Online calculator. Solving inequalities: linear, square and fractional. The solution of the quadratic equations X is equal to

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First, let's recall the basic formulas of degrees and their properties.

Product of a number a happens on itself n times, we can write this expression as a a … a=a n

1. a 0 = 1 (a ≠ 0)

3. a n a m = a n + m

4. (a n) m = a nm

5. a n b n = (ab) n

7. a n / a m \u003d a n - m

Power or exponential equations - these are equations in which the variables are in powers (or exponents), and the base is a number.

Examples of exponential equations:

In this example, the number 6 is the base, it is always at the bottom, and the variable x degree or measure.

Let us give more examples of exponential equations.
2 x *5=10
16x-4x-6=0

Now let's look at how exponential equations are solved?

Let's take a simple equation:

2 x = 2 3

Such an example can be solved even in the mind. It can be seen that x=3. After all, in order for the left and right sides to be equal, you need to put the number 3 instead of x.
Now let's see how this decision should be made:

2 x = 2 3
x = 3

To solve this equation, we removed same grounds(that is, deuces) and wrote down what was left, these are degrees. We got the answer we were looking for.

Now let's summarize our solution.

Algorithm for solving the exponential equation:
1. Need to check the same whether the bases of the equation on the right and on the left. If the grounds are not the same, we are looking for options to solve this example.
2. After the bases are the same, equate degree and solve the resulting new equation.

Now let's solve some examples:

Let's start simple.

The bases on the left and right sides are equal to the number 2, which means we can discard the base and equate their degrees.

x+2=4 The simplest equation has turned out.
x=4 - 2
x=2
Answer: x=2

In the following example, you can see that the bases are different, these are 3 and 9.

3 3x - 9 x + 8 = 0

To begin with, we transfer the nine to the right side, we get:

Now you need to make the same bases. We know that 9=3 2 . Let's use the power formula (a n) m = a nm .

3 3x \u003d (3 2) x + 8

We get 9 x + 8 \u003d (3 2) x + 8 \u003d 3 2 x + 16

3 3x \u003d 3 2x + 16 now it is clear that the bases on the left and right sides are the same and equal to three, which means we can discard them and equate the degrees.

3x=2x+16 got the simplest equation
3x-2x=16
x=16
Answer: x=16.

Let's look at the following example:

2 2x + 4 - 10 4 x \u003d 2 4

First of all, we look at the bases, the bases are different two and four. And we need to be the same. We transform the quadruple according to the formula (a n) m = a nm .

4 x = (2 2) x = 2 2x

And we also use one formula a n a m = a n + m:

2 2x+4 = 2 2x 2 4

Add to the equation:

2 2x 2 4 - 10 2 2x = 24

We gave an example for the same reasons. But other numbers 10 and 24 interfere with us. What to do with them? If you look closely, you can see that on the left side we repeat 2 2x, here is the answer - we can put 2 2x out of brackets:

2 2x (2 4 - 10) = 24

Let's calculate the expression in brackets:

2 4 — 10 = 16 — 10 = 6

We divide the whole equation by 6:

Imagine 4=2 2:

2 2x \u003d 2 2 bases are the same, discard them and equate the degrees.
2x \u003d 2 turned out to be the simplest equation. We divide it by 2, we get
x = 1
Answer: x = 1.

Let's solve the equation:

9 x - 12*3 x +27= 0

Let's transform:
9 x = (3 2) x = 3 2x

We get the equation:
3 2x - 12 3 x +27 = 0

Our bases are the same, equal to three. In this example, it is clear that the first triple has a degree twice (2x) than the second (just x). In this case, you can decide substitution method. The number with the smallest degree is replaced by:

Then 3 2x \u003d (3 x) 2 \u003d t 2

We replace all degrees with x's in the equation with t:

t 2 - 12t + 27 \u003d 0
We get a quadratic equation. We solve through the discriminant, we get:
D=144-108=36
t1 = 9
t2 = 3

back to the variable x.

We take t 1:
t 1 \u003d 9 \u003d 3 x

That is,

3 x = 9
3 x = 3 2
x 1 = 2

One root was found. We are looking for the second one, from t 2:
t 2 \u003d 3 \u003d 3 x
3 x = 3 1
x 2 = 1
Answer: x 1 \u003d 2; x 2 = 1.

On the site you can in the section HELP DECIDE to ask questions of interest, we will definitely answer you.

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It has been necessary to compare values and quantities in solving practical problems since ancient times. At the same time, such words 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 the counting of objects, the measurement and comparison of quantities. For example, the 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 of the triangle lies opposite the larger angle. Archimedes, while calculating the circumference of a circle, found 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-firsts of the diameter.

Symbolically write relationships between numbers and quantities using the > and b signs. Entries in which two numbers are connected by one of the signs: > (greater than), You also met with numerical inequalities in elementary grades. You know that inequalities may or may not be true. For example, \(\frac(1)(2) > \frac(1)(3) \) is a valid numerical inequality, 0.23 > 0.235 is an invalid numerical inequality.

Inequalities that include 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 frequently than problems of solving equations. For example, many economic problems are reduced 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 means to prove or disprove the existence of a certain object, for example, the root of an equation.

Numerical inequalities

Can you compare whole numbers? decimals. 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.

Comparison of 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 than 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 Theorem. If a > b and b > c, then a > c.

Theorem. If the same number is added to both sides of the inequality, then the sign of the inequality does not change.
Consequence. Any term can be transferred 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 inequality sign will be reversed.
Consequence. If both parts of the inequality are divided by the same positive number, then the sign of the inequality does not change. If both parts 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 you solve the problems of evaluating and comparing expression values.

When solving various problems, it is often necessary to add or multiply term by term the left and right parts of inequalities. It is sometimes said that inequalities are added or multiplied. For example, if a tourist walked more than 20 km on the first day, and more than 25 km on the second day, then it can be argued 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 it can be argued that the area of this rectangle is less than 65 cm2.

In considering these examples, the following theorems on addition and multiplication of inequalities:

Theorem. When adding inequalities of the same sign, we get an inequality of the same sign: if a > b and c > d, then a + c > b + d.

Theorem. When multiplying inequalities of the same sign, for which the left and right parts are positive, we get an inequality of the same sign: if a > b, c > d and a, b, c, d - positive numbers, then ac > bd.

Inequalities with the sign > (greater than) and 1/2, 3/4 b, c Along with the strict inequality signs > 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 than b.

Inequalities containing the sign \(\geq \) or the sign \(\leq \) 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 in order to solve a number of applied problems, you have to draw up a mathematical model in the form of an equation or a system of equations. Next, you will find out that mathematical models to solve many problems are inequalities with unknowns. We will introduce the concept of solving an inequality and show how to check whether a given number is a solution to a particular inequality.

Inequalities of the form
\(ax > b, \quad ax where a and b are given numbers, and x is unknown, is called linear inequalities with one unknown.

Definition. The solution of an inequality with one unknown is the value of the unknown for which this inequality turns into a true numerical inequality. To solve an inequality means to find all its solutions or establish that there are none.

You solved the equations by reducing them to the simplest equations. Similarly, when solving inequalities, one tends to reduce them with the help of properties to the form of the simplest inequalities.

Solution of 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 \(a \neq 0 \) are called second degree inequalities with one variable.

Solving the inequality
\(ax^2+bx+c >0 \) or \(ax^2+bx+c \) can be thought of as finding gaps where 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 if the trinomial has roots;
2) if the trinomial has roots, then mark them on the x-axis and draw a schematic parabola through the marked points, the branches of which are directed upwards at a > 0 or downwards at a 0 or lower at a 3) find gaps 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 Solution of inequalities by the method of intervals

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 the function into intervals \((-\infty; -2), \; (-2; 3), \; (3; 5) \) and \( (5; +\infty) \)

Let us find out what are the signs of this function 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 considered intervals 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 not equal numbers. 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 the 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 not equal numbers

Considered method solving inequalities is called the method of intervals.

Let us give examples of solving inequalities by the interval method.

Solve the inequality:

\(x(0.5-x)(x+4) Obviously, the zeros of the function f(x) = x(0.5-x)(x+4) are the points \frac(1)(2) , \; x=-4 \)

Apply to numerical axis zeros of the function and calculate the sign on each interval:

We select those intervals on which the function is less than or equal to zero and write down the answer.

Answer:
\(x \in \left(-\infty; \; 1 \right) \cup \left[ 4; \; +\infty \right) \)

y (x) = e x, whose derivative is equal to the function itself.

The exponent is denoted as , or .

e number

The base of the degree of the exponent is e number. This is an irrational number. It is approximately equal
e ≈ 2,718281828459045...

The number e is determined through the limit of the sequence. This so-called second wonderful limit:
.

Also, the number e can be represented as a series:
.

Exhibitor chart

Exponent plot, y = e x .

The graph shows the exponent, e to the extent X.
y (x) = e x
The graph shows that the exponent increases monotonically.

Formulas

The basic formulas are the same as for exponential function with base e.

;
;
;

Expression of an exponential function with an arbitrary base of degree a through the exponent:
.

Private values

Let y (x) = e x. Then
.

Exponent Properties

The exponent has the properties of an exponential function with a base of degree e > 1 .

Domain of definition, set of values

Exponent y (x) = e x defined for all x .
Its scope is:
- ∞ < x + ∞ .
Its set of meanings:
0 < y < + ∞ .

Extremes, increase, decrease

The exponent is a monotonically increasing function, so it has no extrema. Its main properties are presented in the table.

Inverse function

The reciprocal of the exponent is the natural logarithm.
;
.

Derivative of the exponent

Derivative e to the extent X is equal to e to the extent X :
.
Derivative of the nth order:
.
Derivation of formulas > > >

Integral

Complex numbers

Actions with complex numbers carried out through Euler formulas:
,
where is the imaginary unit:
.

Expressions in terms of hyperbolic functions

; ;
.

Expressions in terms of trigonometric functions

; ;
;
.

Power series expansion

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

What's happened "square inequality"? Not a question!) If you take any quadratic equation and change the sign in it "=" (equal) to any inequality icon ( > ≥ < ≤ ≠ ), we get a quadratic inequality. For example:

1. x2 -8x+12 ≥ 0

2. -x 2 +3x > 0

3. x2 ≤ 4

Well, you get the idea...)

I knowingly linked equations and inequalities here. The fact is that the first step in solving any square inequality - solve the equation from which this inequality is made. For this reason, the inability to decide quadratic equations automatically leads to a complete failure in inequalities. Is the hint clear?) If anything, look at how to solve any quadratic equations. Everything is detailed there. And in this lesson we will deal with inequalities.

The inequality ready for solution has the form: left - square trinomial ax 2 +bx+c, on the right - zero. The inequality sign can be absolutely anything. The first two examples are here are ready for a decision. The third example still needs to be prepared.

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. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

Consider the function y=k/y. The graph of this function is a line, called a hyperbola in mathematics. The general view of the hyperbola is shown in the figure below. (The graph shows a function y equals k divided by x, where k is equal to one.)

It can be seen that the graph consists of two parts. These parts are called branches of the hyperbola. It is also worth noting that each branch of the hyperbola comes closer and closer to the coordinate axes in one of the directions. The coordinate axes in this case are called asymptotes.

In general, any straight lines that the graph of a function infinitely approaches, but does not reach, are called asymptotes. A hyperbola, like a parabola, has axes of symmetry. For the hyperbola shown in the figure above, this is the straight line y=x.

Now let's deal with two general cases of hyperbolas. The graph of the function y = k/x, for k ≠ 0, will be a hyperbola, the branches of which are located either in the first and third coordinate angles, for k>0, or in the second and fourth coordinate angles, for k<0.