这部分需要会什么？
- 解一元一次不等式并表示解集
- 识别端点是否包含与边界线是否实线
- 检验二元线性不等式的公共可行域
核心概念与公式
不等式两边加减同一个数，不等号方向不变；乘除正数方向不变，乘除负数必须反向。例如 \(-2x>6\) 等价于 \(x<-3\)。复合不等式的“且”表示交集，“或”表示并集。
二元不等式先画边界直线，再用不在边界上的测试点判断哪侧满足。\(\le\)、\(\ge\) 包含边界，使用实线；\(<\)、\(>\) 不含边界，使用虚线。可行域需同时满足全部条件。
做题时怎样使用
- 将文字中的至少、至多、超过分别转成对应不等号
- 解不等式；涉及负数乘除时特别检查方向
- 若变量表示人数、件数等，再在解集中选整数，并核对端点
容易混淆的地方
“至少 5”包括 5；“大于 5”不包括 5。图上的某点满足一个条件，并不代表它满足整个不等式组。
用一个例题串起方法
知识点教学例题一项活动收取固定费用 18 元，每位参加者另付 7 元。总预算不超过 60 元，最多可有多少位参加者？
展开解题思路 ＋
设人数为非负整数 \(n\)，则 \(18+7n\le60\)，所以 \(n\le6\)。最大人数为 6，此时费用恰好为 60 元。
本知识点题库
先独立作答，再查看答案与解析。
线性不等式与可行域进阶
A car travels at a speed of at least 40 miles per hour but no more than 50 miles per hour for a certain part of a trip. Which inequality represents this situation, where x is the speed of the car, in miles per hour, on this part of the trip?
参考答案
线性不等式与可行域进阶
\( y>40a \)
For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality, where a is a positive constant?
For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality, where a is a positive constant?
参考答案
线性不等式与可行域进阶
Company A charges a $25.00 reservation fee and $8.50 per hour to rent its lawn equipment. Company B charges a $22.00 reservation fee and $9.25 per hour to rent its lawn equipment. If x represents the number of hours the equipment is rented, what are all the values of x for which Company B's total charge is greater than Company A's total charge?
参考答案
线性不等式与可行域进阶
For a particular tree species, it is recommended to plant a maximum of 111 trees per acre. Based on this recommendation, what is the maximum number of trees of this species that can be planted on 2 acres?
原库参考解析
线性不等式与可行域挑战
The shaded region shown represents the solutions to the inequality \( ry<128 \) , where r is a constant. What is the value of r?
原库参考解析
线性不等式与可行域进阶
For a party, 65 hamburger buns are needed. Hamburger buns are sold in packages of 12. What is the minimum number of packages that should be bought for the party?
原库参考解析
线性不等式与可行域基础
Julissa needs at least 100 hours of flight time to get her private pilot certification. If Julissa already has 86 hours of flight time, what is the minimum number of additional hours of flight time Julissa needs to get her private pilot certification?
原库参考解析
线性不等式与可行域挑战
A small business owner budgets $2,200 to purchase candles. The owner must purchase a minimum of 200 candles to maintain the discounted pricing. If the owner pays $4.90 per candle to purchase small candles and $11.60 per candle to purchase large candles, what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discounted pricing?
原库参考解析
线性不等式与可行域挑战
\( y\le x+7 \)
\( y\ge-2x-1 \)
Which point \( \left(x,\ y\right) \) is a solution to the given system of inequalities in the xy-plane?
原库参考解析
线性不等式与可行域基础
For a literature class, a student's essay needs to have no more than 5,000 words. If a student's essay already has 3,026 words, what is the maximum number of additional words the student can include in the essay?
原库参考解析
线性不等式与可行域挑战
The shaded region shown represents the solutions to the inequality \( ry<60 \) , where r is a constant. What is the value of r?
原库参考解析
线性不等式与可行域挑战
Leo goes to a packing store to buy containers and tape. Leo has $15. Each container costs $1.87 and each roll of tape costs $2.40. Which inequality represents the relationship between the number of containers, c, and the number of rolls of tape, t, Leo can buy?
原库参考解析
