Number we will compose a system of inequalities. Systems of inequalities - Knowledge Hypermarket. Solving systems of inequalities. General information. Solutions
Solving inequalities online
Before solving inequalities, you need to have a good understanding of how equations are solved.
It doesn’t matter whether the inequality is strict () or non-strict (≤, ≥), the first step is to solve the equation by replacing the inequality sign with equality (=).
Let us explain what it means to solve an inequality?
After studying the equations, the following picture emerges in the student’s head: he needs to find values of the variable such that both sides of the equation take same values. In other words, find all points at which equality holds. Everything is correct!
When we talk about inequalities, we mean finding intervals (segments) on which the inequality holds. If there are two variables in the inequality, then the solution will no longer be intervals, but some areas on the plane. Guess for yourself what will be the solution to an inequality in three variables?
How to solve inequalities?
A universal way to solve inequalities is considered to be the method of intervals (also known as the method of intervals), which consists in determining all intervals within the boundaries of which a given inequality will be satisfied.
Without going into the type of inequality, in this case this is not the point, you need to solve the corresponding equation and determine its roots, followed by the designation of these solutions on the number axis.
How to correctly write the solution to an inequality?
When you have determined the solution intervals for the inequality, you need to correctly write out the solution itself. There is an important nuance - are the boundaries of the intervals included in the solution?
Everything is simple here. If the solution to the equation satisfies the ODZ and the inequality is not strict, then the boundary of the interval is included in the solution to the inequality. Otherwise, no.
Considering each interval, the solution to the inequality may be the interval itself, or a half-interval (when one of its boundaries satisfies the inequality), or a segment - the interval together with its boundaries.
Important point
Do not think that only intervals, half-intervals and segments can solve the inequality. No, the solution may also include individual points.
For example, the inequality |x|≤0 has only one solution - this is point 0.
And the inequality |x|
Why do you need an inequality calculator?
The inequalities calculator gives the correct final answer. In most cases, an illustration of a number axis or plane is provided. It is visible whether the boundaries of the intervals are included in the solution or not - the points are displayed as shaded or punctured.
Thanks to online calculator inequalities, you can check whether you correctly found the roots of the equation, marked them on the number axis and checked the fulfillment of the inequality condition on the intervals (and boundaries)?
If your answer differs from the calculator’s answer, then you definitely need to double-check your solution and identify the mistake.
The article covers the topic of inequalities, the definitions of systems and their solutions are examined. Frequent examples of solving systems of equations in school in algebra will be considered.
Definition of a system of inequalities
Systems of inequalities are determined by the definitions of systems of equations, which means that special attention is paid to the records and meaning of the equation itself.
Definition 1
System of inequalities called a record of equations combined by a curly brace with a set of solutions simultaneously for all inequalities included in the system.
Below are examples of inequalities. Two inequalities are given: 2 x − 3 > 0 and 5 − x ≥ 4 x − 11. It is necessary to write one equation under the other, and then combine it using a curly brace:
2 x - 3 > 0, 5 - x ≥ 4 x - 11
In the same way, the definition of systems of inequalities is presented in school textbooks both for using one variable and two.
Main types of system of inequalities
An infinite number of systems of inequalities are created. They are classified into groups that differ in certain characteristics. Inequalities are divided according to the following criteria:
- number of system inequalities;
- number of recording variables;
- type of inequalities.
The number of incoming inequalities can be two or more. The previous paragraph considered an example of solving a system with two inequalities.
2 x - 3 > 0, 5 - x ≥ 4 x - 11
Let's consider solving a system with four inequalities.
x ≥ - 2 , y ≤ 5 , x + y + z ≥ 3 , z ≤ 1 - x 2 - 4 y 2
Solving an inequality separately does not indicate the solution of the system as a whole. To solve the system, it is necessary to use all existing inequalities.
Such systems of inequalities can have one, two, three or more variables. In the last depicted system this is clearly visible; there we have three variables: x, y, z. Equations can contain one variable, as in the example, or several. Based on the examples, the inequality x + 0 · y + 0 · z ≥ − 2 and 0 · x + y + 0 · z ≤ 5 are not considered equivalent. School programs pay attention to solving inequalities with one variable.
When writing a system, the equations can be used different types and with different amounts variables. Most often there are entire inequalities different degrees. When preparing for exams, you may encounter systems with irrational, logarithmic, exponential equations type:
544 - 4 - x 32 - 2 - x ≥ 17 , log x 2 16 x + 20 16 ≤ 1
Such a system includes an exponential and logarithmic equation.
Solving the system of inequalities
Definition 2Let's consider an example of solving systems of equations with one variable.
x > 7, 2 - 3 x ≤ 0
If the value x = 8, then the solution of the system is obvious, since 8 > 7 and 2 − 3 8 ≤ 0 hold. At x = 1 the system will not be solved, since the first numerical inequality during substitution has 1 > 7 . A system with two or more variables is solved in the same way.
Definition 3
Solving a system of inequalities with two or more variables name the values that are the solution to all inequalities when each turns into a correct numerical inequality.
If x = 1 and y = 2 will be the solution to the inequality x + y< 7 x - y < 0 , потому как выражения 1 + 2 < 7 и 1 − 2 < 0 верны. Если подставить числовую пару (3 , 5 , 3) , тогда система не даст значения переменных и неравенство будет неверным 3 , 5 − 3 < 0 .
When solving systems of inequalities, they can give a certain number of answers, or they can give an infinite number. This means that there are many solutions to such a system. If there are no solutions, we say that it has an empty set of solutions. If a solution has a specific number, then the solution set has a finite number of elements. If there are many solutions, then the solution set contains an infinite number of numbers.
Some textbooks give a definition of a particular solution to a system of inequalities, which is understood as a separate solution. A general decision systems of inequalities count all its particular solutions. This definition is rarely used, so they say “solving a system of inequalities.”
These definitions of systems of inequalities and solutions are considered as intersections of sets of solutions to all inequalities of the system. Special attention It is worth paying attention to the section devoted to equivalent inequalities.
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In this lesson we will begin to study 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. At the end we will solve specific examples of systems of linear inequalities.
Subject: Dietal inequalities and their systems
Lesson:Mainconcepts, solving systems of linear inequalities
So far we have solved individual inequalities and applied the interval method to them, these could be linear inequalities, both 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
Find the domain of a function
A function exists when both square roots exist, i.e.
How to solve such a system? It is necessary to find all x that satisfy both the first and second inequalities.
Let us depict 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 arbitrary nature and a set B of arbitrary nature, which intersect.
Definition: The intersection of two sets A and B is the third set that consists of all the elements included in both A and B.
Using specific examples of solving linear systems of inequalities, let us consider how to find intersections of sets of solutions to 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
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 contradictory.
We examined typical support problems to which the solution of any linear system of inequalities can be reduced.
Consider the following system.
7. 
Sometimes a linear system is given double inequality, let's consider such a case.
8. 
We looked at systems of linear inequalities, understood where they come from, looked at the standard systems to which all linear systems can be reduced, and solved some of them.
1. Mordkovich A.G. and others. Algebra 9th grade: Textbook. For general education Institutions.- 4th ed. - M.: Mnemosyne, 2002.-192 p.: ill.
2. Mordkovich A.G. and others. Algebra 9th grade: Problem book for students educational institutions/ A. G. Mordkovich, T. N. Mishustina and others - 4th ed. - M.: Mnemosyne, 2002.-143 p.: ill.
3. Makarychev Yu. N. Algebra. 9th grade: educational. 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. 9th grade. 16th ed. - M., 2011. - 287 p.
5. Mordkovich A. G. Algebra. 9th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich, P. V. Semenov. — 12th ed., erased. - M.: 2010. - 224 p.: ill.
6. Algebra. 9th grade. In 2 parts. Part 2. Problem book for students of general education 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 10-11 grades for entrance exams in computer science, mathematics, Russian language ().
4. Education Center “Teaching Technology” ().
5. College.ru section on mathematics ().
1. Mordkovich A.G. and others. Algebra 9th grade: Problem book for students of general education institutions / A. G. Mordkovich, T. N. Mishustina, etc. - 4th ed. - M.: Mnemosyne, 2002.-143 p.: ill. No. 53; 54; 56; 57.
There are only “X’s” and only the abscissa axis, but now “Y’s” are added and the field of activity expands to the entire coordinate plane. Further in the text, the phrase “linear inequality” is understood in a two-dimensional sense, which will become clear in a matter of seconds.
Besides analytical geometry, the material is relevant for a number of problems of mathematical analysis, economic and mathematical modeling, so I recommend studying this lecture with all seriousness.
Linear inequalities
There are two types of linear inequalities:
1) Strict inequalities: .
2) Lax inequalities: .
Which geometric meaning these inequalities? If linear equation defines a straight line, then a linear inequality defines half-plane.
To understand the following information, you need to know the types of lines on a plane and be able to construct straight lines. If you have any difficulties in this part, read the help Graphs and properties of functions– paragraph about linear function.
Let's start with the simplest linear inequalities. The blue dream of any poor student - coordinate plane, on which there is nothing:
As you know, the x-axis is given by the equation - the “y” is always (for any value of “x”) equal to zero
Let's consider inequality. How to understand it informally? “Y” is always (for any value of “x”) positive. Obviously, this inequality defines the upper half-plane - after all, all the points with positive “games” are located there.
In the event that the inequality is not strict, to the upper half-plane additionally the axis itself is added.
Similarly: the inequality is satisfied by all points of the lower half-plane; a non-strict inequality corresponds to the lower half-plane + axis.
The same prosaic story is with the y-axis:
– the inequality specifies the right half-plane;
– the inequality specifies the right half-plane, including the ordinate axis;
– the inequality specifies the left half-plane;
– the inequality specifies the left half-plane, including the ordinate axis.
In the second step, we consider inequalities in which one of the variables is missing.
Missing "Y":
Or there is no “x”:
These inequalities can be dealt with in two ways: please consider both approaches. Along the way, let’s remember and consolidate school actions with inequalities, already discussed in class Function Domain.
Example 1
Solve linear inequalities:
What does it mean to solve a linear inequality?
Solving a linear inequality means finding a half-plane, whose points satisfy this inequality (plus the line itself, if the inequality is not strict). Solution, usually, graphic.
It’s more convenient to immediately execute the drawing and then comment out everything: 
a) Solve the inequality
Method one
The method is very reminiscent of the story with coordinate axes, which we discussed above. The idea is to transform the inequality - to leave one variable on the left side without any constants, in this case the variable “x”.
Rule: In an inequality, the terms are transferred from part to part with a change of sign, while the sign of the inequality ITSELF does not change(for example, if there was a “less than” sign, then it will remain “less than”).
We move the “five” to the right side with a change of sign:
Rule POSITIVE does not change.
Now draw a straight line (blue dotted line). The straight line is drawn as a dotted line because the inequality strict, and points belonging to this line will certainly not be included in the solution.
What is the meaning of inequality? “X” is always (for any value of “Y”) less than . Obviously, this statement is satisfied by all points of the left half-plane. This half-plane, in principle, can be shaded, but I will limit myself to small blue arrows so as not to turn the drawing into an artistic palette.
Method two
This is a universal method. READ VERY CAREFULLY!
First we draw a straight line. For clarity, by the way, it is advisable to present the equation in the form .
Now select any point on the plane, not belonging to direct. In most cases, the sweet spot is, of course. Let's substitute the coordinates of this point into the inequality: 
Received false inequality (in simple words, this cannot be), this means that the point does not satisfy the inequality .
Key Rule our task:
does not satisfy inequality, then ALL points of a given half-plane do not satisfy this inequality.
– If any point of the half-plane (not belonging to a line) satisfies inequality, then ALL points of a given half-plane satisfy this inequality.
You can test: any point to the right of the line will not satisfy the inequality.
What is the conclusion from the experiment with the point? There is nowhere to go, the inequality is satisfied by all points of the other - left half-plane (you can also check).
b) Solve the inequality
Method one
Let's transform the inequality:
Rule: Both sides of the inequality can be multiplied (divided) by NEGATIVE number, with the inequality sign CHANGING to the opposite (for example, if there was a “greater than or equal” sign, it will become “less than or equal”).
We multiply both sides of the inequality by:
Let's draw a straight line (red), and draw a solid line, since we have inequality non-strict, and the straight line obviously belongs to the solution.
Having analyzed the resulting inequality, we come to the conclusion that its solution is the lower half-plane (+ the straight line itself).
We shade or mark the appropriate half-plane with arrows.
Method two
Let's draw a straight line. Let's choose an arbitrary point on the plane (not belonging to a line), for example, and substitute its coordinates into our inequality: 
Received true inequality, which means that the point satisfies the inequality, and in general, ALL points of the lower half-plane satisfy this inequality.
Here, with the experimental point, we “hit” the desired half-plane.
The solution to the problem is indicated by a red line and red arrows.
Personally, I prefer the first solution, since the second is more formal.
Example 2
Solve linear inequalities:
This is an example for independent decision. Try to solve the problem in two ways (by the way, this is good way checking the solution). The answer at the end of the lesson will only contain the final drawing.
I think that after all the actions done in the examples, you will have to marry them; it will not be difficult to solve the simplest inequality like, etc.
Let us move on to consider the third, general case, when both variables are present in the inequality:
Alternatively, the free term “ce” may be zero.
Example 3
Find half-planes corresponding to the following inequalities:
Solution: The universal solution method with point substitution is used here.
a) Let’s construct an equation for the straight line, and the line should be drawn as a dotted line, since the inequality is strict and the straight line itself will not be included in the solution.
We select an experimental point of the plane that does not belong to a given line, for example, and substitute its coordinates into our inequality:
Received false inequality, which means that the point and ALL points of a given half-plane do not satisfy the inequality. The solution to the inequality will be another half-plane, let’s admire the blue lightning: 
b) Let's solve the inequality. First, let's construct a straight line. This is not difficult to do; we have the canonical direct proportionality. We draw the line continuously, since the inequality is not strict.
Let us choose an arbitrary point of the plane that does not belong to the straight line. I would like to use the origin again, but, alas, it is not suitable now. Therefore, you will have to work with another friend. It is more profitable to take a point with small coordinate values, for example, . Let's substitute its coordinates into our inequality:
Received true inequality, which means that the point and all points of a given half-plane satisfy the inequality . The desired half-plane is marked with red arrows. In addition, the solution includes the straight line itself.
Example 4
Find half-planes corresponding to the inequalities:
This is an example for you to solve on your own. Complete solution, an approximate sample of the final design and the answer at the end of the lesson.
Let's look at the inverse problem:
Example 5
a) Given a straight line. Define the half-plane in which the point is located, while the straight line itself must be included in the solution.
b) Given a straight line. Define half-plane in which the point is located. The straight line itself is not included in the solution.
Solution: There is no need for a drawing here and the solution will be analytical. Nothing difficult:
a) Let's create an auxiliary polynomial 
and calculate its value at point:
. Thus, the desired inequality will have a “less than” sign. By condition, the straight line is included in the solution, so the inequality will not be strict:
b) Let's compose a polynomial and calculate its value at point:
. Thus, the desired inequality will have a “greater than” sign. By condition, the straight line is not included in the solution, therefore, the inequality will be strict: .
Answer:
Creative example for self-study:
Example 6
Given points and a straight line. Among the listed points, find those that, together with the origin of coordinates, lie on the same side of the given line.
A little hint: first you need to create an inequality that determines the half-plane in which the origin of coordinates is located. Analytical solution and answer at the end of the lesson.
Systems of linear inequalities
A system of linear inequalities is, as you understand, a system composed of several inequalities. Lol, well, I gave out the definition =) A hedgehog is a hedgehog, a knife is a knife. But it’s true – it turned out simple and accessible! No, seriously, I don’t want to give any general examples, so let’s move straight to the pressing issues:
What does it mean to solve a system of linear inequalities?
Solve a system of linear inequalities- this means find the set of points on the plane, which satisfy to each inequality of the system.
As the simplest examples, consider the systems of inequalities that determine the coordinate quarters of a rectangular coordinate system (“the picture of the poor students” is at the very beginning of the lesson):
The system of inequalities defines the first coordinate quarter (upper right). Coordinates of any point in the first quarter, for example, 
etc. satisfy to each inequality of this system.
Likewise:
– the system of inequalities specifies the second coordinate quarter (upper left);
– the system of inequalities defines the third coordinate quarter (lower left);
– the system of inequalities defines the fourth coordinate quarter (lower right).
A system of linear inequalities may have no solutions, that is, to be non-joint. Again simplest example: . It is quite obvious that “x” cannot simultaneously be more than three and less than two.
The solution to the system of inequalities can be a straight line, for example: . A swan, a crayfish, without a pike, pulling the cart in two different directions. Yes, things are still there - the solution to this system is the straight line.
But the most common case is when the solution to the system is some plane region. Solution area May be not limited(for example, coordinate quarters) or limited. The limited solution region is called polygon solution system.
Example 7
Solve a system of linear inequalities
In practice, in most cases we have to deal with weak inequalities, so they will be the ones leading the round dances for the rest of the lesson.
Solution: The fact that there are too many inequalities should not be scary. How many inequalities can there be in the system? Yes, as much as you like. The main thing is to adhere to a rational algorithm for constructing a solution area:
1) First we deal with the simplest inequalities. The inequalities define the first coordinate quarter, including the boundary of the coordinate axes. It’s already much easier, since the search area has narrowed significantly. In the drawing, we immediately mark the corresponding half-planes with arrows (red and blue arrows)
2) The second simplest inequality is that there is no “Y” here. Firstly, we construct the straight line itself, and, secondly, after converting the inequality to the form , it immediately becomes clear that all the “X’s” are less than 6. We mark the corresponding half-plane with green arrows. Well, the search area has become even smaller - such a rectangle not limited from above.
3) At the last step we solve the inequalities “with full ammunition”: . We discussed the solution algorithm in detail in the previous paragraph. In short: first we build a straight line, then, using an experimental point, we find the half-plane we need.
Stand up, children, stand in a circle: 
The solution area of the system is a polygon; in the drawing it is outlined with a crimson line and shaded. I overdid it a little =) In the notebook, it is enough to either shade the solution area or outline it bolder with a simple pencil.
Any point of a given polygon satisfies EVERY inequality of the system (you can check it for fun).
Answer: The solution to the system is a polygon.
When applying for a clean copy, it would be a good idea to describe in detail which points you used to construct straight lines (see lesson Graphs and properties of functions), and how half-planes were determined (see the first paragraph of this lesson). However, in practice, in most cases, you will be credited with just the correct drawing. The calculations themselves can be carried out on a draft or even orally.
In addition to the solution polygon of the system, in practice, albeit less frequently, there is an open region. Try to understand the following example yourself. Although, for the sake of accuracy, there is no torture here - the construction algorithm is the same, it’s just that the area will not be limited.
Example 8
Solve the system
The solution and answer are at the end of the lesson. You will most likely have others letter designations vertices of the resulting region. This is not important, the main thing is to find the vertices correctly and construct the area correctly.
It is not uncommon when problems require not only constructing the solution domain of a system, but also finding the coordinates of the vertices of the domain. In the two previous examples, the coordinates of these points were obvious, but in practice everything is far from ice:
Example 9
Solve the system and find the coordinates of the vertices of the resulting region
Solution: Let us depict in the drawing the solution area of this system. The inequality defines the left half-plane with the ordinate axis, and there is no more freebie here. After calculations on the final copy/draft or deep thought processes, we get the following area of solutions: 
Let's look at examples of how to solve a system of linear inequalities.
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To solve a system, you need each of its constituent inequalities. Only the decision was made not to write separately, but together, combining them with a curly brace.
In each of the inequalities of the system, we move the unknowns to one side, the known ones to the other with the opposite sign:
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After simplification, both sides of the inequality must be divided by the number in front of X. We divide the first inequality by a positive number, so the sign of the inequality does not change. We divide the second inequality by a negative number, so the inequality sign must be reversed:
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We mark the solution to the inequalities on the number lines:
In response, we write down the intersection of the solutions, that is, the part where there is shading on both lines.
Answer: x∈[-2;1).
In the first inequality, let's get rid of the fraction. To do this, multiply both sides term by term by the smallest common denominator 2. When multiplied by a positive number, the inequality sign does not change.
In the second inequality we open the brackets. The product of the sum and the difference of two expressions is equal to the difference of the squares of these expressions. On the right side is the square of the difference between the two expressions.
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We move the unknowns to one side, the known ones to the other with the opposite sign and simplify:
We divide both sides of the inequality by the number in front of X. In the first inequality, we divide by a negative number, so the sign of the inequality is reversed. In the second, we divide by a positive number, the inequality sign does not change:
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Both inequalities have a “less than” sign (it doesn’t matter that one sign is strictly “less than”, the other is loose, “less than or equal”). We can not mark both solutions, but use the “ “ rule. The smaller one is 1, therefore the system reduces to the inequality
We mark its solution on the number line:
Answer: x∈(-∞;1].
Opening the parentheses. In the first inequality - . It is equal to the sum of the cubes of these expressions.
In the second, the product of the sum and the difference of two expressions, which is equal to the difference of squares. Since here there is a minus sign in front of the brackets, it is better to open them in two stages: first use the formula, and only then open the brackets, changing the sign of each term to the opposite.
We move the unknowns in one direction, the knowns in the other with the opposite sign:
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Both are greater than signs. Using the “more than more” rule, we reduce the system of inequalities to one inequality. The larger of the two numbers is 5, therefore,
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We mark the solution to the inequality on the number line and write down the answer: 
Answer: x∈(5;∞).
Since in algebra systems of linear inequalities occur not only as independent tasks, but also in the course of solving various kinds of equations, inequalities, etc., it is important to master this topic in time.
Next time we will look at examples of solving systems of linear inequalities in special cases when one of the inequalities has no solutions or its solution is any number.
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