and on the bottom - which act as a solution to the second inequality x>10 We correlate the results of the number lines, find out that both inequalities will be satisfied for x>10.

Answer: (10;+∞).

We do by analogy with the first sample. On a given numerical axis, plot all those values X for which the first exists system inequality, and on the second numerical axis, placed under the first, all those values X, for which the second inequality of the system is satisfied. Let us compare these two results and determine that both inequalities will simultaneously be satisfied for all values X located between 7 and 10, taking into account the signs, we get 7<x≤10

Answer: (7; 10].

The following are solved in the same way. systems of inequalities.

In this lesson, we will begin the study of systems of inequalities. First, we will consider systems of linear inequalities. At the beginning of the lesson, we will consider where and why systems of inequalities arise. Next, we will study what it means to solve a system, and remember the union and intersection of sets. In the end, we will solve specific examples for systems of linear inequalities.

Subject: dietreal inequalities and their systems

Lesson:Mainconcepts, solution of systems of linear inequalities

Until now, we have solved individual inequalities and applied the interval method to them, these could be linear inequalities, and square and rational. Now let's move on to solving systems of inequalities - first linear systems. Let's look at an example where the need to consider systems of inequalities comes from.

Find the scope of a function

Find the scope of a function

The function exists when both square roots exist, i.e.

How to solve such a system? It is necessary to find all x satisfying both the first and second inequalities.

Draw on the x-axis the set of solutions to the first and second inequalities.

The intersection interval of two rays is our solution.

This method of representing the solution of a system of inequalities is sometimes called the roof method.

The solution of the system is the intersection of two sets.

Let's represent this graphically. We have a set A of arbitrary nature and a set B of arbitrary nature that intersect.

Definition: The intersection of two sets A and B is a third set that consists of all the elements included in both A and B.

Consider, using specific examples of solving linear systems of inequalities, how to find intersections of the sets of solutions of individual inequalities included in the system.

Solve the system of inequalities:

Answer: (7; 10].

4. Solve the system

Where can the second inequality of the system come from? For example, from the inequality

We graphically denote the solutions of each inequality and find the interval of their intersection.

Thus, if we have a system in which one of the inequalities satisfies any value of x, then it can be eliminated.

Answer: the system is inconsistent.

We have considered typical support problems, to which the solution of any linear system of inequalities is reduced.

Consider the following system.

7.

Sometimes a linear system is given by a double inequality; consider this case.

8.

We considered systems of linear inequalities, understood where they come from, considered typical systems to which all linear systems reduce, and solved some of them.

1. Mordkovich A.G. and others. Algebra 9th grade: Proc. For general education Institutions. - 4th ed. - M.: Mnemosyne, 2002.-192 p.: ill.

2. Mordkovich A.G. et al. Algebra Grade 9: Taskbook for students of educational institutions / A. G. Mordkovich, T. N. Mishustina et al. - 4th ed. — M.: Mnemosyne, 2002.-143 p.: ill.

3. Yu. N. Makarychev, Algebra. Grade 9: textbook. for general education students. institutions / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, I. E. Feoktistov. - 7th ed., Rev. and additional - M .: Mnemosyne, 2008.

4. Alimov Sh.A., Kolyagin Yu.M., Sidorov Yu.V. Algebra. Grade 9 16th ed. - M., 2011. - 287 p.

5. Mordkovich A. G. Algebra. Grade 9 At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich, P. V. Semenov. - 12th ed., erased. — M.: 2010. — 224 p.: ill.

6. Algebra. Grade 9 At 2 hours. Part 2. Task book for students of educational institutions / A. G. Mordkovich, L. A. Aleksandrova, T. N. Mishustina and others; Ed. A. G. Mordkovich. - 12th ed., Rev. — M.: 2010.-223 p.: ill.

1. Portal of Natural Sciences ().

2. Electronic educational and methodological complex for preparing grades 10-11 for entrance exams in computer science, mathematics, Russian language ().

4. Education Center "Technology of Education" ().

5. College.ru section on mathematics ().

1. Mordkovich A.G. et al. Algebra Grade 9: Taskbook for students of educational institutions / A. G. Mordkovich, T. N. Mishustina et al. - 4th ed. - M .: Mnemosyne, 2002.-143 p.: ill. No. 53; 54; 56; 57.

The system of inequalities.
Example 1. Find the scope of an expression
Decision. There must be a non-negative number under the square root sign, which means that two inequalities must simultaneously hold: In such cases, the problem is said to be reduced to solving the system of inequalities

But we have not met with such a mathematical model (system of inequalities) yet. This means that we are not yet able to complete the solution of the example.

The inequalities that form a system are combined with a curly bracket (the same is the case in systems of equations). For example, the entry

means that the inequalities 2x - 1 > 3 and 3x - 2< 11 образуют систему неравенств.

Sometimes the system of inequalities is written as a double inequality. For example, the system of inequalities

can be written as a double inequality 3<2х-1<11.

In the 9th grade algebra course, we will only consider systems of two inequalities.

Consider the system of inequalities

You can pick up several of its particular solutions, for example x = 3, x = 4, x = 3.5. Indeed, for x = 3 the first inequality takes the form 5 > 3, and the second - the form 7< 11. Получились два верных числовых неравенства, значит, х = 3 - решение системы неравенств. Точно так же можно убедиться в том, что х = 4, х = 3,5 - решения системы неравенств.

At the same time, the value x = 5 is not a solution to the system of inequalities. For x = 5, the first inequality takes the form 9 > 3 - the correct numerical inequality, and the second - the form 13< 11- неверное числовое неравенство .
To solve a system of inequalities means to find all its particular solutions. It is clear that such guessing as demonstrated above is not a method for solving a system of inequalities. In the following example, we will show how one usually argues when solving a system of inequalities.

Example 3 Solve the system of inequalities:

Decision.

a) Solving the first inequality of the system, we find 2x > 4, x > 2; solving the second inequality of the system, we find Zx< 13 Отметим эти промежутки на одной координатной прямой , использовав для выделения первого промежутка верхнюю штриховку, а для второго - нижнюю штриховку (рис. 22). Решением системы неравенств будет пересечение решений неравенств системы, т.е. промежуток, на котором обе штриховки совпали. В рассматриваемом примере получаем интервал
b) Solving the first inequality of the system, we find x > 2; solving the second inequality of the system, we find We mark these gaps on one coordinate line, using the top hatching for the first gap, and the bottom hatching for the second (Fig. 23). The solution of the system of inequalities will be the intersection of the solutions of the inequalities of the system, i.e. the interval where both hatches coincide. In the example under consideration, we get a beam

in) Solving the first inequality of the system, we find x< 2; решая второе неравенство системы, находим Отметим эти промежутки на одной координатной прямой, использовав для первого промежутка верхнюю штриховку, а для второго - нижнюю штриховку (рис. 24). Решением системы неравенств будет пересечение решений неравенств системы, т.е. промежуток, на котором обе штриховки совпали. Здесь такого промежутка нет, значит, система неравенств не имеет решений.

Let us generalize the reasoning carried out in the considered example. Suppose we need to solve a system of inequalities

Let, for example, the interval (a, b) be the solution to the inequality fx 2 > g (x), and the interval (c, d) be the solution to the inequality f 2 (x) > s 2 (x). We mark these gaps on one coordinate line, using the top hatching for the first gap, and the bottom hatching for the second (Fig. 25). The solution of the system of inequalities is the intersection of the solutions of the inequalities of the system, i.e. the interval where both hatches coincide. On fig. 25 is the interval (s, b).

Now we can easily solve the system of inequalities that we got above, in example 1:

Solving the first inequality of the system, we find x > 2; solving the second inequality of the system, we find x< 8. Отметим эти промежутки (лучи) на одной координатной прямой, использовав для первого -верхнюю, а для второго - нижнюю штриховку (рис. 26). Решением системы неравенств будет пересечение решений неравенств системы, т.е. промежуток, на котором обе штриховки совпали, - отрезок . Это - область определения того выражения, о котором шла речь в примере 1.

Of course, the system of inequalities does not have to consist of linear inequalities, as has been the case so far; any rational (and not only rational) inequalities can occur. Technically, working with a system of rational non-linear inequalities is, of course, more difficult, but there is nothing fundamentally new (compared to systems of linear inequalities).

Example 4 Solve the system of inequalities

Decision.

1) Solve the inequality We have
Note the points -3 and 3 on the number line (Fig. 27). They divide the line into three intervals, and on each interval the expression p (x) = (x - 3) (x + 3) retains a constant sign - these signs are indicated in Fig. 27. We are interested in the intervals where the inequality p(x) > 0 is satisfied (they are shaded in Fig. 27), and the points where the equality p(x) = 0 is satisfied, i.e. points x \u003d -3, x \u003d 3 (they are marked in Fig. 2 7 with dark circles). Thus, in fig. 27 shows a geometric model for solving the first inequality.

2) Solve the inequality We have
Note the points 0 and 5 on the number line (Fig. 28). They divide the line into three intervals, and on each interval the expression<7(х) = х(5 - х) сохраняет постоянный знак - эти знаки указаны на рис. 28. Нас интересуют промежутки, на которых выполняется неравенство g(х) >O (shaded in Fig. 28), and the points at which the equality g (x) - O is satisfied, i.e. points x = 0, x = 5 (they are marked in Fig. 28 by dark circles). Thus, in fig. 28 shows a geometric model for solving the second inequality of the system.

3) We mark the solutions found for the first and second inequalities of the system on the same coordinate line, using the upper hatching for the solutions of the first inequality, and the lower hatching for the solutions of the second (Fig. 29). The solution of the system of inequalities will be the intersection of the solutions of the inequalities of the system, i.e. the interval where both hatches coincide. Such an interval is a segment.

Example 5 Solve the system of inequalities:

Decision:

a) From the first inequality we find x >2. Consider the second inequality. The square trinomial x 2 + x + 2 has no real roots, and its leading coefficient (the coefficient at x 2) is positive. This means that for all x the inequality x 2 + x + 2>0 is satisfied, and therefore the second inequality of the system has no solutions. What does this mean for the system of inequalities? This means that the system has no solutions.

b) From the first inequality we find x > 2, and the second inequality holds for any values ​​of x. What does this mean for the system of inequalities? This means that its solution has the form x>2, i.e. coincides with the solution of the first inequality.

Answer:

a) there are no decisions; b) x>2.

This example is an illustration for the following useful

1. If in a system of several inequalities with one variable one inequality has no solutions, then the system has no solutions.

2. If in a system of two inequalities with one variable one inequality is satisfied for any values ​​of the variable , then the solution of the system is the solution of the second inequality of the system.

Concluding this section, let us return to the problem of the conceived number given at the beginning of it and solve it, as they say, according to all the rules.

Example 2(see p. 29). Think of a natural number. It is known that if 13 is added to the square of the conceived number, then the sum will be greater than the product of the conceived number and the number 14. If 45 is added to the square of the conceived number, then the sum will be less than the product of the conceived number and the number 18. What number is conceived?

Decision.

First stage. Drawing up a mathematical model.
The intended number x, as we saw above, must satisfy the system of inequalities

Second phase. Working with the compiled mathematical model. Let's transform the first inequality of the system to the form
x2- 14x+ 13 > 0.

Let's find the roots of the trinomial x 2 - 14x + 13: x 2 \u003d 1, x 2 \u003d 13. Using the parabola y \u003d x 2 - 14x + 13 (Fig. 30), we conclude that the inequality of interest to us is satisfied for x< 1 или x > 13.

Let us transform the second inequality of the system to the form x2 - 18 2 + 45< 0. Найдем корни трехчлена х 2 - 18x + 45: = 3, х 2 = 15.

Lesson and presentation on the topic: "Systems of inequalities. Examples of solutions"

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System of inequalities

Guys, you have studied linear and quadratic inequalities, learned how to solve problems on these topics. Now let's move on to a new concept in mathematics - a system of inequalities. The system of inequalities is similar to the system of equations. Do you remember systems of equations? You studied systems of equations in the seventh grade, try to remember how you solved them.

Let us introduce the definition of a system of inequalities.
Several inequalities with some variable x form a system of inequalities if you need to find all values ​​of x for which each of the inequalities forms a true numerical expression.

Any value of x such that each inequality evaluates to a valid numeric expression is a solution to the inequality. It can also be called a private decision.
What is a private decision? For example, in the answer we received the expression x>7. Then x=8, or x=123, or some other number greater than seven is a particular solution, and the expression x>7 is a general solution. The general solution is formed by a set of particular solutions.

How did we combine the system of equations? That's right, a curly brace, so they do the same with inequalities. Let's look at an example of a system of inequalities: $\begin(cases)x+7>5\\x-3
If the system of inequalities consists of identical expressions, for example, $\begin(cases)x+7>5\\x+7
So, what does it mean to find a solution to a system of inequalities?
A solution to an inequality is a set of partial solutions to an inequality that satisfies both inequalities of the system at once.

We write the general form of the system of inequalities as $\begin(cases)f(x)>0\\g(x)>0\end(cases)$

Let $X_1$ denote the general solution of the inequality f(x)>0.
$X_2$ is the general solution of the inequality g(x)>0.
$X_1$ and $X_2$ are the set of particular solutions.
The solution of the system of inequalities will be the numbers belonging to both $X_1$ and $X_2$.
Let's look at operations on sets. How can we find the elements of a set that belong to both sets at once? That's right, there is an intersection operation for this. So, the solution to our inequality will be the set $A= X_1∩ X_2$.

Examples of solutions to systems of inequalities

Let's see examples of solving systems of inequalities.

Solve the system of inequalities.
a) $\begin(cases)3x-1>2\\5x-10 b) $\begin(cases)2x-4≤6\\-x-4
Decision.
a) Solve each inequality separately.
$3x-1>2; \; 3x>3; \; x>1$.
$5x-10
We mark our intervals on one coordinate line.

The solution of the system will be the segment of the intersection of our intervals. The inequality is strict, then the segment will be open.
Answer: (1;3).

B) We also solve each inequality separately.
$2x-4≤6; 2x≤ 10; x ≤ $5.
$-x-4 -5$.


The solution of the system will be the segment of the intersection of our intervals. The second inequality is strict, then the segment will be open on the left.
Answer: (-5; 5].

Let's summarize what we've learned.
Suppose we need to solve a system of inequalities: $\begin(cases)f_1 (x)>f_2 (x)\\g_1 (x)>g_2 (x)\end(cases)$.
Then, the interval ($x_1; x_2$) is the solution to the first inequality.
The interval ($y_1; y_2$) is the solution to the second inequality.
The solution of a system of inequalities is the intersection of the solutions of each inequality.

Systems of inequalities can consist of inequalities not only of the first order, but also of any other types of inequalities.

Important rules for solving systems of inequalities.
If one of the inequalities of the system has no solutions, then the whole system has no solutions.
If one of the inequalities is satisfied for any values ​​of the variable, then the solution of the system will be the solution of the other inequality.

Examples.
Solve the system of inequalities:$\begin(cases)x^2-16>0\\x^2-8x+12≤0 \end(cases)$
Decision.
Let's solve each inequality separately.
$x^2-16>0$.
$(x-4)(x+4)>0$.

Let's solve the second inequality.
$x^2-8x+12≤0$.
$(x-6)(x-2)≤0$.

The solution to the inequality is a gap.
Let's draw both intervals on one straight line and find the intersection.
The intersection of the intervals is the segment (4; 6].
Answer: (4;6].

Solve the system of inequalities.
a) $\begin(cases)3x+3>6\\2x^2+4x+4 b) $\begin(cases)3x+3>6\\2x^2+4x+4>0\end(cases )$.

Decision.
a) The first inequality has a solution x>1.
Let's find the discriminant for the second inequality.
$D=16-4 * 2 * 4=-16$. $D Recall the rule, when one of the inequalities has no solutions, then the whole system has no solutions.
Answer: There are no solutions.

B) The first inequality has a solution x>1.
The second inequality is greater than zero for all x. Then the solution of the system coincides with the solution of the first inequality.
Answer: x>1.

Problems on systems of inequalities for independent solution

Solve systems of inequalities:
a) $\begin(cases)4x-5>11\\2x-12 b) $\begin(cases)-3x+1>5\\3x-11 c) $\begin(cases)x^2-25 d) $\begin(cases)x^2-16x+55>0\\x^2-17x+60≥0 \end(cases)$
e) $\begin(cases)x^2+36