The program for solving linear, quadratic and fractional inequalities not only gives the answer to the problem, it provides a detailed solution with explanations, i.e. displays the solution process to test knowledge in mathematics and/or algebra.

Moreover, if in the process of solving one of the inequalities it is necessary to solve, for example, a quadratic equation, then its detailed solution is also displayed (it is contained in a spoiler).

This program may be useful for high school students in preparing for tests, to parents to monitor their children’s solutions to inequalities.

This program can be useful for high school students in general education schools when preparing for tests and exams, when testing knowledge before the Unified State Exam, and for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get it done as quickly as possible? homework in mathematics or algebra? In this case, you can also use our programs with detailed solutions.

In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems increases.

Rules for entering inequalities

Any Latin letter can act as a variable.
For example: \(x, y, z, a, b, c, o, p, q\), etc.

Numbers can be entered as whole or fractional numbers.
Moreover, fractional numbers can be entered not only in the form of a decimal, but also in the form of an ordinary fraction.

Rules for entering decimal fractions.
In decimal fractions, the fractional part can be separated from the whole part by either a period or a comma.
For example, you can enter decimals like this: 2.5x - 3.5x^2

Rules for entering ordinary fractions.
Only a whole number can act as the numerator, denominator and integer part of a fraction.

The denominator cannot be negative.

When entering numerical fraction The numerator is separated from the denominator by a division sign: /
The whole part is separated from the fraction by the ampersand sign: &
Input: 3&1/3 - 5&6/5y +1/7y^2
Result: \(3\frac(1)(3) - 5\frac(6)(5) y + \frac(1)(7)y^2 \)

You can use parentheses when entering expressions. In this case, when solving inequalities, the expressions are first simplified.
For example: 5(a+1)^2+2&3/5+a > 0.6(a-2)(a+3)

Select the desired inequality sign and enter the polynomials in the fields below.

Solve the system of inequalities

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A little theory.

Systems of inequalities with one unknown. Numeric intervals

You were introduced to the concept of a system in 7th grade and learned how to solve systems linear equations with two unknowns. Next we will consider systems of linear inequalities with one unknown. Sets of solutions to systems of inequalities can be written using intervals (intervals, half-intervals, segments, rays). You will also become familiar with the notation of number intervals.

If in the inequalities \(4x > 2000\) and \(5x \leq 4000\) the unknown number x is the same, then these inequalities are considered together and they are said to form a system of inequalities: $$ \left\(\begin( array)(l) 4x > 2000 \\ 5x \leq 4000 \end(array)\right. $$

The curly bracket shows that you need to find values ​​of x for which both inequalities of the system turn into correct numerical inequalities. This system is an example of a system of linear inequalities with one unknown.

The solution to a system of inequalities with one unknown is the value of the unknown at which all the inequalities of the system turn into true numerical inequalities. Solving a system of inequalities means finding all solutions to this system or establishing that there are none.

The inequalities \(x \geq -2 \) and \(x \leq 3 \) can be written as a double inequality: \(-2 \leq x \leq 3 \).

Solutions to systems of inequalities with one unknown are various numerical sets. These sets have names. Thus, on the number axis, the set of numbers x such that \(-2 \leq x \leq 3 \) is represented by a segment with ends at points -2 and 3.

-2

3

If \(a is a segment and is denoted by [a; b]

If \(a is an interval and is denoted by (a; b)

Sets of numbers \(x\) satisfying the inequalities \(a \leq x are half-intervals and are denoted respectively [a; b) and (a; b]

Segments, intervals, half-intervals and rays are called numerical intervals.

Thus, numerical intervals can be specified in the form of inequalities.

The solution to an inequality in two unknowns is a pair of numbers (x; y) that turns the given inequality into a true numerical inequality. Solving an inequality means finding the set of all its solutions. Thus, the solutions to the inequality x > y will be, for example, pairs of numbers (5; 3), (-1; -1), since \(5 \geq 3 \) and \(-1 \geq -1\)

Solving systems of inequalities

You have already learned how to solve linear inequalities with one unknown. Do you know what a system of inequalities and a solution to the system are? Therefore, the process of solving systems of inequalities with one unknown will not cause you any difficulties.

And yet, let us remind you: to solve a system of inequalities, you need to solve each inequality separately, and then find the intersection of these solutions.

For example, the original system of inequalities was reduced to the form:
$$ \left\(\begin(array)(l) x \geq -2 \\ x \leq 3 \end(array)\right. $$

To solve this system of inequalities, mark the solution to each inequality on the number line and find their intersection:

-2

3

The intersection is the segment [-2; 3] - this is the solution to the original system of inequalities.

Preliminary information

Definition 1

An inequality of the form $f(x) >(≥)g(x)$, in which $f(x)$ and $g(x)$ are entire rational expressions, is called an entire rational inequality.

Examples of whole rational inequalities are linear, quadratic, and cubic inequalities with two variables.

Definition 2

The value $x$ at which the inequality from the definition of $1$ is satisfied is called the root of the equation.

An example of solving such inequalities:

Example 1

Solve the whole inequality $4x+3 >38-x$.

Solution.

Let's simplify this inequality:

We got a linear inequality. Let's find its solution:

Answer: $(7,∞)$.

In this article we will consider the following methods for solving entire rational inequalities.

Factorization method

This method will be as follows: An equation of the form $f(x)=g(x)$ is written. This equation is reduced to the form $φ(x)=0$ (where $φ(x)=f(x)-g(x)$). Then the function $φ(x)$ is factorized with the minimum possible powers. The rule applies: The product of polynomials equals zero when one of them equals zero. Next, the found roots are marked on the number line and a sign curve is constructed. Depending on the sign of the initial inequality, the answer is written.

Here are examples of solutions this way:

Example 2

Solve by factorization. $y^2-9

Solution.

Let's solve the equation $y^2-9

Using the difference of squares formula, we have

Using the rule that the product of factors is equal to zero, we obtain the following roots: $3$ and $-3$.

Let's draw a curve of signs:

Since the initial inequality has a “less than” sign, we get

Answer: $(-3,3)$.

Example 3

Solve by factorization.

$x^3+3x+2x^2+6 ≥0$

Solution.

Let's solve the following equation:

$x^3+3x+2x^2+6=0$

Let us take out of brackets the common factors from the first two terms and from the last two

$x(x^2+3)+2(x^2+3)=0$

Let's take out the common factor $(x^2+3)$

$(x^2+3)(x+2)=0$

Using the rule that the product of factors is equal to zero, we obtain:

$x+2=0 \ and \ x^2+3=0$

$x=-2$ and "no roots"

Let's draw a curve of signs:

Since the initial inequality has a “greater than or equal” sign, we get

Answer: $(-∞,-2]$.

Method for introducing a new variable

This method is as follows: Write an equation of the form $f(x)=g(x)$. We solve it as follows: we introduce a new variable to obtain an equation, the method of solving which is already known. We subsequently solve it and return to replacement. From it we will find the solution to the first equation. Next, the found roots are marked on the number line and a sign curve is constructed. Depending on the sign of the initial inequality, the answer is written.

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In the article we will consider solving inequalities. We will tell you clearly about how to construct a solution to inequalities, with clear examples!

Before we look at solving inequalities using examples, let’s understand the basic concepts.

General information about inequalities

Inequality is an expression in which functions are connected by relation signs >, . Inequalities can be both numerical and literal.
Inequalities with two signs of the ratio are called double, with three - triple, etc. For example:
a(x) > b(x),
a(x) a(x) b(x),
a(x) b(x).
a(x) Inequalities containing the sign > or or - are not strict.
Solving the inequality is any value of the variable for which this inequality will be true.
"Solve inequality" means that we need to find the set of all its solutions. There are different methods for solving inequalities. For inequality solutions They use the number line, which is infinite. For example, solution to inequality x > 3 is the interval from 3 to +, and the number 3 is not included in this interval, therefore the point on the line is denoted by an empty circle, because inequality is strict.
+
The answer will be: x (3; +).
The value x=3 is not included in the solution set, so the parenthesis is round. The infinity sign is always highlighted with a parenthesis. The sign means "belonging."
Let's look at how to solve inequalities using another example with a sign:
x 2
-+
The value x=2 is included in the set of solutions, so the bracket is square and the point on the line is indicated by a filled circle.
The answer will be: x. The solution set graph is shown below.

Double inequalities

When two inequalities are connected by a word And, or, then it is formed double inequality. Double inequality like
-3 And 2x + 5 ≤ 7
called connected, because it uses And. Entry -3 Double inequalities can be solved using the principles of addition and multiplication of inequalities.

Example 2 Solve -3 Solution We have

Set of solutions (x|x ≤ -1 or x > 3). We can also write the solution using interval notation and the symbol for associations or including both sets: (-∞ -1] (3, ∞). The graph of the solution set is shown below.

To check, let's plot y 1 = 2x - 5, y 2 = -7, and y 3 = 1. Note that for (x|x ≤ -1 or x > 3), y 1 ≤ y 2 or y 1 > y 3 .

Inequalities with absolute value (modulus)

Inequalities sometimes contain moduli. The following properties are used to solve them.
For a > 0 and algebraic expression x:
|x| |x| > a is equivalent to x or x > a.
Similar statements for |x| ≤ a and |x| ≥ a.

For example,
|x| |y| ≥ 1 is equivalent to y ≤ -1 or y ≥ 1;
and |2x + 3| ≤ 4 is equivalent to -4 ≤ 2x + 3 ≤ 4.

Example 4 Solve each of the following inequalities. Graph the set of solutions.
a) |3x + 2| b) |5 - 2x| ≥ 1

Solution
a) |3x + 2|

The solution set is (x|-7/3
b) |5 - 2x| ≥ 1
The solution set is (x|x ≤ 2 or x ≥ 3), or (-∞, 2] )