GBOU Secondary School No. 149, St. Petersburg

Lesson summary

Novikova Olga Nikolaevna

2016

Topic: "System of exponential equations and inequalities."

Lesson objectives:

educational:

generalize and consolidate knowledge about ways to solve exponential equations and inequalities contained in systems of equations and inequalities

developing: activation of cognitive activity; development of skills of self-control and self-esteem, self-analysis of one’s activities.

educational: developing the ability to work independently; make decisions and draw conclusions; nurturing aspiration for self-education and self-improvement.

Lesson type : combined.

Lesson type: workshop lesson.

During the classes

I. Organizational moment (1 minute)

Statement of the goal for the class: Generalize and consolidate knowledge about methods of solving exponential equations and inequalities contained in systems of equations and inequalities based on the properties of the exponential function.

II. Oral work (1 minutes)

Definition of an exponential equation.
Methods for solving exponential equations.
Algorithm for solving exponential inequalities.

III . Checking homework (3 min)

Students are in their places. The teacher checks answers and asks how to solve exponential equations and inequalities. No. 228-231(odd)

IV. Updating basic knowledge. "Brainstorm": (3 min)

Questions are shown on printed sheets on students’ desks “Exponential functions, equations, inequalities” and are offered to students for oral answers from their seats.

1. What function is called exponential?

2. What is the domain of the function y= 0,5x?

3. What is the domain of definition of the exponential function?

4. What is the range of the function y= 0,5x?

5. What properties can a function have?

6. Under what condition is the exponential function increasing?

7. Under what condition is the exponential function decreasing?

8. The exponential function increases or decreases

9. Which equation is called exponential?

Diagnostics of the level of formation of practical skills.

10 task: write down the solution in your notebooks. (7 min)

10. Knowing the properties of an increasing and decreasing exponential function, solve the inequalities

2 3 < 2 X ;
; 3 X < 81 ; 3 X < 3 4

11 . Solve the equation: 3 x = 1

12 . Calculate 7.8 0 ; 9.8 0

13 . Indicate a method for solving exponential equations and solve it:

After completion, the pairs exchange leaves. Evaluating each other. Criteria on the board. Checking against entries on sheets in the file.

Thus, we repeated the properties of the exponential function and methods for solving exponential equations.

The teacher selectively takes and evaluates the work of 2-3 students.

Solution workshop systems exponential equations and inequalities: (23 min)

Let's consider solving systems of exponential equations and inequalities based on the properties of the exponential function.

When solving systems of exponential equations and inequalities, the same techniques are used as when solving systems of algebraic equations and inequalities (substitution method, addition method, method of introducing new variables). In many cases, before applying one or another solution method, it is necessary to transform each equation (inequality) of the system to the simplest possible form.

Examples.

1.

Solution:

Answer: (-7; 3); (1; -1).

2.

Solution:

Let's denote 2 X= u, 3 y= v. Then the system will be written like this:

Let's solve this system using the substitution method:

Equation 2 X= -2 has no solutions, because –2<0, а 2 X> 0.

b)

Answer: (2;1).

№244(1)

Answer: 1.5; 2

Summarizing. Reflection. (5 minutes)

Lesson summary: Today we repeated and generalized the knowledge of methods for solving exponential equations and inequalities contained in systems, based on the properties of the exponential function.

The children, one by one, are asked to select and continue the phrase from the phrases presented below.

Reflection:

today I found out...

it was difficult…

I understand that…

I taught myself...

I could)…

It was interesting to know that...

I was surprised...

I wanted…

Homework. (2 minutes)

No. 240-242 (odd) p.86

“Inequalities with one variable” - You can’t stop learning. Specify the largest integer that belongs to the interval. We learn from examples. The solution to an inequality in one variable is the value of the variable. Linear inequality. Find the mistake. Inequalities. Lesson objectives. Solving an inequality means finding all its solutions. Historical reference.

“Algorithm for solving inequalities” - Function. Task. Happening. Lots of solutions. Solving inequalities. Inequalities. Solution of inequality. Let's consider the discriminant. Let's solve the inequality using the interval method. The simplest linear inequality. Algorithm for solving inequalities. Axis. Now let's solve the quadratic inequality.

“Logarithmic Equations and Inequalities” - Find out whether a number is positive or negative. The purpose of the lesson. Solve the equation. Properties of logarithms. Logarithms. Formulas for transition to a new base. Practicing skills in solving logarithmic equations and inequalities. Definition of logarithm. Calculate. Indicate the process for solving the following equations.

“Proof of inequalities” - Application of the method of mathematical induction. For n=3 we get. Prove that for any n? N Proof. by Bernoulli's theorem, as required. But this clearly proves that our assumption is incorrect. The method is based on the property of non-negativity of a quadratic trinomial, if and. Cauchy-Bunyakovsky inequality.

“Solving inequalities by the interval method” - Solving inequalities by the interval method. 2. Algorithm for solving inequality using the interval method. Given the graph of the function: Solve the inequality:

“Solving irrational equations and inequalities” - Extraneous roots. A set of tasks. Enter the multiplier under the root sign. Working with a task. Irrational equations and inequalities. Updating knowledge. Irrational equation. Definition. Choose those that are irrational. Irrational equations. For what values ​​of A is the equality true. Irrational inequalities.

Methods for solving systems of equations

To begin with, let us briefly recall what methods generally exist for solving systems of equations.

Exist four main ways solutions to systems of equations:

Substitution method: take any of the given equations and express $y$ in terms of $x$, then $y$ is substituted into the system equation, from where the variable $x.$ is found. After this, we can easily calculate the variable $y.$

Addition method: In this method, you need to multiply one or both equations by such numbers that when you add both together, one of the variables “disappears.”

Graphical method: both equations of the system are depicted on the coordinate plane and the point of their intersection is found.

Method of introducing new variables: in this method we replace some expressions to simplify the system, and then use one of the above methods.

Systems of exponential equations

Definition 1

Systems of equations consisting of exponential equations are called systems of exponential equations.

We will consider solving systems of exponential equations using examples.

Example 1

Solve system of equations

Picture 1.

Solution.

We will use the first method to solve this system. First, let's express $y$ in the first equation in terms of $x$.

Figure 2.

Let's substitute $y$ into the second equation:

\ \ \[-2-x=2\] \ \

Answer: $(-4,6)$.

Example 2

Solve system of equations

Figure 3.

Solution.

This system is equivalent to the system

Figure 4.

Let us apply the fourth method of solving equations. Let $2^x=u\ (u >0)$, and $3^y=v\ (v >0)$, we get:

Figure 5.

Let us solve the resulting system using the addition method. Let's add up the equations:

\ \

Then from the second equation, we get that

Returning to the replacement, I received a new system of exponential equations:

Figure 6.

We get:

Figure 7.

Answer: $(0,1)$.

Systems of exponential inequalities

Definition 2

Systems of inequalities consisting of exponential equations are called systems of exponential inequalities.

We will consider solving systems of exponential inequalities using examples.

Example 3

Solve the system of inequalities

Figure 8.

Solution:

This system of inequalities is equivalent to the system

Figure 9.

To solve the first inequality, recall the following theorem on the equivalence of exponential inequalities:

Theorem 1. The inequality $a^(f(x)) >a^(\varphi (x)) $, where $a >0,a\ne 1$ is equivalent to the collection of two systems

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