B) Number line
Consider the number line (Fig. 6):
Consider the set of rational numbers
Each rational number is represented by a certain point on the number axis. So, the numbers are marked in the figure.
Let's prove that .
Proof. Let there be a fraction: . We have the right to consider this fraction irreducible. Since , then - the number is even: - odd. Substituting its expression, we find: , which implies that is an even number. We have obtained a contradiction that proves the statement.
So, not all points on the number axis represent rational numbers. Those points that do not represent rational numbers represent numbers called irrational.
Any number of the form , , is either an integer or an irrational number.
Numeric intervals
Numerical segments, intervals, half-intervals and rays are called numerical intervals.
| Inequality specifying a numerical interval | Designation of a numerical interval | Name of the number interval | It reads like this: |
| a ≤ x ≤ b | [a; b] | Numerical segment | Segment from a to b |
| a< x < b | (a; b) | Interval | Interval from a to b |
| a ≤ x< b | [a; b) | Half-interval | Half-interval from a before b, including a. |
| a< x ≤ b | (a; b] | Half-interval | Half-interval from a before b, including b. |
| x ≥ a | [a; +∞) | Number beam | Number beam from a up to plus infinity |
| x>a | (a; +∞) | Open number beam | Open numerical beam from a up to plus infinity |
| x ≤ a | (- ∞; a] | Number beam | Number ray from minus infinity to a |
| x< a | (- ∞; a) | Open number beam | Open number ray from minus infinity to a |
Let us represent the numbers on the coordinate line a And b, as well as the number x between them.
The set of all numbers that meet the condition a ≤ x ≤ b, called numerical segment or just a segment. It is designated as follows: [ a; b] - It reads like this: a segment from a to b.
The set of numbers that meet the condition a< x < b , called interval. It is designated as follows: ( a; b)
It reads like this: interval from a to b.
Sets of numbers satisfying the conditions a ≤ x< b или a<x ≤ b, are called half-intervals. Designations:
Set a ≤ x< b обозначается так:[a; b), reads like this: half-interval from a before b, including a.
A bunch of a<x ≤ b is indicated as follows:( a; b], reads like this: half-interval from a before b, including b.
Now let's imagine Ray with a dot a, to the right and left of which there is a set of numbers.
a, meeting the condition x ≥ a, called numerical beam.
It is designated as follows: [ a; +∞)-Reads like this: a numerical ray from a to plus infinity.
Set of numbers to the right of a point a, corresponding to the inequality x>a, called open number beam.
It is designated as follows: ( a; +∞)-Reads like this: an open numerical ray from a to plus infinity.
a, meeting the condition x ≤ a, called numerical ray from minus infinity toa .
It is designated as follows:( - ∞; a]-Reads like this: a numerical ray from minus infinity to a.
Set of numbers to the left of the point a, corresponding to the inequality x< a , called open number ray from minus infinity toa .
It is designated as follows: ( - ∞; a)-Reads like this: an open number ray from minus infinity to a.
The set of real numbers is represented by the entire coordinate line. He is called number line. It is designated as follows: ( - ∞; + ∞ )
3) Linear equations and inequalities with one variable, their solutions:
An equation containing a variable is called an equation with one variable, or an equation with one unknown. For example, an equation with one variable is 3(2x+7)=4x-1.
The root or solution of an equation is the value of a variable at which the equation becomes a true numerical equality. For example, the number 1 is a solution to the equation 2x+5=8x-1. The equation x2+1=0 has no solution, because the left side of the equation is always greater than zero. The equation (x+3)(x-4) =0 has two roots: x1= -3, x2=4.
Solving an equation means finding all its roots or proving that there are no roots.
Equations are called equivalent if all the roots of the first equation are roots of the second equation and vice versa, all the roots of the second equation are roots of the first equation or if both equations have no roots. For example, the equations x-8=2 and x+10=20 are equivalent, because the root of the first equation x=10 is also the root of the second equation, and both equations have the same root.
When solving equations, the following properties are used:
If you move a term in an equation from one part to another, changing its sign, you will get an equation equivalent to the given one.
If both sides of an equation are multiplied or divided by the same non-zero number, you get an equation equivalent to the given one.
The equation ax=b, where x is a variable and a and b are some numbers, is called a linear equation with one variable.
If a¹0, then the equation has a unique solution.
If a=0, b=0, then the equation is satisfied by any value of x.
If a=0, b¹0, then the equation has no solutions, because 0x=b is not executed for any value of the variable.
Example 1. Solve the equation: -8(11-2x)+40=3(5x-4)
Let's open the brackets on both sides of the equation, move all terms with x to the left side of the equation, and terms that do not contain x to the right side, we get:
16x-15x=88-40-12
Example 2. Solve the equations:
x3-2x2-98x+18=0;
These equations are not linear, but we will show how such equations can be solved.
3x2-5x=0; x(3x-5)=0. The product is equal to zero, if one of the factors is equal to zero, we get x1=0; x2= .
Answer: 0; .
Factor the left side of the equation:
x2(x-2)-9(x-2)=(x-2)(x2-9)=(x-2)(x-3)(x-3), i.e. (x-2)(x-3)(x+3)=0. This shows that the solutions to this equation are the numbers x1=2, x2=3, x3=-3.
c) Imagine 7x as 3x+4x, then we have: x2+3x+4x+12=0, x(x+3)+4(x+3)=0, (x+3)(x+4)= 0, hence x1=-3, x2=- 4.
Answer: -3; - 4.
Example 3. Solve the equation: ½x+1ç+½x-1ç=3.
Let us recall the definition of the modulus of a number: 
For example: ½3½=3, ½0½=0, ½- 4½= 4.
In this equation, under the modulus sign are the numbers x-1 and x+1. If x is less than –1, then the number x+1 is negative, then ½x+1½=-x-1. And if x>-1, then ½x+1½=x+1. At x=-1 ½x+1½=0.
Thus, 
Likewise 
a) Consider this equation½x+1½+½x-1½=3 for x £-1, it is equivalent to the equation -x-1-x+1=3, -2x=3, x=, this number belongs to the set x £-1.
b) Let -1< х £ 1, тогда данное уравнение равносильно уравнению х+1-х+1=3, 2¹3 уравнение не имеет решения на данном множестве.
c) Consider the case x>1.
x+1+x-1=3, 2x=3, x= . This number belongs to the set x>1.
Answer: x1=-1.5; x2=1.5.
Example 4. Solve the equation:½x+2½+3½x½=2½x-1½.
Let us show a short record of the solution to the equation, revealing the sign of the modulus “over intervals”.
x £-2, -(x+2)-3x=-2(x-1), - 4x=4, x=-2О(-¥; -2]
–2<х£0, х+2-3х=-2(х-1), 0=0, хÎ(-2; 0]
0<х£1, х+2+3х=-2(х-1), 6х=0, х=0Ï(0; 1]
x>1, x+2+3x=2(x-1), 2x=- 4, x=-2П(1; +¥)
Answer: [-2; 0]
Example 5. Solve the equation: (a-1)(a+1)x=(a-1)(a+2), for all values of parameter a.
There are actually two variables in this equation, but consider x to be the unknown and a to be the parameter. It is required to solve the equation for the variable x for any value of the parameter a.
If a=1, then the equation has the form 0×x=0; any number satisfies this equation.
If a=-1, then the equation looks like 0×x=-2; not a single number satisfies this equation.
If a¹1, a¹-1, then the equation has a unique solution.
Answer: if a=1, then x is any number;
if a=-1, then there are no solutions;
if a¹±1, then .
B) Linear inequalities with one variable.
If the variable x is given any numerical value, then we get a numerical inequality expressing either a true or false statement. Let, for example, the inequality 5x-1>3x+2 be given. For x=2 we get 5·2-1>3·2+2 – a true statement (true numerical statement); at x=0 we get 5·0-1>3·0+2 – a false statement. Any value of a variable at which a given inequality with a variable turns into a true numerical inequality is called a solution to the inequality. Solving an inequality with a variable means finding the set of all its solutions.
Two inequalities with the same variable x are said to be equivalent if the sets of solutions to these inequalities coincide.
The main idea of solving the inequality is as follows: we replace the given inequality with another, simpler, but equivalent to the given one; we again replace the resulting inequality with a simpler inequality equivalent to it, etc.
Such replacements are made on the basis of the following statements.
Theorem 1. If any term of an inequality with one variable is transferred from one part of the inequality to another with the opposite sign, while leaving the sign of the inequality unchanged, then an inequality equivalent to the given one will be obtained.
Theorem 2. If both sides of an inequality with one variable are multiplied or divided by the same positive number, leaving the sign of the inequality unchanged, then an inequality equivalent to the given one will be obtained.
Theorem 3. If both sides of an inequality with one variable are multiplied or divided by the same negative number, while changing the sign of the inequality to the opposite, then an inequality equivalent to the given one will be obtained.
An inequality of the form ax+b>0 is called linear (respectively, ax+b<0, ax+b³0, ax+b£0), где а и b – действительные числа, причем а¹0. Решение этих неравенств основано на трех теоремах равносильности изложенных выше.
Example 1. Solve the inequality: 2(x-3)+5(1-x)³3(2x-5).
Opening the brackets, we get 2x-6+5-5x³6x-15,
Numerical intervals include rays, segments, intervals and half-intervals.
Types of numerical intervals
| Name | Image | Inequality | Designation |
|---|---|---|---|
| Open beam | x > a | (a; +∞) | |
| x < a | (-∞; a) | ||
| Closed beam | x ⩾ a | [a; +∞) | |
| x ⩽ a | (-∞; a] | ||
| Line segment | a ⩽ x ⩽ b | [a; b] | |
| Interval | a < x < b | (a; b) | |
| Half-interval | a < x ⩽ b | (a; b] | |
| a ⩽ x < b | [a; b) |
In the table a And b are boundary points, and x- a variable that can take the coordinate of any point belonging to a numerical interval.
Boundary point- this is the point that defines the boundary of the numerical interval. A boundary point may or may not belong to a numerical interval. In the drawings, boundary points that do not belong to the numerical interval under consideration are indicated by an open circle, and those that belong to them are indicated by a filled circle.
Open and closed beam
Open beam is a set of points on a line lying on one side of a boundary point that is not included in this set. The ray is called open precisely because of the boundary point that does not belong to it.
Let's consider a set of points on the coordinate line that have a coordinate greater than 2, and therefore located to the right of point 2:
Such a set can be defined by the inequality x> 2. Open rays are denoted using parentheses - (2; +∞), this entry reads like this: open numeric ray from two to plus infinity.
The set to which the inequality corresponds x < 2, можно обозначить (-∞; 2) или изобразить в виде луча, все точки которого лежат с левой стороны от точки 2:
Closed beam is a set of points on a line lying on one side of a boundary point belonging to a given set. In the drawings, boundary points belonging to the set under consideration are indicated by a filled circle.
Closed number rays are defined by non-strict inequalities. For example, inequalities x 2 and x 2 can be depicted like this:
These closed rays are designated as follows: , it is read like this: a numerical ray from two to plus infinity and a numerical ray from minus infinity to two. The square bracket in the notation indicates that point 2 belongs to the numerical interval.
Line segment
Line segment is the set of points on a line lying between two boundary points belonging to a given set. Such sets are defined by double non-strict inequalities.
Consider a segment of a coordinate line with ends at points -2 and 3:
The set of points that make up a given segment can be specified by the double inequality -2 x 3 or designate [-2; 3], such a record reads like this: a segment from minus two to three.
Interval and half-interval
Interval- this is the set of points on a line lying between two boundary points that do not belong to this set. Such sets are defined by double strict inequalities.
Consider a segment of a coordinate line with ends at points -2 and 3:
The set of points that make up a given interval can be specified by the double inequality -2< x < 3 или обозначить (-2; 3), такая запись читается так: интервал от минус двух до трёх.
Half-interval is the set of points on a line lying between two boundary points, one of which belongs to the set and the other does not. Such sets are defined by double inequalities:
These half-intervals are designated as follows: (-2; 3] and [-2; 3), it is read like this: half-interval from minus two to three, including 3, and half-interval from minus two to three, including minus two.
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Slide captions:
7th grade Number intervals Math teacher: Bakhvalova G.S. Gymnasium No. 52
Lesson objectives: 1.Introduce the concept of a numerical interval; 2. Instill the skills of depicting numerical intervals on a number line and the ability to designate them. 3.Develop logical thinking: analyze, compare. Lesson plan: 1. Updating knowledge: “Coordinate axis.” 2. New topic: “Numerical intervals.” 3. Educational independent work. 4. Lesson summary.
Complete the task: 1. Mark on the number line the points with coordinates: A(-2); AT 5); O(0); C(5); D (-3).
Answer: 1. A(-2); AT 5); O(0); C(3); D(-3). 0 A B C 1 0 D
Complete the task: 2. Compare the numbers: -2 and 5; 5 and 0; -2 and –3; 5 and 3; 0 and –2.
Answer: -2 0; -2 > –3; 5 > 3; 0 > –2. check yourself
Complete the task orally: 3.Which of the given numbers on the number line is to the left: -2 or 5; 5 or 0; -2 or –3; 5 or 3; 0 or –2. CONCLUSION: of two numbers on the number line, the smaller number is located to the left, and the larger number is located to the right.
Let us mark points on the coordinate line with coordinates – 3 and 2. If the point is located between them, then it corresponds to a number that is greater than –3 and less than 2. The reverse is also true: if the number x satisfies the condition - 3Slide 9
The set of all numbers satisfying the condition 3Slide 10
A number x that satisfies the condition -3 ≤x≤ 2 is represented by a point that either lies between the points with coordinates –3 and 2, or coincides with one of them. A set of such numbers is denoted [-3;2]. - 3 2 Write it down in your notebook Write it down in your notebook Write it down in your notebook
A number x that satisfies the condition x≤ 2 is represented by a point that either lies to the left of the point with coordinate 2 or coincides with it. The set of such numbers is denoted by (-∞;2]. 2 Write it down in your notebook Write it down in your notebook Write it down in your notebook
A number x that satisfies the condition x > -3 is represented by a point that either lies to the right of the point with coordinate -3. The set of such numbers denotes (-3; +∞). - 3 Write it down in your notebook Write it down in your notebook Write it down in your notebook
3 5 3 5 3 5 3 5 5 -7 3
Independent work OPTION 1 OPTION 4 OPTION 2 OPTION 3 CHOOSE AN OPTION Help me! And to me, and to me. Choose me! You'll help me, won't you?
OPTION 1 1.Draw numerical intervals on the coordinate line: a). ; b). (-2; + ∞); V). [ 3;5) ; g).(- ∞ ;5 ]. 2. Write down the numerical interval shown in the figure: 3. Which of the numbers -1.6; -1.5; -1; 0; 3; 5.1; 6.5 belong between: a). [-1.5;6.5]; b).(3; + ∞); V). (- ∞ ;1]. 3 7 -5 6 -7 c). A). b). 4. Indicate the largest integer belonging to the interval: a). [-12;-9]; b). (-1;17). THANK YOU!
OPTION 2 1.Draw numerical intervals on the coordinate line: a). [ - 3; 0) ; b). [ - 3 ; + ∞); V). (- thirty) ; g).(- ∞ ; 0) . 2. Write down the numerical interval shown in the figure: 3. Which of the numbers are 2, 2; - 2, 1; -1; 0; 0.5; 1; 8, 9 belong to the interval: a). (- 2 , 2 ; 8 , 9 ]; b).(- ∞ ;0 ] ; c). (1 ;+ ∞) . -5 6 3 7 c). A). b). 4. Indicate the largest integer belonging to the interval: a). [-12;-9) ; b). [ -1;17 ] . 2 Help me!
OPTION 3 1.Draw numerical intervals on the coordinate line: a). (-0.44;5) ; b). (10 ; + ∞); V). [ 0 ; 13) ; d).(- ∞ ; -0.44 ]. 2. Write down the numerical interval shown in the figure: 3. Name all the integers belonging to the interval: a). [- 3 ; 1 ]; b).(- 3; 1); at 3 ; 1) ; G). (- 3 ; 1 ]; . 7 20 -8 6 -7 c). A). b). 4. Indicate the smallest integer belonging to the interval: a). [-12;-9]; b). (-1;17 ] . Thank you, I’m very happy!
OPTION 4 1. Draw numerical intervals on the coordinate line: a). [ -4 ; -0.29 ]; b). (- ∞ ;+ ∞); V). [1.7;5.9); g).(0.01;+ ∞) . 2. Write down the numerical interval shown in the figure: 3. Name all the integers belonging to the interval: a). [- 4 ; 3 ]; b).(-4; 3); at 4 ; 3) ; G). (- 4 ; 3 ]; . -4 -1 -5 25 in). A). b). 4. Indicate the smallest integer belonging to the interval: a). [-12;-9) ; b). (-1;17]. -8 Well done!
Calling the test program If you still have free minutes, call the test program by clicking on the word “CALL” Homework You can solve another OPTION
Homework 1). Draw two number intervals on the same coordinate line such that they have common points (2 examples). 2). Draw two numerical intervals on the same coordinate line such that they do not have common points (2 examples). Shutdown
THANK YOU FOR YOUR WORK!!!
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Basic tutorial. Algebra 8th grade: textbook for educational institutions./ Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov, S.B. Suvorov; edited by S.A. Telyakovsky. – 15th ed., revised. – M.: Education, 2007. ISBN 978-5-09-015964-7.
Didactic purpose of the lesson: creating conditions for conscious learning of new material and incorporating students’ knowledge into the learning process.
Lesson objectives:
- Educational:
- introduce the concept of a numerical interval;
- develop the ability to work with numerical intervals;
- depict on a coordinate line an interval and a set of numbers that satisfy the inequality;
- instill graphic culture skills.
- Educational:
- nurturing interest in mathematics through the use and application of ICT;
- creating conditions for the formation of communication skills.
- Developmental:
- improvement of mental activity: analysis, synthesis, classification;
- development of the ability to independently solve educational problems, development of students’ curiosity, cognitive interest in the subject;
Lesson objectives:
- Know:
- concepts: numerical interval, numerical ray, open numerical ray;
- designation of numerical intervals, their names.
- Be able to:
- depict numerical intervals on a coordinate line;
- write number intervals in mathematical language.
- Learn to do self-analysis of the lesson.
Children's acquired skills:
- the ability to analyze, compare, contrast, and draw appropriate conclusions;
- development of logical thinking, memory, speech, spatial imagination;
- increasing the level of perception, comprehension and memorization;
- fostering an attentive attitude towards others, towards each other, academic discipline;
- the ability to summarize your work, analyze your activities;
Lesson type: lesson of learning new material and primary consolidation.
Forms of organizing children's work: individual, frontal, steam room.
Forms of organizing the work of a teacher:
- the verbal-illustrative method, reproductive method, practical method, problem method, conversation-message are used;
- checking previously studied material, organizing the perception of new information;
- setting the lesson goal for students;
- generalization of what is studied in the lesson and its introduction into the system of previously acquired knowledge.
Equipment: computer, multimedia projector, screen, PC, ruler, pencil, set of colored pencils, Presentation.
Lesson structure and flow:
Lesson steps |
Teacher activities |
Student activity |
| Organizational moment (1 min.) | The teacher checks readiness for the lesson | Students determine readiness for the lesson |
| Checking homework and updating knowledge. (1 min.) | Checking your homework. A word from the consultants. (there are responsible students on each row who check their homework before the start of the lesson). |
They open their notebooks. Report on the completion of homework by students. (If there is no homework, students are given consultation after class) |
| Mental arithmetic (6 min.) Slides 2, 3, 4, 5. |
1. Add the inequalities term by term:
2. Multiply term by term:
3. Read the inequality and name several values of the variable that satisfy this inequality:
4. Between what integers is the number enclosed? |
Student answers:
Students read and name the values of the variable X that satisfy the given inequality. Name the integers between which the number is enclosed. |
| Goal setting (2 min.) Slide 6. |
Today in the lesson we must learn to depict inequalities in the form of intervals and write them down with notations. We will need a ruler, pencil and colored pencils if anyone has them. | Preparing tools |
| Learning new material. (10 min.) Slide 7 Slides 8, 9 Slides 10, 11 |
Studying new material is accompanied by a presentation 1. Introducing the concept of a numerical interval. |
Listen to the teacher's explanation and make notes in their workbooks. |
| Physical exercise (1 min.) | It's time to do some gymnastics to give your head and body a rest from work! 1. Stretch your arms in front of you and twist your hands in one direction or the other. Do it 3 times. 2. Press your fingers against each other, press, and then press again and hold your fingers in this state for 5-7 seconds. 3. Turn your head, 3 times in one direction, three times in the other. 4. Cover your eye with your hand, twist the body in one direction, and then in the other. Do it 3 times. |
Comply with the specified instructions on site. The class attendant conducts physical exercises |
| Students mastering new information (5 min.) | Working with information from the textbook Page 173, table. |
Remember the designation and name of numerical intervals. |
| Primary consolidation of knowledge (14 min.) | 1. No. 812 (a, b, f, g); 2. №815; 3. №816; 4. No. 825 (a, b); 5. No. 827 (a, b). |
At the board and in notebooks. |
| Control and testing of knowledge (2 min.) | №813 | One student is at the board, the rest check the correctness of his answer and the recording of the number interval. |
| Reflection (1 min.) | Guys, please answer the following questions: – What was the most interesting thing in the lesson? |
Answers from the spot |
| Lesson summary (1 min.) | So, let's summarize the lesson. Guys, please answer the question: – What new number intervals did you learn today? |
Answer the question: Open beam, closed beam, Line segment, Interval, Half-interval. |
| Homework (2 min.) | paragraph 33, page 173, know the designation and name of numerical intervals. No. 814, No. 816 (c, d), No. 825 (c). |
Get acquainted with homework, write it down in a diary |