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挑战
Which shaded region represents the solutions to the inequality \( y<−60x+10 \) ?
原库参考解析
线性不等式与可行域进阶
\( y<−2x+14 \)
Which point \( \left(x,\ y\right) \) is a solution to the given inequality in the xy-plane?
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 \( rx+ty\ge-65 \) , where r and t are constants. What is the value of \( r+t \) ?
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 which inequality?
原库参考解析
线性不等式与可行域挑战
\( y>5x−4 \)
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
For a particular tree species, it is recommended to plant a maximum of 92 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
\( y\le x+3 \)
\( y\ge-3x-7 \)
Which point \( (x,\ y) \) 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挑战
A landscaper will plant two types of shrubs, hydrangeas and winter hazels, in a park. There will be no more than 244 total shrubs planted, and the number of hydrangeas planted will be at most three times the number of winter hazels planted. Which of the following systems of inequalities best represents this situation, where h is the number of hydrangeas that will be planted and w is the number of winter hazels that will be planted?
原库参考解析
线性不等式与可行域挑战
原库参考解析
线性不等式与可行域挑战
A model estimates that a gray whale travels 72 to 77 miles each day during its migration. Based on this model, which inequality represents the estimated total number of miles, x, a gray whale could travel in 27 days of its migration?
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 which inequality?
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 $45 to rent the rink and $10.75 per attendee. The budget is $275. What is the greatest number of attendees possible without exceeding the budget?
原库参考解析
