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 newspaper stand sells magazines for $5 each and newspapers for $2 each. The stand needs to sell m magazines and n newspapers for a combined total value of at least $150 each day to cover operational costs. Which inequality represents this situation?
原库参考解析
线性不等式与可行域挑战
\( y<−6x−22 \)
\( y>−4x−7 \)
For which of the following tables are all the values of x and their corresponding values of y solutions to the given system of inequalities?
原库参考解析
线性不等式与可行域挑战
A certain blue whale calf weighed 5,200 pounds when it was born, and its weight increased at an average rate of 250 pounds per day for the first n days after birth, where n is an integer. If the calf's weight was greater than 8,700 pounds but less than 9,700 pounds n days after birth, what is one possible value of n?
原库参考解析
线性不等式与可行域进阶
The shaded region shown represents the solutions to an inequality. Which ordered pair ( x , y ) is a solution to this inequality?
原库参考解析
线性不等式与可行域挑战
In a set of four consecutive odd integers, where the integers are ordered from least to greatest, the first unteger is represented by x. The product of 26 and the third odd integer in the set is at most the value of 46 less than the sum of the first and fourth odd integers in the set. What is the greatest possible value of x?
原库参考解析
线性不等式与可行域进阶
An event planner is planning an ice-skating event. It costs the event planner a onetime fee of $30 to rent the skating rink and $10.25 per attendee. The event planner has a budget of $250. What is the greatest number of attendees possible without exceeding the budget?
原库参考解析
线性不等式与可行域挑战
\( y>−\frac{7}{2}x+17 \)
For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?
原库参考解析
线性不等式与可行域进阶
A soccer team's goal is to earn at least $1,320 by selling coupon books. The team earns $12 from selling each coupon book. Which of the following inequalities describes all possible values for the number of coupon books, n, the team can sell to meet the goal?
原库参考解析
线性不等式与可行域挑战
\( y>17 \)
\( 4x+y<21 \)
The point (x, 54) is a solution to the given system of inequalities in the xy-plane. Which of the following could be the value of x?
原库参考解析
线性不等式与可行域进阶
A botanist estimates the number of oak trees in a forest to be at least 2,900 and not more than 3,100. Which inequality best represents the number of oak trees, x, in this forest?
原库参考解析
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 65 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?
原库参考解析
线性不等式与可行域挑战
If \( x\ge-8 \) represents all solutions to the inequality \( ax-18\le14 \), where a is a constant, what is the greatest possible value of a ?
原库参考解析
