How to solve inequalities?

Not everyone knows how to solve inequalities, which in their structure have similar and distinctive features with equations. An equation is an exercise consisting of two parts, between which there is an equal sign, and between the parts of an inequality there can be a “more” or “less” sign. Thus, before finding a solution to a specific inequality, we must understand that it is necessary to take into account the sign of a number (positive or negative) if it becomes necessary to multiply both parts by an expression. The same fact should be taken into account if it is required to square one in order to solve inequalities, since the squaring is carried out by multiplication.

How to solve a system of inequalities

It is much more difficult to solve systems of inequalities than ordinary inequalities. How to solve class 9 inequalities, we consider specific examples. It should be understood that before solving square inequalities (systems) or any other systems of inequalities, it is necessary to solve each inequality separately, and then compare them.The solution to the inequality system will be either a positive or a negative answer (does the system have a solution or does not have a solution).

The task is to solve a set of inequalities:

Solve each inequality separately

We build a numerical line on which we depict a set of solutions


Since a set is a union of solution sets, this set on the number line must be underlined with at least one line.

Solving inequalities with a module

This example will show how to solve inequalities with a module. So, we have a definition:

We need to solve the inequality:

| x |> 2

Before solving such an inequality, it is necessary to get rid of the module (sign).

We write, based on the definition data:


Now it is necessary to solve each of the systems separately.

We construct one numerical line on which we draw sets of solutions.

As a result, we have an aggregate that unites many solutions.


The solution of quadratic inequalities

Using the numerical line we consider the example of the solution of quadratic inequalities. We have inequality:

We know that the graph of the square triple is a parabola.We also know that the branches of the parabola are directed upwards, if a> 0.

x2-3x-4 <0

Using the theorem of Vieta we find the roots of x1= - 1; x2 = 4

Let's draw a parabola, or rather, its sketch.

Thus, we found that the values ​​of the square trinomial will be less than 0 in the interval from - 1 to 4.


Many people have questions when solving double inequalities of the type g (x) <f (x) <q (x). Before you solve double inequalities, you need to decompose them into simple, and each simple inequality is solved separately. For example, having laid out our example, we obtain as a result the system of inequalities g (x) <f (x) and f (x) <q (x), which should be solved.

In fact, there are several methods for solving inequalities, so you can use the graphical method for solving complex inequalities.

The solution of fractional inequalities

More careful approach requires fractional inequalities. This is due to the fact that in the process of solving some fractional inequalities, the sign may change. Before solving fractional inequalities, it is necessary to know that the interval method is used to solve them. Fractional inequality must be presented in such a way that one side of the sign looks like a fractionally rational expression, and the second - “- 0”.Transforming the inequality in this way, we get f (x) / g (x)> (.

The solution of inequalities by the method of intervals

The interval technique is based on the method of complete induction, that is, it is necessary to go through all possible options to find the solution of inequality. This method of solution may not be necessary for students in the 8th grade, because they need to know how to solve 8th class inequalities, which are the simplest exercises. But for the higher grades, this method is irreplaceable, as it helps to solve fractional inequalities. The solution of inequalities using this technique is based on such a property of a continuous function as the preservation of the sign between the values ​​in which it turns to 0.

Construct a graph of a polynomial. This is a continuous function, acquiring the value of 0 3 times, that is, f (x) will be equal to 0 at points x1, x2and x3, the roots of a polynomial. Between these points, the sign of the function is preserved.

Since to solve the inequality f (x)> 0, we need the sign of the function, we go to the coordinate line, leaving the graph.

f (x)> 0 for x (x1; x2) and with x (x3; )

f (x) x (-; x1) and with x (x2; x3)

The graph clearly shows the solutions of the inequalities f (x) f (x)> 0 (the blue solution for the first inequality, and the red for the second).To determine To determine the sign of a function on an interval, it is enough that you know the sign of the function at one of the points. This method allows you to quickly solve inequalities, in which the left side is factorized, because in such inequalities it is sufficient to simply find the roots.

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