It has been necessary to compare quantities and quantities when solving practical problems since ancient times. At the same time, words such as more and less, higher and lower, lighter and heavier, quieter and louder, cheaper and more expensive, etc. appeared, denoting the results of comparing homogeneous quantities.
The concepts of more and less arose in connection with counting objects, measuring and comparing quantities. For example, mathematicians of Ancient Greece knew that the side of any triangle is less than the sum of the other two sides and that the larger side lies opposite the larger angle in a triangle. Archimedes, while calculating the circumference, established that the perimeter of any circle is equal to three times the diameter with an excess that is less than a seventh of the diameter, but more than ten seventy times the diameter.
Symbolically write relationships between numbers and quantities using the signs > and b. Records in which two numbers are connected by one of the signs: > (greater than), You also encountered numerical inequalities in the lower grades. You know that inequalities can be true, or they can be false. For example, \(\frac(1)(2) > \frac(1)(3)\) is a correct numerical inequality, 0.23 > 0.235 is an incorrect numerical inequality.
Inequalities involving unknowns may be true for some values of the unknowns and false for others. For example, the inequality 2x+1>5 is true for x = 3, but false for x = -3. For an inequality with one unknown, you can set the task: solve the inequality. Problems of solving inequalities in practice are posed and solved no less often than problems of solving equations. For example, many economic problems come down to the study and solution of systems of linear inequalities. In many branches of mathematics, inequalities are more common than equations.
Some inequalities serve as the only auxiliary, allowing you to prove or disprove the existence of a certain object, for example, the root of an equation.
Numerical inequalities
Can you compare integers? decimals. Do you know the rules of comparison? ordinary fractions with the same denominators but different numerators; with the same numerators, but different denominators. Here you will learn how to compare any two numbers by finding the sign of their difference.
Comparing numbers is widely used in practice. For example, an economist compares planned indicators with actual ones, a doctor compares a patient’s temperature with normal, a turner compares the dimensions of a machined part with a standard. In all such cases, some numbers are compared. As a result of comparing numbers, numerical inequalities arise.
Definition. Number a more number b, if difference a-b positive. Number a less number b, if the difference a-b is negative.
If a is greater than b, then they write: a > b; if a is less than b, then they write: a Thus, the inequality a > b means that the difference a - b is positive, i.e. a - b > 0. Inequality a For any two numbers a and b, from the following three relations a > b, a = b, a To compare the numbers a and b means to find out which of the signs >, = or Theorem. If a > b and b > c, then a > c.
Theorem. If you add the same number to both sides of the inequality, the sign of the inequality will not change.
Consequence. Any term can be moved from one part of the inequality to another by changing the sign of this term to the opposite.
Theorem. If both sides of the inequality are multiplied by the same positive number, then the sign of the inequality does not change. If both sides of the inequality are multiplied by the same a negative number, then the sign of inequality will change to the opposite.
Consequence. If both sides of the inequality are divided by the same positive number, then the sign of the inequality will not change. If both sides of the inequality are divided by the same negative number, then the sign of the inequality will change to the opposite.
You know that numerical equalities can be added and multiplied term by term. Next, you will learn how to perform similar actions with inequalities. The ability to add and multiply inequalities term by term is often used in practice. These actions help solve problems of evaluating and comparing the meanings of expressions.
When solving various problems, it is often necessary to add or multiply the left and right sides of inequalities term by term. At the same time, it is sometimes said that inequalities add up or multiply. For example, if a tourist walked more than 20 km on the first day, and more than 25 km on the second, then we can say that in two days he walked more than 45 km. Similarly, if the length of a rectangle is less than 13 cm and the width is less than 5 cm, then we can say that the area of this rectangle is less than 65 cm2.
When considering these examples, the following were used: theorems on addition and multiplication of inequalities:
Theorem. When adding inequalities of the same sign, an inequality of the same sign is obtained: if a > b and c > d, then a + c > b + d.
Theorem. When multiplying inequalities of the same sign, whose left and right sides are positive, an inequality of the same sign is obtained: if a > b, c > d and a, b, c, d are positive numbers, then ac > bd.
Inequalities with the sign > (greater than) and 1/2, 3/4 b, c Along with the signs of strict inequalities > and In the same way, the inequality \(a \geq b \) means that the number a is greater than or equal to b, i.e. .and not less b.
Inequalities containing the \(\geq \) sign or the \(\leq \) sign are called non-strict. For example, \(18 \geq 12 , \; 11 \leq 12 \) are not strict inequalities.
All properties of strict inequalities are also valid for non-strict inequalities. Moreover, if for strict inequalities the signs > were considered opposite and you know that to solve a number of applied problems you have to create a mathematical model in the form of an equation or a system of equations. Next, you will learn that mathematical models for solving many problems are inequalities with unknowns. We will introduce the concept of solving an inequality and show how to check whether given number solving a specific inequality.
Inequalities of the form
\(ax > b, \quad ax in which a and b are given numbers, and x is an unknown, are called linear inequalities with one unknown.
Definition. The solution to an inequality with one unknown is the value of the unknown at which this inequality becomes a true numerical inequality. Solving an inequality means finding all its solutions or establishing that there are none.
You solved the equations by reducing them to the simplest equations. Similarly, when solving inequalities, one tries to reduce them, using properties, to the form of simple inequalities.
Solving second degree inequalities with one variable
Inequalities of the form
\(ax^2+bx+c >0 \) and \(ax^2+bx+c where x is a variable, a, b and c are some numbers and \(a \neq 0 \), called inequalities of the second degree with one variable.
Solution to inequality
\(ax^2+bx+c >0 \) or \(ax^2+bx+c can be considered as finding intervals in which the function \(y= ax^2+bx+c \) takes positive or negative values To do this, it is enough to analyze how the graph of the function \(y= ax^2+bx+c\) is located in the coordinate plane: where the branches of the parabola are directed - up or down, whether the parabola intersects the x-axis and if it does, then at what points.
Algorithm for solving second degree inequalities with one variable:
1) find the discriminant of the square trinomial \(ax^2+bx+c\) and find out whether the trinomial has roots;
2) if the trinomial has roots, then mark them on the x-axis and through the marked points draw a schematic parabola, the branches of which are directed upward for a > 0 or downward for a 0 or at the bottom for a 3) find intervals on the x-axis for which the points parabolas are located above the x-axis (if they solve the inequality \(ax^2+bx+c >0\)) or below the x-axis (if they solve the inequality
\(ax^2+bx+c Solving inequalities using the interval method
Consider the function
f(x) = (x + 2)(x - 3)(x - 5)
The domain of this function is the set of all numbers. The zeros of the function are the numbers -2, 3, 5. They divide the domain of definition of the function into the intervals \((-\infty; -2), \; (-2; 3), \; (3; 5) \) and \( (5; +\infty)\)
Let us find out what the signs of this function are in each of the indicated intervals.
The expression (x + 2)(x - 3)(x - 5) is the product of three factors. The sign of each of these factors in the intervals under consideration is indicated in the table:
In general, let the function be given by the formula
f(x) = (x-x 1)(x-x 2) ... (x-x n),
where x is a variable, and x 1, x 2, ..., x n are numbers that are not equal to each other. The numbers x 1 , x 2 , ..., x n are the zeros of the function. In each of the intervals into which the domain of definition is divided by zeros of the function, the sign of the function is preserved, and when passing through zero its sign changes.
This property is used to solve inequalities of the form
(x-x 1)(x-x 2) ... (x-x n) > 0,
(x-x 1)(x-x 2) ... (x-x n) where x 1, x 2, ..., x n are numbers not equal to each other
Considered method solving inequalities is called the interval method.
Let us give examples of solving inequalities using the interval method.
Solve inequality:
\(x(0.5-x)(x+4) Obviously, the zeros of the function f(x) = x(0.5-x)(x+4) are the points \(x=0, \; x= \frac(1)(2) , \; x=-4 \)We plot the zeros of the function on the number axis and calculate the sign on each interval:
We select those intervals at which the function is less than or equal to zero and write down the answer.
Answer:
\(x \in \left(-\infty; \; 1 \right) \cup \left[ 4; \; +\infty \right) \)
The concept of mathematical inequality arose in ancient times. This happened when primitive man began to need to compare their quantity and size when counting and handling various objects. Since ancient times, Archimedes, Euclid and other famous scientists: mathematicians, astronomers, designers and philosophers used inequalities in their reasoning.
But they, as a rule, used verbal terminology in their works. For the first time, modern signs to denote the concepts of “more” and “less” in the form in which every schoolchild knows them today were invented and put into practice in England. The mathematician Thomas Harriot provided such a service to his descendants. And this happened about four centuries ago.
There are many types of inequalities known. Among them are simple ones, containing one, two or more variables, quadratic, fractional, complex ratios, and even those represented by a system of expressions. The best way to understand how to solve inequalities is to use various examples.
Don't miss the train
To begin with, let’s imagine that a resident rural areas hurries to railway station, which is located at a distance of 20 km from his village. In order not to miss the train leaving at 11 o'clock, he must leave the house on time. At what time should this be done if its speed is 5 km/h? The solution to this practical problem comes down to fulfilling the conditions of the expression: 5 (11 - X) ≥ 20, where X is the departure time.
This is understandable, because the distance that a villager needs to cover to the station is equal to the speed of movement multiplied by the number of hours on the road. A person can arrive early, but he cannot be late. Knowing how to solve inequalities and applying your skills in practice, you will end up with X ≤ 7, which is the answer. This means that the villager should go to the railway station at seven in the morning or a little earlier.
Numerical intervals on a coordinate line
Now let's find out how to map the described relations onto the The inequality obtained above is not strict. It means that the variable can take values less than 7, or it can be equal to this number. Let's give other examples. To do this, carefully consider the four figures presented below.
On the first one you can see graphic image gap [-7; 7]. It consists of a set of numbers placed on a coordinate line and located between -7 and 7, including the boundaries. In this case, the points on the graph are depicted as filled circles, and the interval is recorded using
The second figure is a graphical representation of the strict inequality. In this case, the borderline numbers -7 and 7, shown by punctured (not filled in) dots, are not included in the specified set. And the interval itself is written in parentheses as follows: (-7; 7).
That is, having figured out how to solve inequalities of this type and received a similar answer, we can conclude that it consists of numbers that are between the boundaries in question, except -7 and 7. The next two cases must be evaluated in a similar way. The third figure shows images of the intervals (-∞; -7] U ∪[ \frac(2)(3);∞)\)
Quadratic inequalities with negative and zero discriminant
The algorithm above works when the discriminant is greater than zero, that is, it has \(2\) roots. What to do in other cases? For example, these:
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\(1) x^2+2x+9>0\) |
\(2) x^2+6x+9≤0\) |
\(3)-x^2-4x-4>0\) |
\(4)-x^2-64<0\) |
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\(D=4-36=-32<0\) |
\(D=-4 \cdot 64<0\) |
If \(D<0\), то квадратный трехчлен имеет постоянный знак, совпадающий со знаком коэффициента \(a\) (тем, что стоит перед \(x^2\)).
That is, the expression:
\(x^2+2x+9\) – positive for any \(x\), because \(a=1>0\)
\(-x^2-64\) - negative for any \(x\), because \(a=-1<0\)
If \(D=0\), then the quadratic trinomial for one value \(x\) is equal to zero, and for all others it has a constant sign, which coincides with the sign of the coefficient \(a\).
That is, the expression:
\(x^2+6x+9\) is equal to zero for \(x=-3\) and positive for all other x's, because \(a=1>0\)
\(-x^2-4x-4\) - equal to zero for \(x=-2\) and negative for all others, because \(a=-1<0\).
How to find x at which the quadratic trinomial is equal to zero? We need to solve the corresponding quadratic equation.
Given this information, let's solve the quadratic inequalities:
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1) \(x^2+2x+9>0\) |
The inequality, one might say, asks us the question: “for which \(x\) is the expression on the left greater than zero?” We have already found out above that for any. In the answer you can write: “for any \(x\)”, but it is better to express the same idea in the language of mathematics. |
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Answer: \(x∈(-∞;∞)\) |
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2) \(x^2+6x+9≤0\) |
Question from inequality: “for which \(x\) is the expression on the left less than or equal to zero?” It cannot be less than zero, but it can be equal to zero. And to find out at what claim this will happen, let’s solve the corresponding quadratic equation. |
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Let's assemble our expression according to \(a^2+2ab+b^2=(a+b)^2\). |
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Now the only thing stopping us is the square. Let's think together - what number squared is equal to zero? Zero! This means that the square of an expression is equal to zero only if the expression itself is equal to zero. |
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\(x+3=0\) |
This number will be the answer. |
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Answer: \(-3\) |
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3)\(-x^2-4x-4>0\) |
When is the expression on the left greater than zero? As mentioned above, the expression on the left is either negative or equal to zero; it cannot be positive. So the answer is never. Let's write “never” in the language of mathematics, using the “empty set” symbol - \(∅\). |
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Answer: \(x∈∅\) |
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4) \(-x^2-64<0\) |
When the expression is on the left less than zero? Always. This means that the inequality holds for any \(x\). |
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Answer: \(x∈(-∞;∞)\) |
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