Solving inequalities online
Before solving inequalities, you need to have a good understanding of how equations are solved.
It doesn’t matter whether the inequality is strict () or non-strict (≤, ≥), the first step is to solve the equation by replacing the inequality sign with equality (=).
Let us explain what it means to solve an inequality?
After studying the equations, the student gets the following picture in his head: he needs to find values of the variable such that both sides of the equation take on the same values. In other words, find all points at which equality holds. Everything is correct!
When we talk about inequalities, we mean finding intervals (segments) on which the inequality holds. If there are two variables in the inequality, then the solution will no longer be intervals, but some areas on the plane. Guess for yourself what will be the solution to an inequality in three variables?
How to solve inequalities?
A universal way to solve inequalities is considered to be the method of intervals (also known as the method of intervals), which consists in determining all intervals within the boundaries of which a given inequality will be satisfied.
Without going into the type of inequality, in this case this is not the point, you need to solve the corresponding equation and determine its roots, followed by the designation of these solutions on the number axis.
How to correctly write the solution to an inequality?
Once you have determined the solution intervals for the inequality, you need to correctly write out the solution itself. There is an important nuance - are the boundaries of the intervals included in the solution?
Everything is simple here. If the solution to the equation satisfies the ODZ and the inequality is not strict, then the boundary of the interval is included in the solution to the inequality. Otherwise, no.
Considering each interval, the solution to the inequality may be the interval itself, or a half-interval (when one of its boundaries satisfies the inequality), or a segment - the interval together with its boundaries.
Do not think that only intervals, half-intervals and segments can solve the inequality. No, the solution may also include individual points.
For example, the inequality |x|≤0 has only one solution - this is point 0.
And the inequality |x|
Why do you need an inequality calculator?
The inequalities calculator gives the correct final answer. In most cases, an illustration of a number axis or plane is provided. It is visible whether the boundaries of the intervals are included in the solution or not - the points are displayed as shaded or punctured.
Thanks to online calculator For inequalities, you can check whether you correctly found the roots of the equation, marked them on the number axis and checked on the intervals (and boundaries) whether the condition of the inequality is met?
If your answer differs from the calculator’s answer, then you definitely need to double-check your solution and identify the mistake.
Inequality is a numerical relationship that illustrates the size of numbers relative to each other. Inequalities are widely used in searching 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 correspond to the constant comparison of different objects: higher or lower, further or closer, heavier or lighter. Intuitively or visually, we can understand that one object is larger, taller or heavier than another, but in fact we are always talking about comparing numbers that characterize the corresponding quantities. Objects can be compared on any basis and in any case we can create a numerical inequality.
If the unknown quantities are equal under specific conditions, then we create an equation to determine them numerically. If not, then instead of the “equal” sign we can indicate any other relationship 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. Signs 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, quadratic, logarithmic or exponential unequal relationships are resolved by various methods. However, regardless of the method, any inequality must first be reduced to a standard form. For this, identity transformations are used that are identical to modifications of equalities.
Identical transformations of inequalities
Such transformations of expressions are very similar to ghosting equations, but they have nuances that are important to consider when solving inequalities.
The first identity transformation is identical to a similar operation with equalities. The same number or expression with an unknown x can be added or subtracted to both sides of an unequal relationship, while the sign of the inequality remains the same. Most often, this method is used in a simplified form as transferring terms of an expression through an inequality sign with changing the sign of the number to the opposite one. This means a change in 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 an inequality can be divided or multiplied by the same negative number. The sign of inequality itself will change to the opposite.
The second identical transformation of inequalities has serious differences with the modification of equations. Firstly, when multiplying/dividing by a negative number, the sign of the unequal expression is always reversed. Secondly, you can only divide or multiply parts of a ratio by a number, and not by any expression containing an unknown. The fact is that we cannot know for sure whether a number is greater or less than zero hidden behind the unknown, so the second identity transformation is applied to inequalities exclusively with numbers. Let's look at these rules with examples.
Examples of unleashing inequalities
In algebra assignments, there are a variety of assignments on the topic of inequalities. Let us be given the expression:
6x − 3(4x + 1) > 6.
First, let's open the brackets and move all the unknowns to the left, and all the numbers to the right.
6x − 12x > 6 + 3
We need to divide both sides of the expression by −6, so when we find the unknown x, the inequality sign will change to the opposite.
In solving this inequality we used both identity transformations: Move all the numbers to the right of the sign and divide both sides of the ratio by the negative number.
Our program is a calculator for solving numerical inequalities that do not contain unknowns. The program contains the following theorems for the relationships 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. This way, the calculator will automatically present inequalities for sums, differences, products, or fractions.
Conclusion
In real life, inequalities are as common as equations. Naturally, knowledge about resolving inequalities may not be needed in everyday life. However, in applied sciences, inequalities and their systems are widely used. Eg, various studies problems of the global economy come down to the compilation and untying of systems of linear or quadratic inequalities, and some unequal relations serve as an unambiguous way of proving the existence of certain objects. Use our programs to solve linear inequalities or check your own calculations.
Of the form ax 2 + bx + 0 0, where (instead of the > sign there can, of course, be any other inequality sign). We have all the theoretical facts necessary to solve such inequalities, as we will now see.
Example 1. Solve inequality:
a) x 2 - 2x - 3 >0; b) x 2 - 2x - 3< 0;
c) x 2 - 2x - 3 > 0; d) x 2 - 2x - 3< 0.
Solution,
a) Consider the parabola y = x 2 - 2x - 3, shown in Fig. 117.
Solving the inequality x 2 - 2x - 3 > 0 means answering the question at what values of x the ordinates of the points of the parabola are positive.
We note that y > 0, i.e. the graph of the function is located above the x axis, at x< -1 или при х > 3.
This means that the solutions to the inequality are all points of the open beam(- 00 , - 1), as well as all points of the open beam (3, +00).
Using the sign U (the sign for combining sets), the answer can be written as follows: (-00, - 1) U (3, +00). However, the answer can be written like this: x< - 1; х > 3.
b) Inequality x 2 - 2x - 3< 0, или у < 0, где у = х 2 - 2х - 3, также можно решить с помощью рис. 117: schedule located below the x-axis if -1< х < 3. Поэтому решениями данного неравенства служат все точки интервала (- 1, 3).
c) The inequality x 2 - 2x - 3 > 0 differs from the inequality x 2 - 2x - 3 > 0 in that the answer must also include the roots of the equation x 2 - 2x - 3 = 0, i.e. points x = -1
and x = 3. Thus, the solutions to this non-strict inequality are all points of the ray (-00, - 1], as well as all points of the ray.
Practical mathematicians usually say this: why do we need to carefully construct a parabola graph of a quadratic function when solving the inequality ax 2 + bx + c > 0
y = ax 2 + bx + c (as was done in example 1)? It is enough to make a schematic sketch of the graph, for which you just need to find roots quadratic trinomial (the point of intersection of the parabola with the x-axis) and determine whether the branches of the parabola are directed up or down. This schematic sketch will give a visual interpretation of the solution to the inequality.
Example 2. Solve the inequality - 2x 2 + 3x + 9< 0.
Solution.
1) Find the roots of the square trinomial - 2x 2 + 3x + 9: x 1 = 3; x 2 = - 1.5.
2) The parabola, which serves as a graph of the function y = -2x 2 + 3x + 9, intersects the x axis at points 3 and - 1.5, and the branches of the parabola are directed downward, since the highest coefficient- negative number - 2. In Fig. 118 shows a sketch of the graph.
3) Using fig. 118, we conclude:< 0 на тех промежутках оси х, где график расположен ниже оси х, т.е. на открытом луче (-оо, -1,5) или на открытом луче C, +оо).
Answer: x< -1,5; х > 3.
Example 3. Solve the inequality 4x 2 - 4x + 1< 0.
Solution.
1) From the equation 4x 2 - 4x + 1 = 0 we find .
2) A square trinomial has one root; this means that the parabola serving as the graph of a quadratic trinomial does not intersect the x-axis, but touches it at point . The branches of the parabola are directed upward (Fig. 119.)
3) Using the geometric model presented in Fig. 119, we establish that the given inequality is satisfied only at the point, since for all other values of x the ordinates of the graph are positive.
Answer: .
You probably noticed that in fact, in examples 1, 2, 3, a very specific algorithm solution of quadratic inequalities, let's formalize it.
Algorithm for solving the quadratic inequality ax 2 + bx + 0 0 (ax 2 + bx + c< 0)
The first step of this algorithm is to find the roots of a quadratic trinomial. But the roots may not exist, so what can we do? Then the algorithm is not applicable, which means we need to think differently. The key to these arguments is given by the following theorems.
In other words, if D< 0, а >0, then the inequality ax 2 + bx + c > 0 holds for all x; on the contrary, the inequality ax 2 + bx + c< 0 не имеет решений.
Proof.
Schedule functions y = ax 2 + bx + c is a parabola whose branches are directed upward (since a > 0) and which does not intersect the x axis, since the quadratic trinomial has no roots by condition. The graph is shown in Fig. 120. We see that for all x the graph is located above the x axis, which means that for all x the inequality ax 2 + bx + c > 0 holds, which is what needed to be proven.
In other words, if D< 0, а < 0, то неравенство ах 2 + bх + с < 0 выполняется при всех х; напротив, неравенство ах 2 + bх + с >0 has no solutions.
Proof. The graph of the function y = ax 2 + bx +c is a parabola, the branches of which are directed downward (since a< 0) и которая не пересекает ось х, так как корней у квадратного трехчлена по условию нет. График представлен на рис. 121. Видим, что при всех х график расположен ниже оси х, а это значит, что при всех х выполняется неравенство ах 2 + bх + с < 0, что и требовалось доказать.
Example 4. Solve inequality:
a) 2x 2 - x + 4 >0; b) -x 2 + 3x - 8 >0.
a) Find the discriminant of the square trinomial 2x 2 - x + 4. We have D = (-1) 2 - 4 2 4 = - 31< 0.
The leading coefficient of the trinomial (number 2) is positive.
This means, according to Theorem 1, for all x the inequality 2x 2 - x + 4 > 0 holds, that is, the solution to the given inequality is the whole (-00, + 00).
b) Find the discriminant of the square trinomial - x 2 + 3x - 8. We have D = 32 - 4 (- 1) (- 8) = - 23< 0. Старший коэффициент трехчлена (число - 1) отрицателен. Следовательно, по теореме 2, при всех х выполняется неравенство - х 2 + Зx - 8 < 0. Это значит, что неравенство - х 2 + Зх - 8 0 не выполняется ни при каком значении х, т. е. заданное неравенство не имеет решений.
Answer: a) (-00, + 00); b) no solutions.
In the following example, we will introduce another method of reasoning that is used to solve quadratic inequalities.
Example 5. Solve the inequality 3x 2 - 10x + 3< 0.
Solution. Let's decompose quadratic trinomial 3x 2 - 10x + 3 for multipliers. The roots of the trinomial are the numbers 3 and , so using ax 2 + bx + c = a (x - x 1)(x - x 2), we get 3x 2 - 10x + 3 = 3(x - 3) (x - )
Let us mark the roots of the trinomial on the number line: 3 and (Fig. 122).
Let x > 3; then x-3>0 and x->0, and therefore the product 3(x - 3)(x - ) is positive. Next, let< х < 3; тогда x-3< 0, а х- >0. Therefore, the product 3(x-3)(x-) is negative. Finally, let x<; тогда x-3< 0 и x- < 0. Но в таком случае произведение
3(x -3)(x -) is positive.
Summarizing the reasoning, we come to the conclusion: the signs of the square trinomial 3x 2 - 10x + 3 change as shown in Fig. 122. We are interested in at what x the square trinomial takes negative values. From Fig. 122 we conclude: the square trinomial 3x 2 - 10x + 3 takes negative values for any value of x from the interval (, 3)
Answer (, 3), or< х < 3.
Comment. The reasoning method we used in Example 5 is usually called the method of intervals (or the method of intervals). It is actively used in mathematics to solve rational inequalities In 9th grade we will study the interval method in more detail.
Example 6. At what values of the parameter p quadratic equation x 2 - 5x + p 2 = 0:
a) has two different roots;
b) has one root;
c) has no roots?
Solution. The number of roots of a quadratic equation depends on the sign of its discriminant D. In this case, we find D = 25 - 4p 2.
a) The quadratic equation has two different roots, if D>0, then the problem is reduced to solving the inequality 25 - 4р 2 > 0. Let's multiply both sides of this inequality by -1 (not forgetting to change the sign of the inequality). We obtain the equivalent inequality 4p 2 - 25< 0. Далее имеем 4 (р - 2,5) (р + 2,5) < 0.
The signs of the expression 4(p - 2.5) (p + 2.5) are shown in Fig. 123.
We conclude that inequality 4(p - 2.5)(p + 2.5)< 0 выполняется для всех значений р из интервала (-2,5; 2,5). Именно при этих значениях параметра р данное квадратное уравнение имеет два различных корня.
b) quadratic equation has one root if D - 0.
As we established above, D = 0 at p = 2.5 or p = -2.5.
It is for these values of the parameter p that this quadratic equation has only one root.
c) A quadratic equation has no roots if D< 0. Решим неравенство 25 - 4р 2 < 0.
We get 4p 2 - 25 > 0; 4 (p-2.5)(p + 2.5)>0, whence (see Fig. 123) p< -2,5; р >2.5. For these values of the parameter p, this quadratic equation has no roots.
Answer: a) at p (-2.5, 2.5);
b) at p = 2.5 or = -2.5;
c) at p< - 2,5 или р > 2,5.
Mordkovich A. G., Algebra. 8th grade: Textbook. for general education institutions. - 3rd ed., revised. - M.: Mnemosyne, 2001. - 223 p.: ill.
Help for schoolchildren online, Mathematics for 8th grade download, calendar and thematic planning
Inequalities are called linear the left and right sides of which are linear functions with respect to the unknown quantity. These include, for example, inequalities:
2x-1-x+3; 7x0;
5 >4 - 6x 9- x< x + 5 .
1) Strict inequalities: ax +b>0 or ax+b<0
2) Non-strict inequalities: ax +b≤0 or ax+b≫ 0
Let's analyze this task. One of the sides of the parallelogram is 7 cm. What must be the length of the other side so that the perimeter of the parallelogram is greater than 44 cm?
Let the required side be X cm. In this case, the perimeter of the parallelogram will be represented by (14 + 2x) cm. The inequality 14 + 2x > 44 is a mathematical model of the problem of the perimeter of a parallelogram. If we replace the variable in this inequality X on, for example, the number 16, then we obtain the correct numerical inequality 14 + 32 > 44. In this case, they say that the number 16 is a solution to the inequality 14 + 2x > 44.
Solving the inequality name the value of a variable that turns it into a true numerical inequality.
Therefore, each of the numbers is 15.1; 20;73 act as a solution to the inequality 14 + 2x > 44, but the number 10, for example, is not its solution.
Solve inequality means to establish all its solutions or to prove that there are no solutions.
The formulation of the solution to the inequality is similar to the formulation of the root of the equation. And yet it is not customary to designate the “root of inequality.”
The properties of numerical equalities helped us solve equations. Similarly, the properties of numerical inequalities will help solve inequalities.
When solving an equation, we change it to another, more simple equation, but equivalent to the given one. The answer to inequalities is found in a similar way. When changing an equation to an equivalent equation, they use the theorem about transferring terms from one side of the equation to the opposite and about multiplying both sides of the equation by the same non-zero number. When solving an inequality, there is a significant difference between it and an equation, which lies in the fact that any solution to an equation can be verified simply by substitution into the original equation. In inequalities, this method is absent, since it is not possible to substitute countless solutions into the original inequality. Therefore, there is an important concept, these arrows<=>is a sign of equivalent, or equivalent, transformations. The transformation is called equivalent, or equivalent, if they do not change the set of solutions.
Similar rules for solving inequalities.
If we move any term from one part of the inequality to another, replacing its sign with the opposite one, we obtain an inequality equivalent to this one.
If both sides of the inequality are multiplied (divided) by the same positive number, we obtain an inequality equivalent to this one.
If both sides of the inequality are multiplied (divided) by the same negative number, replacing the inequality sign with the opposite one, we obtain an inequality equivalent to the given one.
Using these rules Let us calculate the following inequalities.
1) Let's analyze the inequality 2x - 5 > 9.
This linear inequality, we will find its solution and discuss the basic concepts.
2x - 5 > 9<=>2x>14(5 was moved to the left side with the opposite sign), then we divided everything by 2 and we have x > 7. Let us plot the set of solutions on the axis x
We have obtained a positively directed beam. We note the set of solutions either in the form of inequality x > 7, or in the form of the interval x(7; ∞). What is a particular solution to this inequality? For example, x = 10 is a particular solution to this inequality, x = 12- this is also a particular solution to this inequality.
There are many partial solutions, but our task is to find all the solutions. And there are usually countless solutions.
Let's sort it out example 2:
2) Solve inequality 4a - 11 > a + 13.
Let's solve it: A move it to one side 11 move it to the other side, we get 3a< 24, и в результате после деления обеих частей на 3 the inequality has the form a<8 .
4a - 11 > a + 13<=>3a< 24 <=>a< 8 .
We will also display the set a< 8 , but already on the axis A.
We either write the answer in the form of inequality a< 8, либо A(-∞;8), 8 does not turn on.
inequality solution in mode online solution almost any given inequality online. Mathematical inequalities online to solve mathematics. Find quickly inequality solution in mode online. The website www.site allows you to find solution almost any given algebraic, trigonometric or transcendental inequality online. When studying almost any branch of mathematics at different stages you have to decide inequalities online. To get an answer immediately, and most importantly an accurate answer, you need a resource that allows you to do this. Thanks to the site www.site solve inequality online will take a few minutes. The main advantage of www.site when solving mathematical inequalities online- this is the speed and accuracy of the response provided. The site is able to solve any algebraic inequalities online, trigonometric inequalities online, transcendental inequalities online, and inequalities with unknown parameters in mode online. Inequalities serve as a powerful mathematical apparatus solutions practical problems. With the help mathematical inequalities it is possible to express facts and relationships that may seem confusing and complex at first glance. Unknown quantities inequalities can be found by formulating the problem in mathematical language in the form inequalities And decide received task in mode online on the website www.site. Any algebraic inequality, trigonometric inequality or inequalities containing transcendental features you can easily decide online and get the exact answer. Studying natural Sciences, you inevitably face the need solutions to inequalities. In this case, the answer must be accurate and must be obtained immediately in the mode online. Therefore for solve mathematical inequalities online we recommend the site www.site, which will become your indispensable calculator for solving algebraic inequalities online, trigonometric inequalities online, and transcendental inequalities online or inequalities with unknown parameters. For practical problems of finding online solutions to various mathematical inequalities resource www.. Solving inequalities online yourself, it is useful to check the received answer using online solution inequalities on the website www.site. You need to write the inequality correctly and instantly get online solution, after which all that remains is to compare the answer with your solution to the inequality. Checking the answer will take no more than a minute, it’s enough solve inequality online and compare the answers. This will help you avoid mistakes in decision and correct the answer in time when solving inequalities online either algebraic, trigonometric, transcendental or inequality with unknown parameters.
