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 geologist estimates that the volume of a boulder is greater than 14.6 cubic feet but less than 18.6 cubic feet. The geologist also estimates that the boulder weighs 160 pounds per cubic foot of volume. Which inequality represents this situation, where x represents the estimated total weight, in pounds, of the boulder?
原库参考解析
线性不等式与可行域挑战
\( y<−5x−21 \)
\( y>−3x−8 \)
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 plant nursery classifies a perennial flower species as tall if its typical height when fully grown is more than 100 centimeters. Each of the following inequalities represents the possible heights h, in centimeters, for a specific perennial flower species when fully grown. Which inequality represents the possible heights h, in centimeters, for a tall perennial flower species?
原库参考解析
线性不等式与可行域挑战
Cristina was elected to the city council. Each city council person is required to attend at least 21 of the 27 scheduled council meetings in a year to keep their position. Which inequality represents the number of city council meetings, x, Cristina must attend in a year to keep her position on the city council?
原库参考解析
线性不等式与可行域挑战
To cut a lawn, Antwan charges a fee of $10.00 for his equipment and $8.50 per hour spent cutting a lawn. Taylor charges a fee of $7.00 for her equipment and $9.50 per hour spent cutting a lawn. If x represents the number of hours spent cutting a lawn, what are all the values of x for which Taylor's total charge is greater than Antwan's total charge?
原库参考解析
线性不等式与可行域进阶
An employee earns $30 per hour for the first 40 hours worked in a week. For each hour worked over 40 hours, the employee earns $45 per hour. What is the least number of whole hours that the employee must work in a week to earn at least $1,610?
原库参考解析
线性不等式与可行域挑战
\( y\le x+15 \)
\( y\ge-4x-15 \)
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
The shaded region shown in the graph represents all the solutions to which inequality?
原库参考解析
线性不等式与可行域进阶
When several resistors are connected in series in a circuit, the total resistance of the circuit is the sum of the resistances of these resistors. A certain circuit consists of 8 resistors connected in series, where the resistance of each resistor is positive. This circuit has a total resistance of 130 ohms. The total resistance of 3 resistors in this circuit is 90 ohms. Which inequality best represents all possible values of the resistance x, in ohms, of one of the other 5 resistors in this circuit?
原库参考解析
线性不等式与可行域进阶
Sabrina has a goal to save at least $240. Sabrina has already saved $180. If x represents the additional amount of money, in dollars, Sabrina needs to save to meet her goal, which inequality represents this situation?
原库参考解析
线性不等式与可行域挑战
\( y<42−7x \)
\( \frac{y}{7}>10 \)
Which inequality represents the x values for all solutions (x, y) that satisfy 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
The shaded region shown represents the solutions to the inequality -19y < c, where c is a constant. What is the value of c?
原库参考解析
