Solving systems of linear inequalities graphically

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In this lesson, we will start exploring systems of inequalities. First, we will consider the systems 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 the system, and recall the union and intersection of sets. At the end, we will solve specific examples for systems of linear inequalities.

Theme: The dietReal inequalities and their systems

Lesson:Mainconcepts, solution of systems of linear inequalities

Until now, we have solved individual inequalities and applied the method of intervals to them, it 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 domain 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 ox axis the set of solutions to the first and second inequalities.

The interval of intersection of two rays is our solution.

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

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

Let's depict this graphically. We have a set A of an arbitrary nature and a set B of an arbitrary nature, which 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 at specific examples solutions of linear systems of inequalities, how to find the 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 does the second inequality of the system come from? For example, from the inequality

Let us graphically designate the solutions to 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.

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

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

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

2. Mordkovich A.G. and others. Algebra 9th grade: Problem book for students of educational institutions / A. G. Mordkovich, T. N. Mishustina et al. - 4th ed. - M .: Mnemosina, 2002.-143 p .: ill.

3. Makarychev Yu. N. Algebra. Grade 9: textbook. for general education students. institutions / Yu. N. Makarychev, NG Mindyuk, KI Neshkov, IE Feoktistov. - 7th ed., Rev. and add. - M .: Mnemosina, 2008.

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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.

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1. Portal of Natural Sciences ().

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

4. Education Center "Teaching Technology" ().

5. Section College.ru in mathematics ().

1. Mordkovich A.G. and others. Algebra 9th grade: Problem book for students of educational institutions / A. G. Mordkovich, T. N. Mishustina et al. - 4th ed. - M.: Mnemozina, 2002.-143 p .: ill. No. 53; 54; 56; 57.

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 square 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 the systems of equations? Systems of equations you studied in 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 one needs to find all the values of x for which each of the inequalities forms the correct numeric expression.

Any value of x that makes each inequality a valid numeric expression is a solution to the inequality. It can also be called a private solution.
What is a particular solution? 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 common decision... The general solution is formed by many particular solutions.

How did we combine the system of equations? That's right, with a curly brace, so they do the same with inequalities. Let's consider an example of a system of inequalities: $ \ begin (cases) x + 7> 5 \\ x-3
If the system of inequalities consists of the same 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 particular solutions to an inequality that satisfy both inequalities of the system at once.

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

We denote by $ X_1 $ the general solution of the inequality f (x)> 0.
$ X_2 $ is a general solution to the inequality g (x)> 0.
$ X_1 $ and $ X_2 $ are a set of particular solutions.
The solution to the system of inequalities will be numbers belonging to both $ X_1 $ and $ X_2 $.
Let's remember set operations. How can we find elements of a set that belong to both sets at once? That's right, there is an intersection operation for that. 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
Solution.
a) Solve each inequality separately.
$ 3x-1> 2; \; 3x> 3; \; x> 1 $.
$ 5x-10
Let's mark our intervals on one coordinate line.

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

Let's summarize the knowledge gained.
Suppose it is necessary to solve the 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 $) is a solution to the first inequality.
The interval ($ y_1; y_2 $) is the solution to the second inequality.
The solution to a system of inequalities is the intersection of solutions to 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) $
Solution.
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 inequality is the gap.
Let's draw both intervals on one straight line and find the intersection.
Intersection of intervals - 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 ) $.

Solution.
a) The first inequality has a solution x> 1.
Let us 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 the 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

The system of inequalities it is customary to call any set of two or more inequalities containing an unknown quantity.

This formulation is clearly illustrated, for example, by such systems of inequalities:

Solve the system of inequalities - means to find all the values of the unknown variable for which each inequality of the system is realized, or to prove that there are no such .

Hence, for each individual system inequalities calculate the unknown variable. Further, from the resulting values, it selects only those that are true for both the first and second inequalities. Therefore, when the chosen value is substituted, both inequalities of the system become correct.

Let's analyze the solution to several inequalities:

Place one pair of number lines under the other; on the top we will apply the value x for which the first inequalities about ( x> 1) becomes true, and at the bottom, the value NS, which are a solution to the second inequality ( NS> 4).

Comparing the data on numeric straight lines, note that the solution for both inequalities will NS> 4. Answer, NS> 4.

Example 2.

Calculating the first inequality we get -3 NS< -6, или x> 2, the second - NS> -8, or NS < 8. Затем делаем по аналогии с предыдущим примером. На верхнюю числовую прямую наносим все те значения NS at which the first system inequality, and on the lower number line, all those values NS, at which the second inequality of the system is realized.

Comparing the data, we find that both inequalities will be realized for all values NS placed from 2 to 8. Sets of values NS denote double inequality 2 < NS< 8.

Example 3. Find