The seventh grade mathematical inequality calculation problem the first part
1. A car driving at a constant speed is 50 kilometers away from a place at 1 1 :20. What conditions do you need to meet to pass a place before 12 :00?
Let the speed be x km/h.
From the time point of view, if the car wants to pass A before 12: 00, it will take less than 2/3 hours to drive 50 kilometers at this speed, that is,
Let the speed be x km/h.
From the distance, if the car wants to pass through A before 12: 00, it will take more than 50 kilometers to drive at this speed for 2/3 hours, that is,
2. Definition of inequality: use? & lt？ Or? & gt？ The formula representing the relationship between size is called inequality, like a+2? Is this how A-2 is used? The formula of symbolic representation of inequality relation is also an inequality.
Attention:? & lt？ 、? & gt？ 、? 、? 、? Is not an equal sign.
Exercise questions:
Which of the following formulas are inequalities? What is not inequality? Why?
-2 & lt； 5 x+3 >6 4x-2y？ 0 a-2b a+b？ c
5m+3 = 8 8+4 & lt； seven
3. Solution of inequality
What have we learned? The value of the unknown quantity that makes both sides of the equation equal is the solution of the equation? Similar to an equation, the value of the unknown quantity that can make the inequality hold is called the solution of the inequality.
Method of substitution is a simple and practical method to test whether a value is the solution of inequality.
Exercise questions:
Is x=78 the solution of inequality? What about x=75? What about x=72?
Determine which of the following numbers is the solution of inequality:
76 , 73 , 79 , 80, 74.9 , 75, 75. 1, 90 , 60
Can you find other ways to solve this inequality? How many solutions are there to this inequality? Can you tell me his solution set?
4. Solution set of inequality
Generally speaking, all of an inequality with unknowns is deconstructed into the solution set of this inequality. The process of finding the solution set of inequality is called solving inequality.
Think about it:
Is the solution of inequality the same as the solution set of inequality?
Is solving inequality the same as solving inequality?
Exercise questions:
1, the following statement is true ()
A.x=3 is 2x+1>; 5' s solution
B.x=3 is 2x+1>; Unique solution of 5
C.x=3 is not 2x+1>; 5' s solution
D.x=3 is 2x+1>; 5-piece solution
5. Representation method of solution set
: Use a formula (such as x>2), that is, the simplest form of inequality (such as x>A or x < A) to express it.
For example, the solution set of inequality can be expressed by inequality x > 75.
Exercise questions:
Solution set of inequality;
⑴x+2 & gt； 6⑵3x & gt； 9⑶x-3 & gt； 0
Mark an interval on the number axis with the number axis. In this interval, the values corresponding to points are all solutions of inequalities.
note:
1. Steps to express inequality solution set with number axis:
① Draw several axes; ② Fixed boundary points; 3 set the direction.
2. Use the number axis to represent the solution set of inequality, and remember the following rules:
Greater than drawing to the right and less than drawing to the left; There is an equal sign (? ,? ) draw a solid point,
Draw a hollow circle without the equal sign (>,<).
Exercise questions:
6. One-dimensional linear inequality
We know that 2x+ 1=5 is a linear equation, so how to treat the inequality 2x+1>; How should I name it?
An unknown number, the unknown number is 1, which is similar to the inequality of one-dimensional linear equation and is called one-dimensional linear inequality.
The second part of the seventh grade mathematical inequality calculation problem
1, the following is a linear inequality ()
A.4x-2y？ 0
B.x？ - 1 1
C.x2- 1？ 0
D.
To judge whether a formula is a linear inequality, four conditions must be met:
① There is only one unknown in the formula;
② The number of unknowns is1;
(3) The formula is connected with an inequality symbol.
④ The denominator does not contain unknowns.
2. There are the following mathematical expressions:
①- 1 & lt； 0; ②3m-2n & gt； 0; ③x = 4； ④x？ 7; ⑤5x+4 = x+5；
⑥x2+xy+y2； ⑦x+2 & gt； y+3； ⑧x2 & gt； 4; ⑨3x-2 & gt； 4x-3； ⑩3+5 & lt； 7;
Where () is an inequality.
Is a one-dimensional linear inequality () (only fill in the serial number)
3, the following statement is wrong ()
A. inequality x
B. inequality x
C.x=-4 is the inequality-3x >; 9 solution
D.x>5 is the inequality x+3 >; 6-piece solution
4, expressed by inequality:
The sum of (1)a and 1 is positive;
(2) The sum of 2 times y and 1 is less than 3;
(3) The sum of 3 times y and 2 times x is not negative.
(4) The product of x times 3 plus 2 is at most 5.
5. Use the number axis to represent the solution set of the following inequalities:
⑴x & gt； - 1; ⑵ x？ - 1; ⑶x & lt； - 1; ⑷ x？ - 1.
6. Write the solution set of inequality according to the figure below:
7. Can you find the fitness inequality-1? Integer solution of x<4? What is the maximum integer value of x?
7, the properties of the equation
The basic property of the equation is 1: add (or subtract) the same number or algebraic expression on both sides of the equation, and the results are still equal.
If a=b, then a? c=b？ c
Basic property 2: Multiply or divide the same number on both sides of the equation (the divisor is not 0), and the results are still equal.
If a=b, then ac=bc or (c? 0)
8. The essence of inequality
Do inequalities have similar properties?
If 5 > 3
Then 5+2 _ _ _ 3+2, 5-2 _ _ _ 3-2
If-1
Then-1+2 _ _ 3+2,-1-3 _ _ 3-3.
Attribute 1: If a >；; B, then a+c > B+c or a-c > b-c
That is, add (or subtract) the same number or the same algebraic expression on both sides of the inequality, and the direction of the inequality remains unchanged.
Guess 1: Both sides of the inequality are multiplied (or divided) by the same non-zero number. Has the direction of inequality changed?
If 6 > 2
So 6? 5 ____ 2? 5 ,
6 ? (-5)____2? (-5),
6? 5 ____ 2? 5 ,
6 ? (-5)____2? (-5)
If -2
So -2？ 6____3? 6,
-2? (- 6)____3? ( - 6),
-2? 2____3? 2,
-2? (- 4)____3? ( - 4)
Conjecture 2: Both sides of inequality are multiplied by (or divided by) the same non-zero number. Has the direction of inequality changed?
Both sides of inequality 7 >:4 are multiplied by the same number, and the results are compared with >; & lt fill in the blanks
Conclusion: When the same number is multiplied by a positive number, the direction of inequality remains unchanged; when the same number is multiplied by a negative number, the direction of inequality changes; when the same number is multiplied by 0, it is equal to.
The third part of the seventh grade mathematical inequality calculation problem
Example 1:
1. Is the derivation of the following question correct? why
(1) Because 7.5 >: 5.7, so -7.5.
(2) because a+8 >; 4, so a & gt-4;
(3) because 4a >；; 4b, so a & gtb；;
④ because-1>; -2，so-a- 1 & gt； -a-2；
(5) Because of 3>2, 3a & gt2a.
Step 2 fill in the blanks
( 1)∵0 & gt； 1,
a a+ 1；
(2)∫(a- 1)2 & gt； 0,
(a- 1)2-2 -2
(3) if x+1>; 0, add-1 on both sides, and you get _ _ _ _ _ _ _
(4) If 2x & gt-6, if both sides are divided by 2, the result is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
(5) If -0.5 x? 1, and both sides are multiplied by -2 to get _ _ _ _ _ _ _ _ _ _ _ _ _
3. As we all know.
4. Basic properties of inequality (abstract)
Add (or subtract) the same number or the same formula on both sides of (1) inequality, and the direction of inequality remains unchanged.
(2) Both sides of the inequality are multiplied by (or divided by) the same positive number, and the direction of the inequality remains unchanged.
(3) When both sides of the inequality are multiplied by (or divided by) the same negative number, the direction of the inequality changes.
5. Addressing inequality:
( 1)x-7 & lt； 8(2)3x & lt； 2x-3
6. The sum of three consecutive positive numbers and odd numbers is less than 30. How many groups are there? Write it out separately.
7, if the inequality x-a? 0 has only three positive integer solutions. Find the range of positive integer a.
8. Given the solution of equation 3x-m= x- 5, find the range of m 。