- Property 1. If a> b and b> c, then a> c (Example: 8> 4 and 4> 3 => 8> 3)
- Property 2. If a> b, then a + const> b + const. Const-arbitrary number (Example: x - 3> 0<=>x - 3 + 8> 0 + 8)
- Property 3. If a> b and m> 0, then am> bm;
If a> b and m< 0, то am < bm. m-произвольное число.
The meaning of property 3 is as follows:
- if both sides of the inequality are multiplied by the same positive number, then the inequality sign should be preserved;
- if both sides of the inequality are multiplied by the same negative number, then the sign of the inequality should be changed (the sign “<” на “>”, The sign“> ”to“<”);(для нестрогих неравенств)
Property 3, in particular, implies that, multiplying both sides of the inequality a> b by -1, we obtain: -a< -b.
- Property 4. If a> b and c> d, then a + c> b + d (Example: 8> 4 and 3> 2 => 8 + 3> 4 + 2)
- Property 5. If a, b, c, d are positive numbers and a> b, c> d then ac> bd (Example: 8> 4 and 3> 2 => 8 * 3> 4 * 2)
Linear inequalities
Definition. Solving a one-variable inequality is the value of a variable that turns it into a true numeric inequality.
Consider, for example, the inequality 2x + 5< 7.
We are interested in such numbers x for which 2x + 5< 7— верное числовое неравенство.
Let's simplify our inequality.
1) According to property 2 the same number “-5” was added to both sides of the inequality, and we got:
2x + 5 - 5< 7 - 5.
We got a simpler inequality.
2) Based on properties 3 you can divide both parts of it by a positive number 2, the resulting inequality:
What does it mean? This means that the solution to the inequality is any number x that is less than 1. Thus, the set of solutions to this inequality is the set of numbers x< 1 (или иначе в виде числовой прямой (-∞;1])
The properties allow the following rules to be followed when solving inequalities:
- Rule 1. Any term of the inequality can be transferred from one part of the inequality to another with the opposite sign, without changing the sign of the inequality.
- Rule 2. Both sides of the inequality can be multiplied or divided by the same positive number without changing the sign of the inequality.
- Rule 3. Both sides of an inequality can be multiplied or divided by the same negative number, while reversing the sign of the inequality.
We apply these rules to solve linear inequalities, i.e. inequalities reduced to the form
where a and b are any numbers, with one exception: a ≠ 0.
If a = 0, then we consider 2 cases:
1) If b> 0, then x can be any number
2) If b< 0, то решения нет
Example 1:
Solve inequality
Zx - 5 ≥ 7x - 15.
Solution.
We are guided by rule 1 we transfer the 7x term to the left side of the inequality, and the -5 term to the right side of the inequality, not forgetting to change the signs of both the 7x term and the -5 term. Then we get:
Zx - 7x ≥ -15 + 5
According to rule 3 divide both sides of the last inequality by the same negative number -4, remembering to change the sign of the inequality. We get:
This is the solution to the given inequality.
As we agreed, to write the solution, you can use the designation of the corresponding interval of the number line: (-∞; 2,5].
Answer: (- ∞; 2,5].
Example 2:
Solve inequality
3x + 2> 2 (x + 3) + x
Solution.
3x + 2> 2x + 6 + x
Guided by rule 1
3x - 2x - x> 6 - 2
We get a contradiction.
There is no solution.
Example 4:
Solve inequality
2 (x - 1) + 3> 2x - 5
Solution.
Let's expand the brackets in the second part of the inequality:
2x - 2 + 3> 2x - 5
Guided by rule 1 , we transfer the terms "with x" to the left side of the inequality, and "without x" to the right:
2x - 2x> 2 - 5 - 3
We get the correct inequality.
In this case, you can take any number x, since the solution does not depend on it.
The answer is the whole number line.
In conclusion, we note that, using the properties of numerical inequalities and rules, in this section we learned to solve not any inequality with a variable, but only one that, after a number of simple transformations (such as those performed in the examples from this section) takes the form ax > b, such inequalities are called linear ... Next, we will explore methods for solving more complex inequalities.
After receiving the initial information about inequalities with variables, we turn to the question of their solution. We will analyze the solution of linear inequalities with one variable and all methods for their resolution with algorithms and examples. Only linear equations with one variable will be considered.
What is Linear Inequality?
First, you need to define a linear equation and find out its standard form and how it will differ from others. From the school course we have that inequalities do not have a fundamental difference, therefore it is necessary to use several definitions.
Definition 1
Linear inequality with one variable x is an inequality of the form ax + b> 0, when instead of> any inequality sign is used< , ≤ , ≥ , а и b являются действительными числами, где a ≠ 0 .
Definition 2
Inequalities a x< c или a · x >c, with x being a variable, and a and c some numbers, is called linear inequalities in one variable.
Since nothing is said about whether the coefficient can be equal to 0, then a strict inequality of the form 0 x> c and 0 x< c может быть записано в виде нестрогого, а именно, a · x ≤ c , a · x ≥ c . Такое уравнение считается линейным.
Their differences are:
- the notation a · x + b> 0 in the first, and a · x> c - in the second;
- the admissibility of the equality to zero of the coefficient a, a ≠ 0 - in the first, and a = 0 - in the second.
It is believed that the inequalities a x + b> 0 and a x> c are equivalent, because they are obtained by transferring a term from one part to another. Solving the inequality 0 x + 5> 0 will lead to the fact that it will need to be solved, and the case a = 0 will not work.
Definition 3
It is believed that linear inequalities in one variable x are considered to be inequalities of the form a x + b< 0 , a · x + b >0, a x + b ≤ 0 and a x + b ≥ 0 where a and b are real numbers. Instead of x, there can be an ordinary number.
Based on the rule, we have that 4 x - 1> 0, 0 z + 2, 3 ≤ 0, - 2 3 x - 2< 0 являются примерами линейных неравенств. А неравенства такого плана, как 5 · x >7, - 0, 5 · y ≤ - 1, 2 are called converging to linear.
How to solve linear inequality
The main way to solve such inequalities is reduced to equivalent transformations in order to find elementary inequalities x< p (≤ , >, ≥), p which is some number, for a ≠ 0, and of the form a< p (≤ , >, ≥) for a = 0.
To solve an inequality with one variable, you can apply the method of intervals or plot it graphically. Any of them can be applied in isolation.
Using equivalent transformations
To solve a linear inequality of the form ax + b< 0 (≤ , >, ≥), it is necessary to apply equivalent inequality transformations. The coefficient may or may not be zero. Let's consider both cases. To find out, it is necessary to adhere to a scheme consisting of 3 points: the essence of the process, the algorithm, the solution itself.
Definition 4
Algorithm for solving linear inequality a x + b< 0 (≤ , >, ≥) for a ≠ 0
- the number b will be transferred to the right-hand side of the inequality with the opposite sign, which will allow us to arrive at an equivalent a x< − b (≤ , > , ≥) ;
- both sides of the inequality will be divided by a number not equal to 0. Moreover, when a is positive, then the sign remains, when a is negative, it changes to the opposite.
Let's consider the application of this algorithm to the solution of examples.
Example 1
Solve an inequality of the form 3 x + 12 ≤ 0.
Solution
This linear inequality has a = 3 and b = 12. Hence, the coefficient a at x is not equal to zero. Let's apply the above algorithms and decide.
It is necessary to transfer the term 12 to the other part of the inequality with a change in the sign in front of it. Then we obtain an inequality of the form 3 x ≤ - 12. It is necessary to divide both parts by 3. The sign does not change, since 3 is a positive number. We get that (3 x): 3 ≤ (- 12): 3, which gives the result x ≤ - 4.
An inequality of the form x ≤ - 4 is equivalent. That is, the solution for 3 x + 12 ≤ 0 is any real number that is less than or equal to 4. The answer is written as an inequality x ≤ - 4, or a numerical interval of the form (- ∞, - 4].
The entire algorithm described above is written as follows:
3 x + 12 ≤ 0; 3 x ≤ - 12; x ≤ - 4.
Answer: x ≤ - 4 or (- ∞, - 4].
Example 2
Indicate all available solutions to the inequality - 2, 7 · z> 0.
Solution
From the condition we see that the coefficient a at z is equal to - 2, 7, and b is explicitly absent or equal to zero. The first step of the algorithm can be skipped over to the second.
We divide both sides of the equation by the number - 2, 7. Since the number is negative, it is necessary to reverse the sign of inequality. That is, we get that (- 2, 7 z): (- 2, 7)< 0: (− 2 , 7) , и дальше z < 0 .
We write the entire algorithm in a short form:
- 2, 7 · z> 0; z< 0 .
Answer: z< 0 или (− ∞ , 0) .
Example 3
Solve the inequality - 5 x - 15 22 ≤ 0.
Solution
By condition, we see that it is necessary to solve the inequality with the coefficient a at the variable x, which equals - 5, with the coefficient b, which corresponds to the fraction - 15 22. It is necessary to solve the inequality following the algorithm, that is: transfer - 15 22 to another part with the opposite sign, divide both parts by - 5, change the sign of the inequality:
5 x ≤ 15 22; - 5 x: - 5 ≥ 15 22: - 5 x ≥ - 3 22
At the last transition, for the right side, the rule for dividing numbers with different signs is used 15 22: - 5 = - 15 22: 5, after which we perform division of an ordinary fraction by a natural number - 15 22: 5 = - 15 22 1 5 = - 15 1 22 5 = - 3 22.
Answer: x ≥ - 3 22 and [- 3 22 + ∞).
Consider the case when a = 0. Linear expression of the form a x + b< 0 является неравенством 0 · x + b < 0 , где на рассмотрение берется неравенство вида b < 0 , после чего выясняется, оно верное или нет.
Everything is based on the definition of the solution to inequality. For any value of x, we obtain a numerical inequality of the form b< 0 , потому что при подстановке любого t вместо переменной x , тогда получаем 0 · t + b < 0 , где b < 0 . В случае, если оно верно, то для его решения подходит любое значение. Когда b < 0 неверно, тогда линейное уравнение не имеет решений, потому как не имеется ни одного значения переменной, которое привело бы верному числовому равенству.
We consider all judgments in the form of an algorithm for solving linear inequalities 0 x + b< 0 (≤ , > , ≥) :
Definition 5
Numerical inequality of the form b< 0 (≤ , >, ≥) is true, then the original inequality has a solution for any value, and it is false if the original inequality has no solutions.
Example 4
Solve the inequality 0 x + 7> 0.
Solution
This linear inequality 0 x + 7> 0 can take any value of x. Then we obtain an inequality of the form 7> 0. The last inequality is considered true, which means that any number can be its solution.
Answer: interval (- ∞, + ∞).
Example 5
Find a solution to the inequality 0 x - 12, 7 ≥ 0.
Solution
When substituting the variable x of any number, we get that the inequality will take the form - 12, 7 ≥ 0. It is wrong. That is, 0 x - 12, 7 ≥ 0 has no solutions.
Answer: no solutions.
Consider the solution to linear inequalities where both coefficients are equal to zero.
Example 6
Determine an inequality that has no solution from 0 x + 0> 0 and 0 x + 0 ≥ 0.
Solution
Substituting any number instead of x, we obtain two inequalities of the form 0> 0 and 0 ≥ 0. The first is not true. Hence, 0 x + 0> 0 has no solutions, and 0 x + 0 ≥ 0 has an infinite number of solutions, that is, any number.
Answer: the inequality 0 x + 0> 0 has no solutions, and 0 x + 0 ≥ 0 has solutions.
This method is considered in the school mathematics course. The interval method is able to resolve various types of inequalities, including linear ones.
The method of intervals is used for linear inequalities when the value of the coefficient x is not equal to 0. Otherwise, you will have to calculate using another method.
Definition 6
The spacing method is:
- introduction of the function y = a x + b;
- searching for zeros to split the domain into intervals;
- definition of signs for understanding them at intervals.
Let's put together an algorithm for solving linear equations a x + b< 0 (≤ , >, ≥) for a ≠ 0 using the method of intervals:
- finding the zeros of the function y = a x + b to solve an equation of the form a x + b = 0. If a ≠ 0, then the solution is the only root that takes the notation x 0;
- construction of a coordinate line with an image of a point with a coordinate x 0, with strict inequality, the point is denoted by a punctured one, with a non-strict one - filled;
- determination of the signs of the function y = a · x + b at intervals, for this it is necessary to find the values of the function at points in the interval;
- the solution of the inequality with signs> or ≥ on the coordinate line is added shading over the positive interval,< или ≤ над отрицательным промежутком.
Let's consider several examples of solving a linear inequality using the interval method.
Example 6
Solve the inequality - 3 x + 12> 0.
Solution
From the algorithm it follows that first you need to find the root of the equation - 3 x + 12 = 0. We get that - 3 x = - 12, x = 4. It is necessary to draw a coordinate line where we mark point 4. It will be punctured since the inequality is severe. Consider the drawing below.
It is necessary to identify the signs in between. To determine it on the interval (- ∞, 4), it is necessary to calculate the function y = - 3 x + 12 at x = 3. From here we get that - 3 3 + 12 = 3> 0. The sign in between is positive.
Determine the sign from the interval (4, + ∞), then substitute the value x = 5. We have that - 3 5 + 12 = - 3< 0 . Знак на промежутке является отрицательным. Изобразим на числовой прямой, приведенной ниже.
We solve the inequality with the> sign, and the shading is performed over the positive interval. Consider the drawing below.
It can be seen from the drawing that the sought solution has the form (- ∞, 4) or x< 4 .
Answer: (- ∞, 4) or x< 4 .
To understand how to represent graphically, it is necessary to consider 4 linear inequalities as an example: 0.5 x - 1< 0 , 0 , 5 · x − 1 ≤ 0 , 0 , 5 · x − 1 >0 and 0.5 x - 1 ≥ 0. Their solutions will be the values x< 2 , x ≤ 2 , x >2 and x ≥ 2. To do this, we will depict the graph of the linear function y = 0.5 · x - 1, shown below.
It's clear that
Definition 7
- solution of the inequality 0.5 x - 1< 0 считается промежуток, где график функции y = 0 , 5 · x − 1 располагается ниже О х;
- the solution 0, 5 · x - 1 ≤ 0 is considered to be the interval where the function y = 0, 5 · x - 1 is lower than O x or coincides;
- the solution 0, 5 · x - 1> 0 is considered to be the interval, the function is located above O x;
- the solution 0, 5 · x - 1 ≥ 0 is considered to be the interval where the graph is higher than O x or coincides.
The meaning of the graphical solution of inequalities is to find the intervals that must be depicted on the graph. In this case, we find that the left side has y = a x + b, and the right - y = 0, and coincides with O x.
Definition 8The plotting of the function y = a x + b is performed:
- while solving the inequality a x + b< 0 определяется промежуток, где график изображен ниже О х;
- during the solution of the inequality a · x + b ≤ 0, the interval is determined, where the graph is displayed below the O x axis or coincides;
- during the solution of the inequality a x + b> 0, the interval is determined, where the graph is depicted above O x;
- while solving the inequality a · x + b ≥ 0, the interval is determined where the graph is above O x or coincides.
Example 7
Solve the inequality - 5 x - 3> 0 using the graph.
Solution
It is necessary to plot a linear function - 5 x - 3> 0. This line is decreasing because the coefficient at x is negative. To determine the coordinates of the point of its intersection with O x - 5 · x - 3> 0, we get the value - 3 5. Let's draw it graphically.
The solution to the inequality with the> sign, then it is necessary to pay attention to the interval above O x. We highlight the necessary part of the plane in red and get that
The required spacing is part of the red O x. Hence, an open number ray - ∞, - 3 5 will be a solution to the inequality. If by condition they had a non-strict inequality, then the value of the point - 3 5 would also be a solution to the inequality. And it would be the same as Oh.
Answer: - ∞, - 3 5 or x< - 3 5 .
The graphical solution is used when the left side will correspond to the function y = 0 x + b, that is, y = b. Then the line will be parallel to O x or coinciding at b = 0. These cases show that the inequality may not have solutions, or any number may be the solution.
Example 8
Determine from the inequalities 0 x + 7< = 0 , 0 · x + 0 ≥ 0 то, которое имеет хотя бы одно решение.
Solution
The representation y = 0 x + 7 is y = 7, then a coordinate plane with a straight line parallel to O x and located above O x will be given. Hence, 0 x + 7< = 0 решений не имеет, потому как нет промежутков.
The graph of the function y = 0 x + 0, y = 0 is considered, that is, the straight line coincides with O x. Hence, the inequality 0 x + 0 ≥ 0 has a set of solutions.
Answer: the second inequality has a solution for any value of x.
Inequalities Reducing to Linear
Solving inequalities can be reduced to solving a linear equation, which are called inequalities that reduce to linear.
These inequalities were considered in the school course, since they were a special case of solving inequalities, which led to the opening of parentheses and the reduction of similar terms. For example, consider that 5 - 2 x> 0.7 (x - 1) + 3 ≤ 4 x - 2 + x, x - 3 5 - 2 x + 1> 2 7 x.
The above inequalities are always reduced to the form of a linear equation. Then the brackets are opened and similar terms are given, transferred from different parts, changing the sign to the opposite.
Reducing the inequality 5 - 2 x> 0 to a linear one, we represent it in such a way that it has the form - 2 x + 5> 0, and to reduce the second we obtain that 7 (x - 1) + 3 ≤ 4 x - 2 + x. It is necessary to open the brackets, bring similar terms, move all terms to the left and bring similar terms. It looks like this:
7x - 7 + 3 ≤ 4x - 2 + x 7x - 4 ≤ 5x - 2 7x - 4 - 5x + 2 ≤ 0 2x - 2 ≤ 0
This brings the solution to a linear inequality.
These inequalities are considered as linear, since they have the same principle of solution, after which it is possible to reduce them to elementary inequalities.
To solve this kind of inequality of this kind, it is necessary to reduce it to a linear one. This should be done in this way:
Definition 9
- expand brackets;
- collect variables on the left, and numbers on the right;
- bring similar terms;
- divide both sides by the coefficient of x.
Example 9
Solve the inequality 5 (x + 3) + x ≤ 6 (x - 3) + 1.
Solution
We expand the parentheses, then we get an inequality of the form 5 x + 15 + x ≤ 6 x - 18 + 1. After reducing similar terms, we have that 6 x + 15 ≤ 6 x - 17. After transferring the terms from the left to the right, we get that 6 x + 15 - 6 x + 17 ≤ 0. Hence, it has an inequality of the form 32 ≤ 0 from the one obtained in the calculation of 0 x + 32 ≤ 0. It can be seen that the inequality is incorrect, which means that the inequality given by the condition has no solutions.
Answer: no solutions.
It is worth noting that there are many inequalities of another kind, which can be reduced to a linear one or an inequality of the kind shown above. For example, 5 2 x - 1 ≥ 1 is an exponential equation that reduces to a linear solution 2 x - 1 ≥ 0. These cases will be considered when solving inequalities of this type.
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Linear inequality with one variable is an inequality that can be reduced to the form:
ax > b or ax < b.
Where x is a variable, a- coefficient, and b- free member.
If a> 0, then, dividing both sides of the inequality by a, we get:
These inequalities determine all values of the variable x for which this inequality will be true. Both inequalities can be depicted using number spans:
Note that in strict inequalities, the value to which the variable is compared is not included in the set of values of the variable itself. In lax inequalities, it will be included in the set of admissible values:
If a < 0, то, разделив обе части неравенства
ax > b or ax < b
on a and changing the sign in them to the opposite, we get:
We have already considered all possible values of these inequalities above.
If a= 0, then the inequality takes the form:
0 · x > b or 0 x < b
In the first case: 0 x > b, x∈ (-∞; + ∞) if b negative number, otherwise the inequality has no solutions. In the second case: 0 x < b, x∈ (-∞; + ∞) if b a positive number, otherwise the inequality has no solutions.
Equivalent inequalities
Equivalent inequalities are inequalities that have the same set of solutions. Inequalities without solutions are also considered to be equivalent.
An inequality equivalent to a given one will be obtained if:
- Transfer the term from one part of the inequality to the other, changing the sign of the term to the opposite.
- Multiply or divide both sides of the inequality by the same positive number.
- Multiply or divide both sides of an inequality by the same negative number, while reversing the sign of the inequality.
Solving inequalities
Solving an inequality with one variable means finding all the values of this variable for which the given inequality is true, or making sure that the variable does not have such values.
All inequalities in one variable are solved in the same way using transformations that can be performed in any order. List of possible transformations that can be used to solve inequalities:
- exemption from fractional members,
- expansion of brackets,
- transfer of all terms containing a variable to one part, and the rest to another (terms with variables, as a rule, are transferred to the left side of the inequality),
- bringing similar members,
- dividing both sides of the inequality by the coefficient of the variable.
Example 1.
8x - 2 > 14
Solution: Move -2 to the right side:
8x > 14 + 2
8x > 16
Divide both sides of the inequality by -8:
8x : (-8) < 16: (-8)
x < -2
Celebrating many meanings x on the coordinate line:
Answer: (-∞; -2)
Example 2. Solve the inequality and depict the set of solutions on the coordinate line:
6(y+ 12) 3 ( y - 4)
Solution: First, we expand the brackets:
6y+ 72 3 y - 12
Move 72 to the right side, and 3 y to the left and do the reduction of similar terms:
6y - 3y-12 - 72
3y-84
We divide both sides of the inequality by the coefficient of the unknown (by 3):
(3y): 3 (- 84): 3
y-28
Celebrating many meanings y on the coordinate line:
Answer: [-28; +∞)
Linear inequalities are called the left and right parts of which are linear functions with respect to an unknown quantity. These include, for example, inequalities:
2x-1-x + 3; 7x0;
5 > 4 - 6x 9- x< x + 5 .
1) Strict inequalities: ax + b> 0 or ax + b<0
2) Lax inequalities: ax + b≤0 or ax + b≫ 0
Let's analyze such a task... One side of the parallelogram is 7cm. How long must the other side be so that the perimeter of the parallelogram is more than 44 cm?
Let the desired side be NS cm. In this case, the parallelogram perimeter will be represented by (14 + 2x) cm. The inequality 14 + 2x> 44 is a mathematical model of the parallelogram perimeter problem. If in this inequality we replace the variable NS on, for example, the number 16, then we obtain the correct numerical inequality 14 + 32> 44. In this case, they say that the number 16 is a solution to the inequality 14 + 2x> 44.
Solving inequality refers to the value of a variable that turns it into a true numerical inequality.
Therefore, each of the numbers 15.1; 20; 73 are the solution to the inequality 14 + 2x> 44, and the number 10, for example, is not its solution.
Solve inequality means to establish all its solutions or to prove that there are no solutions.
The formulation of the solution to the inequality is similar to the formulation of the root of the equation. And yet it is not customary to denote "the root of inequality".
The properties of numerical equalities helped us solve equations. Likewise, the properties of numerical inequalities will help solve inequalities.
Solving the equation, we change it with another, simpler equation, but equivalent to the given one. The answer and inequalities are found in a similar way. When changing an equation to an equation equivalent to it, use the theorem on the transfer of terms from one side of the equation to the opposite and on the multiplication of both sides of the equation by the same nonzero number. When solving an inequality, there is a significant difference between it and the equation, which consists in the fact that any solution to the equation can be verified by simply substituting it into the original equation. In inequalities, this method is absent, since it is not possible to substitute an infinite number of solutions into the original inequality. Therefore, there is an important concept, these arrows<=>is a sign of equivalent, or equivalent, transformations. The transformation are called equivalent, or equivalent if they do not change the set of decisions.
Similar rules for solving inequalities.
If we move any term from one part of the inequality to the other, replacing its sign with the opposite one, then we obtain an inequality equivalent to the given one.
If both sides of the inequality are multiplied (divided) by the same positive number, then we obtain an inequality equivalent to the given one.
If both sides of the inequality are multiplied (divided) by the same negative number, replacing the sign of the inequality with the opposite one, then we obtain an inequality equivalent to the given one.
Using these regulations we calculate the following inequalities.
1) Let us examine the inequality 2x - 5> 9.
it linear inequality, we will find its solution and discuss the basic concepts.
2x - 5> 9<=>2x> 14(5 was moved to the left with the opposite sign), then we divided everything by 2 and we have x> 7... Let's plot a lot of solutions on the axis x
We have obtained a positively directed beam. We mark the set of solutions either in the form of the inequality x> 7, or in the form of an interval х (7; ∞). And what is a particular solution to this inequality? For example, x = 10 is a particular solution to this inequality, x = 12 is also a particular solution to this inequality.
There are many particular solutions, but our task is to find all solutions. And the solutions, as a rule, are countless.
Let's analyze example 2:
2) Solve inequality 4a - 11> a + 13.
Let's solve it: a move to one side, 11 move to the other side, we get 3a< 24, и в результате после деления обеих частей на 3 the inequality has the form a<8 .
4a - 11> a + 13<=>3a< 24 <=>a< 8 .
We will also display the set a< 8 , but already on the axis a.
We either write the answer in the form of the inequality a< 8, либо a(-∞;8), 8 does not turn on.
What do you need to know about inequality icons? Inequality with icon more (> ), or smaller (< ) are called strict. With icons more or equal (≥ ), less than or equal to (≤ ) are called not strict. Icon not equal (≠ ) stands apart, but examples with such an icon also have to be solved all the time. And we will decide.)
The icon itself has little effect on the decision process. But at the end of the solution, when choosing the final answer, the meaning of the icon appears in full force! What we will see below, with examples. There are jokes there ...
Inequalities, like equality, are faithful and unfaithful. Everything is simple here, no tricks. Let's say 5 > 2 - correct inequality. 5 < 2 is incorrect.
This kind of preparation works for inequalities any kind and terribly simple.) You just need to correctly perform two (only two!) elementary actions. These actions are familiar to everyone. But, which is characteristic, the jambs in these actions are the main mistake in solving inequalities, yes ... Therefore, it is necessary to repeat these actions. These actions are called as follows:
Identical transformations of inequalities.
Identical transformations of inequalities are very similar to identity transformations of equations. Actually, this is the main problem. Differences slip past the head and ... have arrived.) Therefore, I will highlight these differences. So, the first identical transformation of inequalities:
1. To both sides of the inequality, you can add (subtract) the same number, or expression. Anyone. This will not change the inequality sign.
In practice, this rule is applied as a transfer of terms from the left side of the inequality to the right side (and vice versa) with a change in sign. With a sign change, not inequality! The one-to-one rule is the same as the rule for equations. But the following identical transformations in inequalities differ significantly from those in the equations. So I highlight them in red:
2. Both sides of the inequality can be multiplied (divided) by the samepositivenumber. Anypositive Will not change.
3. Both sides of the inequality can be multiplied (divided) by the samenegative number. Anynegativenumber. Inequality sign from thiswill change to the opposite.
You remember (hopefully ...) that the equation can be multiplied / divided by just about anything. And for any number, and for the expression with x. If only not to zero. It is neither hot nor cold for him, the equation.) It does not change. But inequalities are more sensitive to multiplication / division.
A good example for a long memory. Let's write an inequality that is beyond doubt:
5 > 2
Multiply both sides by +3, we get:
15 > 6
Any objections? There are no objections.) And if we multiply both sides of the original inequality by -3, we get:
15 > -6
And this is an outright lie.) Complete lies! Deception of the people! But it is worth changing the sign of inequality to the opposite, as everything falls into place:
15 < -6
About lies and deceit - I'm not just swearing.) "Forgot to change the inequality sign ..."- this is home error in solving inequalities. This trivial and uncomplicated rule has hurt so many people! Forgotten ...) So I swear. Maybe it will be remembered ...)
Those who are especially careful will notice that inequality cannot be multiplied by an expression with an x. Respect attentive!) Why not? The answer is simple. We do not know the sign of this expression with an x. It can be positive, negative ... Therefore, we do not know what sign of inequality to put after multiplication. Should I change it or not? Unknown. Of course, this limitation (prohibition of multiplication / division of inequality by an expression with x) can be bypassed. If you really need to. But this is a topic for other lessons.
That's all the identical transformations of inequalities. Let me remind you once again that they work for any inequalities. And now you can move on to specific types.
Linear inequalities. Solution, examples.
Linear inequalities are inequalities in which x is in the first degree and there is no division by x. Type:
x + 3 > 5x-5
How are these inequalities resolved? They can be solved very easily! Namely: with the help we reduce the most confused linear inequality straight to the answer. That's the whole solution. I will highlight the main points of the solution. To avoid stupid mistakes.)
We solve this inequality:
x + 3 > 5x-5
We solve in the same way as for a linear equation. With only one difference:
We closely follow the sign of inequality!
The first step is the most common. With x - to the left, without x - to the right ... This is the first identical transformation, simple and trouble-free.) Only the signs of the transferred members do not forget to change.
The inequality sign remains:
x-5x > -5-3
Here are similar ones.
The inequality sign remains:
4x > -8
It remains to apply the last identical transformation: divide both sides by -4.
Divide by negative number.
The inequality sign will be reversed:
NS < 2
This is the answer.
This is how all linear inequalities are solved.
Attention! Point 2 is drawn white, i.e. unpainted. Empty inside. This means that she is not included in the answer! I drew her on purpose so healthy. Such a point (empty, not healthy!)) In mathematics is called puncture point.
The rest of the numbers on the axis can be marked, but not necessary. Extraneous numbers that are not related to our inequality can be confusing, yes ... You just need to remember that the increase in numbers goes along the arrow, i.e. numbers 3, 4, 5, etc. are to the right two, and numbers 1, 0, -1, etc. - to the left.
Inequality x < 2 - strict. X is strictly less than two. If in doubt, the check is simple. We plug a dubious number into the inequality and think: "Is two less than two? Of course not!" Exactly. Inequality 2 < 2 wrong. A deuce does not work in response.
Is one good? Of course. Less ... And zero is good, and -17, and 0.34 ... Yes, all numbers that are less than two are good! And even 1.9999 .... At least a little, but less!
So let's mark all these numbers on the number axis. How? There are options here. The first option is shading. Hover the mouse over the picture (or touch the picture on the tablet) and see that the shaded area of all the x's matching the condition x < 2 ... That's all.
Let's consider the second option using the second example:
NS ≥ -0,5
Draw the axis, mark the number -0.5. Like this:
Did you notice the difference?) Well, yes, it's hard not to notice ... This point is black! Painted. This means that -0.5 is included in the answer. Here, by the way, someone can be checked and confused. We substitute:
-0,5 ≥ -0,5
How so? -0.5 is not more than -0.5! And there is more icon ...
It's OK. In a non-strict inequality, anything that fits the badge is good. AND equals good and more good. Therefore, -0.5 is included in the response.
So, we marked -0.5 on the axis, it remains to mark all the numbers that are greater than -0.5. This time I mark the area of suitable x values bow(from the word arc) rather than hatching. Hover the cursor over the picture and see this bow.
There is not much difference between shading and arches. Do as the teacher said. If there is no teacher, draw arches. In more complex tasks, the shading is less clear. You can get confused.
This is how linear inequalities are drawn on the axis. We pass to the next feature of inequalities.
Recording the answer for inequalities.
The equations were good.) We found x, and wrote down the answer, for example: x = 3. In inequalities, there are two forms of recording answers. One - in the form of final inequality. Good for simple cases. For example:
NS< 2.
This is a complete answer.
Sometimes it is required to write the same thing, but in a different form, at numerical intervals. Then the recording starts to look very scientific):
х ∈ (-∞; 2)
Under the icon ∈ the word is hiding "belongs".
The record is read like this: x belongs to the interval from minus infinity to two not including. It is quite logical. X can be any number from all possible numbers from minus infinity to two. There can be no deuce X, which is what the word tells us "not including".
And where is it in the answer you can see that "not including"? This fact is noted in the answer round parenthesis immediately after the two. If two were included, the parenthesis would be square. Like this:]. The following example uses such a parenthesis.
Let's write down the answer: x ≥ -0,5 at intervals:
x ∈ [-0.5; + ∞)
Read: x belongs to the interval from minus 0.5, including, to plus infinity.
Infinity can never be turned on. It is not a number, it is a symbol. Therefore, in such records, infinity is always adjacent to a parenthesis.
This form of writing is convenient for complex answers consisting of several intervals. But - just for the final answers. In intermediate results, where a further solution is expected, it is better to use the usual form, in the form of a simple inequality. We will deal with this in the relevant topics.
Popular jobs with inequalities.
The linear inequalities themselves are simple. Therefore, often, tasks become more complicated. So, to think it was necessary. It’s not very pleasant, if you’re not used to it.) But it’s useful. I will show examples of such tasks. Not for you to learn them, this is superfluous. And in order not to be afraid when meeting with such examples. Think a little - and everything is simple!)
1. Find any two solutions to the inequality 3x - 3< 0
If it's not very clear what to do, remember the main rule of mathematics:
If you don't know what is needed, do what you can!)
NS < 1
So what? Nothing special. What are they asking us? We are asked to find two specific numbers that solve an inequality. Those. fit the answer. Two any numbers. Actually, this is embarrassing.) A couple of 0 and 0.5 are suitable. A pair of -3 and -8. Yes, these couples are endless! What is the correct answer ?!
The answer is: everything! Any pair of numbers, each less than one, would be the correct answer. Write what you want. Let's go further.
2. Solve the inequality:
4x - 3 ≠ 0
Quests in this form are rare. But, as auxiliary inequalities, when finding the GDV, for example, or when finding the domain of definition of a function, they are often encountered. Such a linear inequality can be solved as an ordinary linear equation. Only everywhere, except for the "=" sign ( equals) put the sign " ≠ " (not equal). So you will approach the answer, with an inequality sign:
NS ≠ 0,75
In more complex examples, it is better to do it differently. Make inequality equal. Like this:
4x - 3 = 0
Calmly solve it, as taught, and get the answer:
x = 0.75
The main thing is, at the very end, when writing down the final answer, do not forget that we have found the X, which gives equality. And we need - inequality. Therefore, we just don't need this X.) And we need to write it down with the correct icon:
NS ≠ 0,75
This approach results in fewer errors. Those who automatically solve the equations. And for those who do not solve the equations, inequalities, in fact, are useless ...) Another example of a popular task:
3. Find the smallest integer solution to the inequality:
3 (x - 1) < 5x + 9
First, we just solve the inequality. We open the parentheses, transfer them, give similar ones ... We get:
NS > - 6
Wrong !? Did they follow the signs !? And behind the signs of the members, and behind the sign of inequality ...
Thinking again. We need to find a specific number that matches both the answer and the condition "smallest integer". If it doesn't immediately dawn, you can just take any number and estimate. Is two more than minus six? Of course! Is there a suitable smaller number? Of course. For example, zero is greater than -6. And even less? We need the smallest possible! Minus three is more than minus six! You can already grasp the pattern and stop stupidly sorting out numbers, right?)
We take a number closer to -6. For example, -5. The answer is executed, -5 > - 6. Can you find another number, less than -5, but more than -6? You can, for example, -5.5 ... Stop! We are told whole solution! Doesn't roll -5.5! And minus six? Uh-uh! The inequality is strict, minus 6 is not less than minus 6!
Therefore, the correct answer is -5.
I hope everything is clear with the choice of value from the general solution. Another example:
4. Solve the inequality:
7 < 3x + 1 < 13
How! This expression is called triple inequality. Strictly speaking, this is an abbreviated notation for a system of inequalities. But you still have to solve such triple inequalities in some tasks ... It is solved without any systems. For the same identical transformations.
It is necessary to simplify, to bring this inequality to a pure xx. But ... What is where to transfer !? Now is the time to remember that the shift left-right is abbreviated form the first identical transformation.
And the full form sounds like this: You can add / subtract any number or expression to both sides of the equation (inequality).
There are three parts here. So we will apply identical transformations to all three parts!
So, let's get rid of the 1 in the middle of the inequality. Subtract one from the entire middle part. So that inequality does not change, we subtract 1 from the remaining two parts. Like this:
7 -1< 3x + 1-1 < 13-1
6 < 3x < 12
Better, right?) It remains to divide all three parts into three:
2 < NS < 4
That's all. This is the answer. X can be any number from two (not including) to four (not including). This answer is also written down at intervals, such records will be in square inequalities. There they are the most common thing.
At the end of the lesson, I will repeat the most important thing. Success in solving linear inequalities depends on the ability to transform and simplify linear equations. If at the same time watch out for the sign of inequality, there will be no problems. Which is what I wish you. No problem.)
If you like this site ...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Instant validation testing. Learning - with interest!)
you can get acquainted with functions and derivatives.