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
In the xy-plane, the point \( (6,\ 7) \) is a solution to which of the following systems of inequalities?
原库参考解析
线性不等式与可行域进阶
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
A baker has a total of 15 cups of flour and enough other ingredients to make banana pancakes and apple pancakes. The baker needs 1.25 cups of flour for a batch of banana pancakes and 0.75 cups of flour for a batch of apple pancakes. If the baker makes 8 batches of banana pancakes, what is the greatest number of whole batches of apple pancakes the baker can make?
原库参考解析
线性不等式与可行域进阶
For a certain classification, the positive mass m, in grams, of an object must be less than 510 grams. Which inequality represents this situation?
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 \( ry<128 \), where r is a constant. What is the value of r?
原库参考解析
线性不等式与可行域进阶
In a set of three consecutive integers, where the integers are ordered from least to greatest, the first integer is represented by x. The sum of 4 and the second integer is less than the product of 17 and the third integer. Which inequality represents this situation?
原库参考解析
线性不等式与可行域挑战
The graphs of \( y=-3x+7 \) and \( y=x-5 \) are shown. Point P (not shown) has coordinates \( (3,−5) \) and lies in the shaded region. The coordinates of P satisfy which of the following inequalities?
I. \( y<−3x+7 \)
II. \( y>x−5 \)
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
A car travels at a speed of at least 25 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
The graphs of \( y=-3x+8 \) and \( y=x-4 \) are shown. Point P (not shown) has coordinates \( (3,−3) \) and lies in the shaded region. The coordinates of P satisfy which of the following inequalities?
I. \( y<−3x+8 \)
II.\( y>x−4 \)
原库参考解析
线性不等式与可行域进阶
To win a game show, a contestant needs to score at least 70 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 9 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挑战
To win a game show, a contestant needs to score at least 80 total points from two rounds. Correct responses in the first round are worth 4 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
