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
To win a game show, a contestant needs to score at least 40 total points from two rounds. Correct responses in the first round are worth 3 points each, and correct responses in the second round are worth 5 points each. Which inequality models this situation, where \( f \) is the number of correct responses in the first round and s is the number of correct responses in the second round?
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?
参考答案
线性不等式与可行域挑战
\( y<−3x+18 \)
Which point (x, y) is a solution to the given inequality in the xy-plane?
Which point (x, y) is a solution to the given inequality in the xy-plane?
参考答案
线性不等式与可行域挑战
\( y>13 \)
\( 4x+y<19 \)
The point (x, 48) is a solution to the given system of inequalities in the xy-plane. Which of the following must not be the value of x?
\( 4x+y<19 \)
The point (x, 48) is a solution to the given system of inequalities in the xy-plane. Which of the following must not be the value of x?
参考答案
线性不等式与可行域进阶
An arborist measured and recorded the heights of red maple trees in a certain area. Of the red maple tree heights recorded in this area, the minimum was 46 feet and the maximum was 57 feet. Which inequality is true for all possible values of \( t \), where \( t \) represents the recorded height, in feet, of a red maple tree in this area?
参考答案
线性不等式与可行域挑战
For the first 10 seconds of an experiment that lasts 100 seconds, a particle travels at a speed that is at least 4 meters per second and is at most 8 meters per second. For the remainder of the experiment, the particle travels at a constant speed of 12 meters per second. Which of the following is a possible number of meters the particle traveled in the first 70 seconds of the experiment?
参考答案
线性不等式与可行域进阶
\( y\le9-x \)
\( y\ge\frac{1}{8}x \)
Which shaded region represents the solutions to the given system of inequalities?
\( y\ge\frac{1}{8}x \)
Which shaded region represents the solutions to the given system of inequalities?
参考答案
线性不等式与可行域挑战
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 9 resistors connected in series, where the resistance of each resistor is positive. This circuit has a total resistance of 110 ohms. The total resistance of 4 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?
参考答案
线性不等式与可行域基础
A researcher tested the muscle strength of 4 house mice. The scores for the house mice are 1, 3, 4, and 7. Which inequality is true for all values of s, where s represents any score obtained by one of these house mice?
参考答案
线性不等式与可行域进阶
\( y\le11-x \)
\( y\ge\frac{4}{7}x \)
Which shaded region represents the solutions to the given system of inequalities?
\( y\ge\frac{4}{7}x \)
Which shaded region represents the solutions to the given system of inequalities?
参考答案
线性不等式与可行域挑战
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 9 resistors connected in series, where the resistance of each resistor is positive. This circuit has a total resistance of 80 ohms. The total resistance of 4 resistors in this circuit is 50 ohms. Which inequality best represents all possible values of the resistance x, in ohms, of one of the other 5 resistors in this circuit?
参考答案
线性不等式与可行域基础
For a party, 52 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?
参考答案
