Lesson and presentation on the topic: "Systems of inequalities. Examples of solutions"

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System of inequalities

Let us introduce the definition of a system of inequalities.
Several inequalities with some variable x form a system of inequalities if you need to find all the values ​​of x for which each of the inequalities forms a true numeric expression.

Any value of x for which each inequality takes the correct numerical expression is a solution to the inequality. Can also be called a private solution.
What is a private solution? For example, in the answer we received the expression x>7. Then x=8, or x=123, or any other number greater than seven is a particular solution, and the expression x>7 is general solution. The general solution is formed by many private solutions.

How did we combine the system of equations? That's right, a curly brace, and so they do the same with inequalities. Let's look at an example of a system of inequalities: $\begin(cases)x+7>5\\x-3
If the system of inequalities consists of identical expressions, for example, $\begin(cases)x+7>5\\x+7
So, what does it mean: to find a solution to a system of inequalities?
A solution to an inequality is a set of partial solutions to an inequality that satisfy both inequalities of the system at once.

We write the general form of the system of inequalities as $\begin(cases)f(x)>0\\g(x)>0\end(cases)$

Let us denote $Х_1$ as the general solution to the inequality f(x)>0.
$X_2$ is the general solution to the inequality g(x)>0.
$X_1$ and $X_2$ are a set of particular solutions.
The solution to the system of inequalities will be numbers belonging to both $X_1$ and $X_2$.
Let's remember operations on sets. How do we find elements of a set that belong to both sets at once? That's right, there is an intersection operation for this. So, the solution to our inequality will be the set $A= X_1∩ X_2$.

Examples of solutions to systems of inequalities

Solve the system of inequalities.
a) $\begin(cases)3x-1>2\\5x-10 b) $\begin(cases)2x-4≤6\\-x-4
Solution.
a) Solve each inequality separately.
$3x-1>2; \; 3x>3; \; x>1$.
$5x-10
Let's mark our intervals on one coordinate line.

The solution of the system will be the segment of intersection of our intervals. The inequality is strict, then the segment will be open.
Answer: (1;3).

B) We will also solve each inequality separately.
$2x-4≤6; 2x≤ 10; x ≤ $5.
$-x-4 -5$.


The solution of the system will be the segment of intersection of our intervals. The second inequality is strict, then the segment will be open on the left.
Answer: (-5; 5].

Let's summarize what we have learned.
Let's say it is necessary to solve the system of inequalities: $\begin(cases)f_1 (x)>f_2 (x)\\g_1 (x)>g_2 (x)\end(cases)$.
Then, the interval ($x_1; x_2$) is the solution to the first inequality.
Interval ($y_1; y_2$) is the solution to the second inequality.
The solution to a system of inequalities is the intersection of the solutions to each inequality.

Systems of inequalities can consist of not only first-order inequalities, but also any other types of inequalities.

Important rules for solving systems of inequalities.
If one of the inequalities of the system has no solutions, then the entire system has no solutions.
If one of the inequalities is satisfied for any values ​​of the variable, then the solution of the system will be the solution of the other inequality.

Examples.
Solve the system of inequalities:$\begin(cases)x^2-16>0\\x^2-8x+12≤0 \end(cases)$
Solution.
Let's solve each inequality separately.
$x^2-16>0$.
$(x-4)(x+4)>0$.

Let's solve the second inequality.
$x^2-8x+12≤0$.
$(x-6)(x-2)≤0$.

The solution to the inequality is the interval.
Let's draw both intervals on the same line and find the intersection.
The intersection of intervals is the segment (4; 6].
Answer: (4;6].

Solve the system of inequalities.
a) $\begin(cases)3x+3>6\\2x^2+4x+4 b) $\begin(cases)3x+3>6\\2x^2+4x+4>0\end(cases )$.

Solution.
a) The first inequality has a solution x>1.
Let's find the discriminant for the second inequality.
$D=16-4 * 2 * 4=-16$. $D Let us remember the rule: when one of the inequalities has no solutions, then the entire system has no solutions.
Answer: There are no solutions.

B) The first inequality has a solution x>1.
Second inequality greater than zero for all x. Then the solution of the system coincides with the solution of the first inequality.
Answer: x>1.

Problems on systems of inequalities for independent solution

System of inequalities It is customary to call any set of two or more inequalities containing an unknown quantity.

This formulation is clearly illustrated, for example, by the following systems of inequalities:

Solve the system of inequalities - means to find all values ​​of an unknown variable at which each inequality of the system is realized, or to justify that such do not exist .

This means that for each individual system inequalities We calculate the unknown variable. Next, from the resulting values, selects only those that are true for both the first and second inequalities. Therefore, when substituting the selected value, both inequalities of the system become correct.

Let's look at the solution to several inequalities:

Let's place a pair of number lines one below the other; put the value on the top x, for which the first inequality about ( x> 1) become true, and at the bottom - the value X, which are the solution to the second inequality ( X> 4).

By comparing the data on number lines, note that the solution for both inequalities will X> 4. Answer, X> 4.

Example 2.

Calculating the first inequality we get -3 X< -6, или x> 2, second - X> -8, or X < 8. Затем делаем по аналогии с предыдущим примером. На верхнюю числовую прямую наносим все те значения X, at which the first is realized inequality system, and to the lower number line, all those values X, at which the second inequality of the system is realized.

Comparing the data, we find that both inequalities will be implemented for all values X, placed from 2 to 8. Set of values X denote double inequality 2 < X< 8.

Example 3. We'll find

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.

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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 as a decimal, but also as an ordinary fraction.

Rules for entering decimal fractions.
In decimals fractional part can be separated from the whole 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: /
Whole part separated from the fraction by an ampersand: &
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 right sign inequalities and enter the polynomials in the boxes below.

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A little theory.

Systems of inequalities with one unknown. Numeric intervals

You became acquainted with the concept of a system in 7th grade and learned to solve systems of 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 \(5x \leq 4000\) unknown number x are 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 .$$

Brace shows that it is necessary to find such values ​​of x for which both inequalities of the system turn into correct numerical inequalities. This system- 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 become 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: \(-2 \leq x \leq 3 \).

The solutions to systems of inequalities with one unknown are different number sets. These sets have names. Yes, on 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.

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 reverses this inequality into the correct 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

Decide linear inequalities with one unknown you have already learned. 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:

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