"Inequalities with one variable" - You can't stop learning. Specify the largest integer that belongs to the interval. We learn from examples. The solution of an inequality with one variable is the value of the variable. Linear inequality. Find the mistake. Inequalities. Lesson goals. To solve an inequality means to find all its solutions. Historical reference.
"Algorithm for solving inequalities" - Function. Task. Happening. Lots of solutions. Solution of inequalities. Inequalities. Solution of inequality. Consider the discriminant. Let's solve the inequality by the interval method. Protozoa linear inequality. Algorithm for solving inequalities. Axis. Now let's solve the quadratic inequality.
"Logarithmic Equations and Inequalities" - Find out if a number is positive or negative. The purpose of the lesson. Solve the equation. Properties of logarithms. Logarithms. Formulas for transition to a new basis. Practicing skills in solving logarithmic equations and inequalities. Definition of a logarithm. Calculate. Specify the course of 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, which clearly proves that our assumption is wrong. The method is based on the property of non-negativity square trinomial, if and. Cauchy-Bunyakovsky inequality.
"Solving inequalities by the method of intervals" - Solving inequalities by the method of intervals. 2. Algorithm for solving the inequality by the interval method. The graph of the function is given: Solve the inequality:
"Solution of irrational equations and inequalities" - Extraneous roots. A set of tasks. Enter a multiplier under the root sign. Working with a task. Irrational equations and inequalities. Knowledge update. Irrational equation. Definition. Choose the ones that are irrational. Irrational equations. For what values of A is the equality true? Irrational inequalities.
Ways to solve systems of equations
To begin with, let us briefly recall what methods of solving systems of equations generally exist.
Exist four main ways solutions of systems of equations:
Substitution method: take any of these equations and express $y$ in terms of $x$, then $y$ is substituted into the equation of the system, from where the variable $x.$ is found. After that, we can easily calculate the variable $y.$
Addition method: in this method it is necessary to multiply one or both equations by such numbers that when both are added together, one of the variables "disappears".
Graphical method: both equations of the system are displayed on the coordinate plane and the point of their intersection is found.
The method of introducing new variables: in this method, we do the replacement of some expressions to simplify the system, and then apply one of the above methods.
Systems of exponential equations
Definition 1
Systems of equations consisting of exponential equations, are called a system of exponential equations.
We will consider the solution of systems of exponential equations using examples.
Example 1
Solve a 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.
Substitute $y$ into the second equation:
\ \ \[-2-x=2\] \ \
Answer: $(-4,6)$.
Example 2
Solve a system of equations
Figure 3
Solution.
This system is equivalent to the system
Figure 4
We apply the fourth method for solving equations. Let $2^x=u\ (u >0)$ and $3^y=v\ (v >0)$, we get:
Figure 5
We solve the resulting system by the addition method. Let's add the equations:
\ \
Then from the second equation, we get that
Returning to the replacement, got new system 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 a system of exponential inequalities.
We will consider the solution of 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 equivalence theorem for exponential inequalities:
Theorem 1. The inequality $a^(f(x)) >a^(\varphi (x)) $, where $a >0,a\ne 1$ is equivalent to the set of two systems
\ \ \
Answer: $(-4,6)$.
Example 2
Solve a system of equations
Figure 3
Solution.
This system is equivalent to the system
Figure 4
We apply the fourth method for solving equations. Let $2^x=u\ (u >0)$ and $3^y=v\ (v >0)$, we get:
Figure 5
We solve the resulting system by the addition method. Let's add 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 a system of exponential inequalities.
We will consider the solution of 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 equivalence theorem for exponential inequalities:
Theorem 1. The inequality $a^(f(x)) >a^(\varphi (x)) $, where $a >0,a\ne 1$ is equivalent to the set of two systems
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