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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 solve 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.
Inequality is a numerical ratio that illustrates the magnitude of numbers relative to each other. Inequalities are widely used in the search for quantities in applied sciences. Our calculator will help you deal with such a difficult topic as solving linear inequalities.
What is inequality
Unequal ratios in real life correlate with the constant comparison of different objects: higher or lower, farther or closer, heavier or lighter. Intuitively or visually, we can understand that one object is larger, higher or heavier than another, but in fact it is always a matter of comparing numbers that characterize the corresponding quantities. You can compare objects on any basis, and in any case, we can make a numerical inequality.
If the unknown quantities under specific conditions are equal, then for their numerical determination we make an equation. If not, then instead of the "equal" sign, we can indicate any other ratio between these quantities. Two numbers or mathematical objects can be greater than ">", less than "<» или равны «=» относительно друг друга. В этом случае речь идет о строгих неравенствах. Если же в неравных соотношениях присутствует знак равно и числовые элементы больше или равны (a ≥ b) или меньше или равны (a ≤ b), то такие неравенства называются нестрогими.
Inequality signs in their modern form were invented by the British mathematician Thomas Harriot, who in 1631 published a book on unequal ratios. Greater than ">" and less than "<» представляли собой положенные на бок буквы V, поэтому пришлись по вкусу не только математикам, но и типографам.
Solving inequalities
Inequalities, like equations, come in different types. Linear, square, logarithmic or exponential unequal ratios are unleashed by various methods. However, regardless of the method, any inequality must first be reduced to a standard form. For this, identical transformations are used, which are identical to the modifications of equalities.
Identity transformations of inequalities
Such transformations of expressions are very similar to the ghost of equations, but they have nuances that are important to consider when untying inequalities.
The first identity transformation is identical to the analogous operation with equalities. To both sides of the unequal ratio, you can add or subtract the same number or expression with an unknown x, while the inequality sign remains the same. Most often, this method is used in a simplified form as the transfer of the terms of the expression through the inequality sign with the change of the sign of the number to the opposite. This refers to the change of the sign of the term itself, that is, + R when transferred through any inequality sign will change to - R and vice versa.
The second transformation has two points:
- Both sides of an unequal ratio are allowed to be multiplied or divided by the same positive number. The sign of the inequality itself will not change.
- Both sides of the inequality are allowed to be divided or multiplied by the same a negative number. The sign of the inequality itself will change to the opposite.
The second identical transformation of inequalities has serious differences with the modification of equations. First, when multiplying/dividing by a negative number, the sign of an unequal expression always reverses. Secondly, dividing or multiplying parts of a relation is allowed only by a number, and not by any expression containing an unknown. The fact is that we cannot know for sure whether a number greater or less than zero is hidden behind the unknown, so the second identical transformation is applied to inequalities exclusively with numbers. Let's look at these rules with examples.
Examples of Untying Inequalities
In algebra assignments, there are a variety of assignments on the topic of inequalities. Let's give us an expression:
6x − 3(4x + 1) > 6.
First, open the brackets and move all unknowns to the left, and all numbers to the right.
6x − 12x > 6 + 3
We need to divide both parts of the expression by −6, so when finding an unknown x, the inequality sign will change to the opposite.
In solving this inequality, we used both identical transformations: Move all numbers to the right of the sign and divide both sides of the ratio by a negative number.
Our program is a solution calculator numerical inequalities, which do not contain unknowns. The program contains the following theorems for the ratios of three numbers:
- if A< B то A–C< B–C;
- if A > B, then A–C > B–C.
Instead of subtracting terms A-C, you can specify any arithmetic operation: addition, multiplication, or division. Thus, the calculator will automatically present the inequalities of sums, differences, products or fractions.
Conclusion
In real life, inequalities are as common as equations. Naturally, in everyday life, knowledge about the resolution of inequalities may not be needed. However, in applied sciences, inequalities and their systems are widely used. For example, various studies of the problems of the global economy are reduced to the compilation and unleashing of systems of linear or square inequalities, and some unequal relationships serve as an unambiguous way of proving the existence of certain objects. Use our programs to solve linear inequalities or check your own calculations.
Inequality is an expression with, ≤, or ≥. For example, 3x - 5 To solve an inequality means to find all values of the variables for which this inequality is true. Each of these numbers is a solution to the inequality, and the set of all such solutions is its many solutions. Inequalities that have the same set of solutions are called equivalent inequalities.
Linear inequalities
The principles for solving inequalities are similar to the principles for solving equations.Principles for solving inequalities
For any real numbers a, b, and c :
The principle of adding inequalities: If a Multiplication principle for inequalities: If a 0 is true, then ac If a bc is also true.
Similar statements also apply for a ≤ b.
When both sides of an inequality are multiplied by a negative number, the sign of the inequality needs to be reversed.
First-level inequalities, as in Example 1 (below), are called linear inequalities.
Example 1 Solve each of the following inequalities. Then draw a set of solutions.
a) 3x - 5 b) 13 - 7x ≥ 10x - 4
Solution
Any number less than 11/5 is a solution.
The set of solutions is (x|x
To make a check, we can plot y 1 = 3x - 5 and y 2 = 6 - 2x. Then it can be seen from here that for x 
The solution set is (x|x ≤ 1), or (-∞, 1]. The graph of the solution set 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. Record -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 the spacing notation and the symbol for associations or inclusions of both sets: (-∞ -1] (3, ∞). The graph of the set of solutions is shown below. 
To test, draw 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 modules. The following properties are used to solve them.
For a > 0 and an 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. Plot the set of solutions.
a) |3x + 2| b) |5 - 2x| ≥ 1
Solution
a) |3x + 2|
b) |5 - 2x| ≥ 1
The solution set is (x|x ≤ 2 or x ≥ 3), or (-∞, 2] )