Presentation "solving systems of inequalities with one variable." Presentation for the lesson "Inequalities. Solving systems of inequalities" presentation for an algebra lesson (8th grade) on the topic Lesson presentation of inequalities systems of inequalities
Solving Linear Inequalities
8th grade
10? 2) Is the number -6 a solution to the 4x12 inequality? 3) Is the inequality 5x-154x+14 strict? 4) Is there an integer that belongs to the interval [-2.8;-2.6]? 5) For any value of the variable a, is the inequality a² +4 o true? 6) Is it true that when both sides of an inequality are multiplied or divided by a negative number, the sign of the inequality does not change?" width="640" Test. (yes - 1, no - 0)
1 ) Is the number 12 a solution to the 2x10 inequality?
2) Is the number -6 a solution to the 4x12 inequality?
3) Is the inequality 5x-154x+14 strict?
4) Is there an integer that belongs to the interval [-2.8;-2.6]?
5) For any value of the variable a, is the inequality a² +4 o true?
6) Is it true that when both sides of an inequality are multiplied or divided by a negative number, the sign of the inequality does not change?
Solve linear inequality:
3x – 5 ≥ 7x - 15
3x – 7x ≥ -15 + 5
-4x ≥ -10
x ≤ 2.5
Answer: (-∞; 2.5].
- Move the terms, changing the signs of the terms
2. Give similar terms on the left and right sides of the inequality.
3. Divide both sides by -4, remembering to change the inequality sign.
50x 62x+31-12x 50x 50x-50x -31 0*x -31 Answer: x 0 No. 2. 3(7-4y) 3y-7 21 -12y 3y-7 -12y + 3y -7-21 -9y - 28 y Answer: (3 1/9 ;+ ∞)" width="640" Find the error in solving inequalities. Explain why the mistake was made. Write down the correct solution in your notebook.
№ 1.
31(2x+1)-12x 50x
62x+31-12x 50x
50x-50x -31
Answer: x 0
№ 2.
3(7-4y) 3y-7
21 -12y 3y-7
-12y + 3y -7-21
-9y - 28
Answer: (3 1/9 ;+ ∞)
Indicate the letter of the correct answer
Restore the solution to the inequality
To use presentation previews, create a Google account and log in to it: https://accounts.google.com
Slide captions:
Algebra 8th grade General lesson “Inequalities. Solving systems of inequalities with one variable.” x -3 x 1
Lesson objectives: 1. Educational: Repeat and generalize students’ knowledge on the topic “Inequalities with one variable and their systems” Continue developing skills to work using algorithm 2. Developmental: Develop the ability to highlight the main thing; generalize existing knowledge, expand the understanding of the scope of application of knowledge on the topic, continue the formation of control and self-control skills 3. Educational: Foster mental activity, independence
Test questions 1. How are numerical intervals designated on a number line? Name them. 2. What is called the solution to an inequality? Is the solution to the inequality 3 x – 11 >1 the number 5, the number 2? What does it mean to solve inequality? 3. How to find the intersection of two sets of numbers? union of two sets? 4. What is called the solution to a system of inequalities? Is the number 3 a solution to the system of inequalities? number 5? What does it mean to solve a system of inequalities?
Instead of asterisks, insert the signs “⋂” and “∪” 1) 1. [ -2; 3) (1; 5] = [ -2; 5] 2. [-2; 3) (1; 5] = (1; 3) 2) 1. = [ 3; 5] 2. = 3) 1 . [-2; 3] = 2 . [-2; 3] = [-2; 6 ] 4) 1. [-2; 1) (3; 5] = 2 . [-2; 1) (3; 5] = [-2; 1) ∪ (3; 5]
Instead of asterisks, insert the signs “⋂” and “∪” 1) 1. [ -2; 3) ∪ (1; 5] = [ -2; 5] 2. [-2; 3) ⋂ (1; 5] = (1; 3) 2) 1. ⋂ [ 3; 7 ] = [ 3; 5] 2. ∪ [ 3; 7] = 3) 1 . [-2; 3] ⋂ [ 1; 6] = 2 . [-2; 3] ∪ = [-2; 6 ] 4) 1. [-2; 1) (3; 5] = 2 . [-2; 1) ∪ (3; 5] = [-2; 1) ∪ (3; 5]
Matrix test 1 (a;c) 2 [a;c] 3 (a;+ ) 4 (– ; a ] 5 [a;c) 6 (a;c ] 7 [a; + ) 8 (– ;a) a≤ x≤ b x ≥ a x a a≤ x
Matrix test 1 (a;c) 2 [a;c] 3 (a;+ ) 4 (– ; a ] 5 [a;c) 6 (a;c ] 7 [a; + ) 8 (– ;a) a≤ x≤ b + x ≥ a + x a + a≤ x
Establish a correspondence between inequality and numerical interval Inequality Numerical interval 1 x ≥ 12 1. (– ; – 0.3) 2 – 4
Answers: 13; 24; 31; 46; 52; 65.
Find the mistake in solving the inequality and explain why the mistake was made “Mathematics teaches you to overcome difficulties and correct your own mistakes”
Solving systems of inequalities with one variable Solving a system of inequalities means finding all its solutions or proving that there are no solutions. The solution to a system of inequalities with one variable is the value of the variable for which each of the inequalities of the system is true
x > 210:7, x ≤ 40 0:5; 7x > 210, 5x ≤ 40 0; x > 30, x ≤ 80. x 30 80 Answer: (30;80 ] We solve the system of inequalities.
Solve every inequality in the system. 2. Graphically depict the solutions to each inequality on the coordinate line. 3. Find the intersection of solutions to inequalities on the coordinate line. 4. Write the answer as a number interval. Algorithm for solving systems of inequalities with one variable
We solve the system of inequalities. -2 Answer: there are no solutions 3 x To solve a system of inequalities means to find all its solutions or prove that there are no solutions.
Preparation for the OGE 1. What system of inequalities corresponds to this numerical interval? 2. It is known that x [- 3; 5) . Which of the following inequalities corresponds to this? 3. What is the smallest integer solution to this system? 16; 2) - 8; 3) 6; 4) 8.
4. 5. Evaluation criteria: 3 points – 3 tasks correct; 4 points – 4 tasks correct; 5 points – 5 tasks correct.
Answers: 1. B 2. C 3. 1 4. 1 5. 2
Where can systems of inequalities be applied? Find the domain of definition of the function: Solution: The denominator is equal to zero if: This means that x = 2 Y = must be excluded from the domain of definition of the function
Problem: A passenger car travels more than 240 km on a forest road in 8 hours, and less than 324 km on a highway in 6 hours. Within what limits can its speed vary?
V t S x km/h 8 h 8 x > 2 4 0 6 x 2 4 0 , 6 x
We solve systems of inequalities 1) 2) -1 44 3) 4) 5) 6)
Thank you for your attention! Good luck! Homework: prepare for the test, No. 958,956.
Good luck everyone!!!
Is the statement true: if x >2 and y >14, then x + y>16? Is the statement true: if x >2 and y >14, then x y
To use presentation previews, create a Google account and log in to it: https://accounts.google.com
Slide captions:
Systems of linear inequalities with one unknown. Author Eremeeva Elena Borisovna teacher of mathematics MBOU secondary school No. 26, Engels
Verbal counting. 1.Name the general solution 4 -2 0 -5 2. Solve the inequalities: a) 3x > 15 b) -5x ≤ -15 3. What comparison sign does positive numbers show?
Is the number in parentheses a solution to the system of inequalities? 2 x + 3 > 0, (-1) 7 – 4 x > 0. Solution: Substitute the number -1 into the system instead of the variable x. 2 (-1) + 3 > 0, -2 + 3 > 0, 1 > 0, true 7 – 4 (-1) > 0; 7 + 4 > 0; 11 > 0. true Answer: The number -1 is the solution of the system.
Training task No. 53 (b) 5x > 10, (3) 6x + 1 10, 15 > 10, correct 6 3
Solving systems of inequalities with one unknown.
Solve the system of inequalities. 13x – 10 6x – 4. Solution: 1) Solve the first inequality of the system 13x – 10
2) Solve the second inequality of the system 10x – 8 > 6x – 4 10x –6x > – 4 + 8 4x > 4 x > 1 3) Solve the simplest system x 1 1 (1; 3) Answer: (1; 3)
Training exercises. No. 55(e;h) f) 5x + 3 2. Solution: 1)5x + 3 2 5x 2 – 7 5x – 5 x
No. 55 (h) 7x 5 + 3x. Solution: 1) 7x 5 + 3x 7x - x 5 – 2 6x 3 x
Additional task No. 58 (b) Find all x, for each of which the functions y = 0.4x + 1 and y = - 2x + 3 simultaneously take positive values. Let's compose and solve the system of inequalities 0.4x + 1 > 0, 0.4x > -1, x > - 2.5 - 2x + 3 > 0 - 2x > -3; X
Homework. No. 55 (a, c, d, g) Optional task No. 58 (a).
On the topic: methodological developments, presentations and notes
Lesson summary "Solving linear inequalities with one unknown"
Lesson type: learning new material Purpose: to develop with students an algorithm for solving linear inequalities with one unknown. Tasks: developing skills for solving linear inequalities with one unknown...
Plan – summary of an algebra lesson “Inequalities with one unknown. Systems of inequalities"
Plan – summary of an algebra lesson “Inequalities with one unknown. Systems of inequalities." Algebra 8th grade. Textbook for general education institutions. Sh.A. Alimov, Yu.M. Kolyagin, Yu.V. Sidorov and others. Purpose...
- Alekseeva Tatyana Alekseevna
- BOU VO "Gryazovets comprehensive boarding school for students with hearing impairments"
- Mathematic teacher
- repeat numerical intervals, their intersection,
- formulate an algorithm for solving systems of inequalities with one variable,
- learn how to correctly write down a solution,
- speak correctly, beautifully,
- listen attentively.
- Repetition:
- warm-up,
- mathematical lottery.
- Learning new material.
- Consolidation.
- Lesson summary.
What kinds of inequalities are there?
Strict, non-strict, simple, double.
_____________________________ What number intervals do you know? _____________________________
- Number lines,
- number intervals,
- half-intervals,
- number rays,
- open rays.
How many ways are there to indicate number intervals? List.
- Using inequality,
- using brackets,
- verbal name of the interval,
- image on a coordinate line
1. Mathematical
Test yourself (3;6) [ 1.5 ; 5 ]
2. Mathematical
Check yourself 0; 1; 2; 3. -6; -5; -4; -3; -2; 0.
3. Mathematical
Test yourself smallest -7 largest 7 smallest -5 largest -3
4. Mathematical
Test yourself - 2 < X < 3 - 1 < Х < 4
- For correct oral answers,
- for finding the intersection of sets,
- for 2 math tasks lotteries,
- for help in the group,
- for the answer at the board.
Evaluate yourself during the warm-up
II. Learning a new topic Solving systems of inequalities with one variable Task No. 1- Solve the inequalities (in draft),
- draw the solution on the coordinate line:
- 2х – 1 > 6,
- 5 – 3x > - 13;
Check yourself
2х – 1 > 6,
5 – 3x > - 13
– 3x > - 13 – 5
– 3x > - 18
Answer: (3.5;+∞)
Answer: (-∞;6)
Task No. 2 Solve the system: 2x – 1 > 6, 5 – 3x > - 13. 1. Let us solve both inequalities simultaneously, writing the solution in parallel in the form of a system, and depict the set of solutions to both inequalities in one and the same the same coordinate line. solution 2x – 1 > 6 2x > 1 + 6 2x > 7 5– 3x > - 13 – 3x > - 13 – 5 – 3x > - 18 x > 3.5 2. let's find the intersection X< 6 two numerical intervals: ///////////// 3,5 6 3. Let's write the answer as a numerical interval Answer: x (3.5; 6) Answer: x (3.5; 6) is a solution to this system. Definition. The solution to a system of inequalities in one variable is called the value of the variable at which each of the inequalities of the system is true.
See the definition in the textbook on page 184 in paragraph 35
“Solving systems of inequalities
with one variable..."
Working with the textbook
Let's talk about what we did to solve the system...- We solved the first and second inequalities, writing the solution in parallel as a system.
- We depicted the set of solutions to each inequality on one coordinate line.
- We found the intersection of two numerical intervals.
- Write down the answer as a number interval.
- Solve the first and second inequalities, writing their solutions in parallel in the form of a system,
- depict the set of solutions to each inequality on the same coordinate line,
- find the intersection of two solutions - two numerical intervals,
- write the answer as a number interval.
Rate yourself on
learning new things...
- For independent solution of inequalities,
- for writing down the solution to the system of inequalities,
- for correct oral answers when formulating the solution and definition algorithm,
- for working with the textbook.
See tutorial
page 188 to "3" No. 876
on "4" and "5" No. 877
Independent work
Examination № 876 a) X>17; b) X<5; c)0<Х<6;
№ 877
a) (6;+∞);
b) (-∞;-1);
d) decisions
No;
e) -1 < X < 3;
e)8<х< 20.
d) decisions
- For 1 mistake - “4”,
- for 2-3 mistakes - “3”,
- for correct answers - “5”.
Rate yourself on
independent
work
IV. RESULT OF THE LESSON Today in class we... ___________________________ Today in class we... ___________________________- Repeated number intervals;
- became acquainted with the definition of a solution to a system of two linear inequalities;
- formulated an algorithm for solving systems of linear inequalities with one variable;
- solved systems of linear inequalities based on an algorithm.
- Has the goal of the lesson been achieved?
- For repetition,
- for learning new material,
- for independent work.
Set yourself
grade for the lesson
HOMEWORK No. 878, No. 903, No. 875 (additional on “4” and “5”)
