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
\( y<−6x−21 \)
\( y>−4x−10 \)
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进阶
A model estimates that a gray whale travels 74 to 79 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 49 days of its migration?
原库参考解析
线性不等式与可行域进阶
A student needs at least 190 signatures from classmates to run for class president. The student collects 30 signatures on Monday and collects enough additional signatures on Tuesday to run for class president. Which inequality represents the number of additional signatures, s, that the student collects on Tuesday?
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 \( rx+ty\ge-44 \), 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
A school is ordering tablet computers for its students. The tablets cost $164 each, and 6% tax is added to the total cost of the order. If the school can spend no more than 44,800 on the tablets, including tax, what is the maximum number of tablets that the school can order?
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 \( rx+ty≥−36 \), 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
In the xy-plane, which of the following does NOT contain any points that are part of the solution set to \( 2x−6y>12 \) ?
原库参考解析
线性不等式与可行域进阶
\( y\le5x-21 \)
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进阶
A shopper is purchasing oranges that cost $0.67 each. If the shopper has a coupon for $2 off the entire purchase and there is no sales tax. Which inequality represents the number of oranges, x, that the shopper can purchase for no more than $12?
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 integer is represented by x. The product of 28 and the third odd integer in the set is at most the value of 50 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
A team of researchers plans to spend no more than 310 hours in total collecting observation data for two projects. For one of the projects, the team will observe at least 13 penguins and will spend 5 hours observing the behavior of each penguin. For the other project, the team will observe at least 16 seals and will spend 6 hours observing the behavior of each seal. Based on this plan, what is the maximum number of penguins the team can observe?
原库参考解析
线性不等式与可行域进阶
\( y<−4x+12 \)
Which point (x, y) is a solution to the given inequality in the xy-plane?
原库参考解析
