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The examination of the inequality in the senior high school entrance examination shows that the traditional questions in the past have been obviously reduced or disappeared, and replaced by more dynamic application questions. As we all know, the foothold of ability lies in application. Inequality, like other knowledge, will have a broader market once it is applied. For example, in the senior high school entrance examination questions in Hebei province, the score of mathematical application questions is as high as 7 1, accounting for 59%, including two inequality application questions, and the senior high school entrance examination papers in Guangzhou even take inequality application questions as the finale questions. This paper only discusses the characteristics and solutions of inequality application problems for reference only.
First, the application of single inequality
Example 1, (Hebei Province) In a knowledge contest of "Man and Nature", there were 25 questions in the contest, and each question gave four answers, only one of which was correct. Ask students to choose the correct answer, with 4 points for each question, and 2 points for not choosing or choosing wrong. If the students score no less than 60 points in this competition,
Comments: One of the difficulties in the application of inequality is to distinguish the similarities and differences between inequality and equation application. How to list inequalities should be good at grasping the mathematical meanings of the words "not less than" and "at least" in the problem. In this question, the sentence "2 points backward" should be understood as no choice or no mistake, but actually 6 points should be deducted. So if you choose the right question X, you won't choose or choose the wrong question entitled (25-x), then there will be.
100-6(25 x)≥60
Solution: x≥ 18x= 19, that is, he at least chose the right question 19.
Example 2. The scoring rules for football matches in a city are: win a game, get 3 points, draw a game, get 1 point, and lose a game, get 0 point. A team should play 15 games, and has lost 3 games. If it wants to score 22 points, then the team must win at least ().
A, 3 B, 4 C, 5 D, 6.
Comments: There are three situations in the result of a football match: victory, draw and loss. If it is assumed that X will win and the rest will be tied, then
3x+ 12-x≥22 minus x≥5.
Why can't you list the equation: 3x+ 12-x=22, because the actual score is less than or equal to 3x+ 12-x (it is possible to lose in future competitions), so
3x+ 12-x≥ Actual score =22
Example 3: A refrigerator sold in a shopping mall (Dongcheng District, Beijing) costs 2 190 yuan each, and the daily power consumption is 1 kWh. Although the price of energy-saving refrigerator B is higher than that of refrigerator A 10%, it consumes 0.55 kWh of electricity every day. Now refrigerator A is on sale (the price after 10% discount is the original price). Ask the mall for at least a discount.
Solution: Now refrigerator A is discounted by X, so it is more cost-effective. According to the meaning of the question:
2 190×+365× 10×0.4≤2 190×( 1+ 10%)+365× 10×0.55×0.4
Deduce that x≤8
That is, the sales of Type A refrigerators in shopping malls should be at least 20% off, so that consumers can buy them economically.
Comments: This example provides decision-making for the purchase and sale behavior in marketing by solving inequality.
Second, the application of comprehensive equations and functional inequalities
Example 4: A shopping mall (Shanxi Province) plans to invest a sum of money to buy a batch of goods out of stock. Through market research, it is found that if it is sold at the beginning of the month, the profit is 15%, and the principal and interest can be used to reinvest in other commodities, then the profit at the end of the month is 10%. If you sell it at the end of the month, you can make a profit of 30%, but you have to pay the cost of warehouse storage. 700 yuan, according to the financial situation of shopping malls, how to get more profits through buying and selling?
Solution: If you invest X yuan and sell it at the beginning of the month, you can make a profit of y 1 yuan at the end of the month; At the end of the month, you can make a profit of y2 yuan. The title means: y1=15% x+10% (x+15% x) = 0.265x.
y2=x 0.3-700=0.3x-700
(1) When y 1=y2, the solution is: 0.265x=0.3x-700, and x = 2,000 yuan is deduced.
(2) When y 1 < y2, the solution is: 0.265x < 0.3x-700, after deducting that x > 2000 yuan.
(3) when y 1 > y2, the solution is: 0.265x > 0.3x < 700, and that x < 2000 yuan is deducted.
That is, when the shopping mall invests 2000 yuan, the two business models have the same profit, when the investment exceeds 2000 yuan, the second model has more profit, and when the investment is less than 2000 yuan, the first model has more profit.
Comments: (1) got two function formulas, (2) and (3) used the solution of inequality to make the optimal decision on management mode, and the application of inequality was well played here.
Example 5 (Chongqing) In order to protect the Yangtze River and reduce soil erosion, a county in our city decided to return farmland to forests, and began to plant trees on sloping wasteland from 1995, and the sloping wasteland was transformed every year, with more trees in the same area than the previous year. Due to natural disasters, tree survival rate and human factors, the same number of new sloping wasteland will be produced every year. The following table shows the statistical data of slope wasteland area and tree planting area in three years (1995, 1996, 1997). Assuming that all the slopes and wasteland are planted with trees, there will be no soil erosion. When can trees be planted on the sloping wasteland in the county?
1995
1996
1997
Annual planting area (mu)
100
1400
1800
After planting trees, the organ will report the area (mu) of slope wasteland.
25200
24000
22400
(2) In order to save water, the city stipulates that when the daily water consumption of our factory does not exceed 20 tons, the water price is 4 yuan per ton; When the daily water consumption exceeds 20 tons, the excess will be charged by 40 yuan per ton. It is known that the daily water consumption of this factory is not less than 20 tons, and the daily water consumption of this factory is t tons, and the profit earned on that day is W yuan. Find the relationship between w and t; The factory strengthens management and actively saves water, so that the daily water consumption is not more than 25 tons and not less than 20 tons, seeking the daily profit range of the factory.
Solution: (1) let y=kx+b, according to the meaning of the question:
Push out,
Derive y=-x+204.
When x= 10 and y= 194, that is, per ton 10 yuan, the profit of producing beverages with 1 ton of water is 194 yuan.
(2) y=-x+204 from (1), when x=4, y=200, when x=40, y= 164.
∴w=200×20+ 164(t-20)= 164t+720(t≥20)
∴t=, from 20≤t≤25, that is
20≤≤25,
Deduce that 4000≤W≤4820
That is, the daily profit of the factory is not less than 4000 yuan and not more than 4820 yuan.
Evaluation; This topic involves the synthesis of inequalities, equations and functions.
Example 7. When a class in Jinan arranges the venue for the Spring Festival Gala, it is necessary to cut the right-angled triangular colored paper into short colored strips with different lengths, as shown on the right. In Rt△ABC, ∠ c = 90, AC=30cm, AB=50cm, and rectangular color bars with the width of 1cm are cut out in turn.
A, B, 25, 26, 27
Analysis: Let n rectangular strips be cut, as shown in the figure, and let the length EF=x of the nth rectangle be x, then
,
X=(30-n), and x≥5,
That is, (30-n)≥5n≤26.
∴n=26, so choose (c).
This is an inequality application problem of comprehensive equation.
Example 8. One ticket for a garden in Suzhou 10 yuan, which can be used once. Considering people's different needs, in order to attract more tourists, the garden not only keeps the original way of selling tickets, but also introduces the way of "purchasing individual annual tickets" (individual annual tickets can be used by ticket holders for one year from the date of purchase). Annual tickets are divided into three categories: A, B and C: Class B annual tickets for 60 yuan, and ticket holders need to buy each ticket for 2 yuan when entering the park; Class C annual ticket per 40 yuan. Ticket holders are required to purchase each ticket for 3 yuan when entering the park.
(1) If you only choose one way to buy tickets, and you plan to spend 80 yuan to buy tickets for the garden every year, then try to find out the way to buy tickets that will make you enter the garden the most times through calculation.
(2) Is it more cost-effective to buy class A tickets when you have to enter the park at least several times a year?
Analysis: The problem (1) is to buy tickets in 80 yuan, and choose the best way to enter the park with the most times by comparing the three forms, obviously excluding class A annual tickets;
If you choose class B annual ticket, the number of times is
= 10 (times);
If you choose class C annual ticket, the number is
= 13 times;
If you don't buy an annual ticket, you can only buy and enter.
=8 times
After the above comparison, the number of times to enter the park by purchasing Class C annual tickets is 13.
Question (2) concerns inequality. If it is more economical to buy at least X class A air tickets, there are:
Therefore, it is more cost-effective to enter the park at least 30 times a year and buy class A tickets.
This example takes tourism as the background and provides economic consumption decision for tourism with the help of inequality knowledge. Application problems like this inequality are rarely seen in the previous middle school entrance examination questions.
Ex. 9: At the beginning of ticket checking at (Guangzhou) station, a passenger queued up in the waiting room to check in. After the check-in began, passengers still came to check in. Assuming that passengers increase at a fixed speed, the ticket checking speed at the ticket gate is also fixed. If you open a ticket gate, it will take 30 minutes to complete the queuing of all passengers. If you open two ticket gates, it only takes 10 minutes to check all the passengers in line. If you want to check all the passengers waiting in line within 5 minutes, at least how many ticket gates must be opened at the same time so that passengers who arrive later can check on the spot?
Analysis: In this case, it is difficult to establish a mathematical model, which involves the number of passengers, the speed of ticket checking and the number of ticket gates opened. Then make clear the equality relation (equation) and inequality relation (inequality). Let's first set the number of passengers entering the station one after another as X people per minute, and check Y people per minute at each ticket gate. To open n ticket gates at the same time in 5 minutes, the question means:
①-② Simplification: y=2x,
Substituting into ①, a=30(y-x)=30x.
Substitute y=2x and a=30x into ③,
35x≤ 10nxn≥3.5n=4
In this case, with the background of passenger check-in, using mathematical knowledge such as equations and inequalities, in order to ease the passenger flow, reasonably arrange the ticket gate, provide high-quality and efficient services for passengers, and make the best decision. This kind of application problem with inequality as the finale of the middle school entrance examination has never appeared before, and it is the first case in the history of the middle school entrance examination.
Example 10. Taxi starting price in a city (Shaanxi) 10 yuan (i.e. fare within 5km 10 yuan). After reaching or exceeding 5km, the fare will increase by 1 for every increase of 1km (less than 1km, it is 1 0.2 yuan).
Analysis: If the distance from A to B is about xkm, then:
16 < 10+(x-5)× 1.2≤ 17.2 10 < x≤ 1 1
That is, the length from a to b is about 10 to 1 1 km.
Example 1 1. According to the regulations of a communication company in a city, the call charge in the business network is: 0.5 yuan 3 minutes before the call, and 0. 1 yuan per minute for calls over 3 minutes (if it is less than 1 minute, it will be calculated as 1 minute). A person's one-time phone bill is 1.65438+.
Analysis: Let this person speak for x minutes, and then:
1 < 0.5+(x-3)×0. 1≤ 1. 18 < x≤9 .
That is, this person's talk time is about 8 minutes to 9 minutes.
Example 12. Twenty laid-off workers from a city (Gansu Province) contracted 50 mu of land in the suburbs to start a farm. These lands can be planted with vegetables, tobacco leaves or wheat. The number of workers and the output value of these crops per mu are predicted as follows:
Crop variety
Number of employees per mu of land
Estimated output value per mu of land
vegetables
1 100 yuan
tobacco leaf
750 yuan
wheat
600 yuan
Please design a planting plan, so that crops can be planted on every mu of land, and 20 employees have jobs. It is estimated that the total output value of crops is the most.
Analysis: Let's plant x mu, y mu and (50-x-y) mu of vegetables, tobacco leaves and wheat respectively.
x+y+(50-x-y)=203x+y=90y=90-3x
Assuming that the estimated total output value is W (yuan), then
w = 1 100 x+750(90-3x)+600(50-x-90+3x)
Then substitute y=90-3x.
W=50x+43500
This is a linear function about X, and its maximum value depends on the range of X. The key lies in how to list the inequalities about X. After reviewing the questions repeatedly, I found that there are
y=90-3x≥00