作为章末总结，对不等式一章的知识进行讲课梳理，重点内容可以举简单案例，讲解风趣自然。---The image contains content from a chapter titled "第二章 不等式" (Chapter Two: Inequalities), presenting various aspects of inequalities, including their properties, interval notation, and methods for solving different types of inequalities.
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**I. Overall Structure (Mind Map Description)**
* **Type:** Mind Map / Flowchart.
* **Main Element:** Central node "不等式" (Inequalities).
* **Branches:**
* "不等式的性质" (Properties of Inequalities)
* Sub-branches: "基本性质" (Basic Properties) leading to "比较实数大小的方法" (Method for comparing real numbers); "不等式的性质" (Properties of Inequalities) leading to "传递性" (Transitivity), "加法性质" (Addition Property), "乘法性质" (Multiplication Property), and "......" (ellipsis indicating more).
* "区间" (Intervals)
* "不等式的解法" (Methods for Solving Inequalities)
* Sub-branches: "一元一次不等式(组)的解法" (Solving linear inequalities (systems)), "一元二次不等式的解法" (Solving quadratic inequalities), "含绝对值的不等式的解法" (Solving inequalities involving absolute values).
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**II. Knowledge Points**
**一、不等式的性质 (Properties of Inequalities)**
1. **实数的基本性质 (Basic properties of real numbers):**
设 a, b ∈ R, 则
* `a - b > 0 ⇔ a > b`
* `a - b = 0 ⇔ a = b`
* `a - b < 0 ⇔ a < b`
**说明 (Explanation):** From this, methods for comparing the size of real numbers emerge. It transforms the comparison of two numbers into comparing their difference with 0.
2. **不等式的性质 (Properties of inequalities):**
* **性质 1 (传递性 - Transitivity):** 如果 `a > b`, 且 `b > c`, 则 `a > c`.
* **性质 2 (加法性质 - Addition Property):** 如果 `a > b`, 且 `c ∈ R`, 则 `a + c > b + c`.
* **性质 3 (乘法性质 - Multiplication Property):** 如果 `a > b`, 且 `c > 0`, 则 `ac > bc`. 如果 `a > b`, 且 `c < 0`, 则 `ac < bc`.
**不等式的其他常用性质 (Other common properties of inequalities):**
| Property Description | Condition/Result | Name/Description |
| :----------------------------------------------------------- | :--------------------------------------------- | :------------------- |
| 如果 `a + b > c`, 则 `a > c - b` | `a > c - b` | 移项 (Transposition) |
| 如果 `a > b`, 且 `c > d`, 则 `a + c > b + d` | `a + c > b + d` | 同向不等式可加性 |
| 如果 `a > b > 0`, 且 `c > d > 0`, 则 `ac > bd` | `ac > bd` | 两个或几个同向是正数的同向不等式，把它们的同边分别相乘，所得不等式与原不等式同向 |
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**二、区间 (Intervals)**
1. **定义及表示 (Definition and representation):**
设 a, b 为任意实数, 且 `a < b`。我们规定:
* **闭区间 (Closed Interval):** 满足不等式 `a ≤ x ≤ b` 的实数 x 的集合叫闭区间, 表示为 `[a, b]`.
* **开区间 (Open Interval):** 满足不等式 `a < x < b` 的实数 x 的集合叫开区间, 表示为 `(a, b)`.
* **左半开区间 (Left-Half-Open Interval):** 满足不等式 `a < x ≤ b` 的实数 x 的集合叫左半开区间, 表示为 `(a, b]` (or `(-∞, b]`).
* **右半开区间 (Right-Half-Open Interval):** 满足不等式 `a ≤ x < b` 的实数 x 的集合叫右半开区间, 表示为 `[a, b)` (or `[a, +∞)`).
2. **画法 (Drawing method):**
各种区间表示的集合如下: **注意端点的取舍!** (Note the inclusion/exclusion of endpoints!)
| 不等式 (Inequality) | 数轴表示 (Number Line Representation) | 集合表示 (Set Notation) | 区间表示 (Interval Notation) |
| :------------------ | :----------------------------------------------------------------------------------------------------------- | :-------------------------- | :--------------------------- |
| `a ≤ x ≤ b` | Line with solid circle at `a` and `b`, shaded segment between `a` and `b`. Labels: `a`, `b`, `x`. | `{x | a ≤ x ≤ b}` | `[a, b]` (闭区间 - Closed Interval) |
| `a < x < b` | Line with hollow circle at `a` and `b`, shaded segment between `a` and `b`. Labels: `a`, `b`, `x`. | `{x | a < x < b}` | `(a, b)` (开区间 - Open Interval) |
| `a ≤ x < b` | Line with solid circle at `a` and hollow circle at `b`, shaded segment between `a` and `b`. Labels: `a`, `b`, `x`. | `{x | a ≤ x < b}` | `[a, b)` (右半开区间 - Right-Half-Open Interval) |
| `a < x ≤ b` | Line with hollow circle at `a` and solid circle at `b`, shaded segment between `a` and `b`. Labels: `a`, `b`, `x`. | `{x | a < x ≤ b}` | `(a, b]` (左半开区间 - Left-Half-Open Interval) |
| `x ≥ a` | Line with solid circle at `a`, shaded segment extending to the right. Labels: `a`, `x`. | `{x | x ≥ a}` | `[a, +∞)` |
| `x > a` | Line with hollow circle at `a`, shaded segment extending to the right. Labels: `a`, `x`. | `{x | x > a}` | `(a, +∞)` |
| `x ≤ b` | Line with solid circle at `b`, shaded segment extending to the left. Labels: `b`, `x`. | `{x | x ≤ b}` | `(-∞, b]` |
| `x < b` | Line with hollow circle at `b`, shaded segment extending to the left. Labels: `b`, `x`. | `{x | x < b}` | `(-∞, b)` |
| `R` | Line with no specific points, implying entire number line is shaded. Labels: `0`, `x`. | `R` (Real Numbers) | `(-∞, +∞)` |
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**三、不等式的解法 (Methods for Solving Inequalities)**
1. **一元二次不等式 (Quadratic Inequalities):**
Only contains one unknown, and the highest power of the unknown is 2. This is called a quadratic inequality. The general form is `ax^2 + bx + c > 0`, where `a ≠ 0`. ">" can also be replaced with "<", "≥" and "≤".
Solving quadratic inequalities usually involves two methods:
**方法一 (Method 1): 利用因式分解法求解 (Solving by Factoring).**
The basis for equivalent transformations is:
* `ab > 0 ⇔ {a > 0, b > 0} 或 {a < 0, b < 0}`
* `ab < 0 ⇔ {a > 0, b < 0} 或 {a < 0, b > 0}`
**说明 (Explanation):** The steps for solving quadratic inequalities using the factoring method are: first, transform the given inequality so that the right side becomes 0, then factor the left side, and then use the above basis for equivalent transformations to convert it into a system of two linear inequalities, and solve them.
**方法二 (Method 2): 利用一元二次函数性质求解 (Solving by using properties of quadratic functions).**
Assume the inequality `ax^2 + bx + c > 0` and the equation `ax^2 + bx + c = 0` (a, b, c are constants, and `a > 0`).
If the coefficient of `x^2`, `a`, is less than 0, multiply both sides of the inequality by -1, making the quadratic function `f(x) = ax^2 + bx + c (a > 0)`'s graph an upward-opening parabola, then this property can be applied.
Let's assume `a > 0`. The relationship between the function `f(x) = ax^2 + bx + c`, the equation `ax^2 + bx + c = 0`, and the inequality `ax^2 + bx + c > 0` (or `< 0`) is as follows:
| Discriminant | `Δ > 0` | `Δ = 0` | `Δ < 0` |
| :------------- | :-------------------------------------------------------------------------- | :---------------------------------------------------------------------------- | :----------------------------------------------- |
| `Δ = b^2 - 4ac` | | | |
| 二次函数 `y = ax^2 + bx + c (a > 0)` 的图象 (Graph of quadratic function) | Parabola opening upwards, intersecting x-axis at two distinct points `x1` and `x2`. Origin (0) shown on x-axis. | Parabola opening upwards, tangent to x-axis at one point `x1=x2`. Origin (0) shown on x-axis. | Parabola opening upwards, entirely above x-axis. Origin (0) shown on x-axis. |
| 一元二次方程 `ax^2 + bx + c = 0 (a > 0)` 的根 (Roots of quadratic equation) | `x_{1,2} = (-b ± √(b^2 - 4ac)) / (2a)` (其中 `x1 < x2`) | `x1 = x2 = -b / (2a)` | 无实根 (No real roots) |
| `ax^2 + bx + c > 0 (a > 0)` 的解集 (Solution set) | `{x | x < x1 或 x > x2}` | `{x | x ≠ -b / (2a)}` | `R` (All real numbers) |
| `ax^2 + bx + c < 0 (a > 0)` 的解集 (Solution set) | `{x | x1 < x < x2}` | `Ø` (Empty set) | `Ø` (Empty set) |
**说明 (Explanation):** The steps for solving quadratic inequalities using the properties of quadratic functions are: first, find the roots of the corresponding quadratic equation, then determine the solution set of the quadratic inequality based on the graph. This method embodies the mathematical idea of combining number and shape, and fundamentally reveals the relationship between quadratic equations, quadratic functions, and quadratic inequalities. Especially when the solution set of a quadratic inequality is a special set (such as R, Ø, etc.), this method is more intuitive and concise.
2. **含绝对值的不等式 (Inequalities involving absolute values):**
Inequalities containing unknowns within an absolute value are called inequalities involving absolute values.
Basis for solving inequalities involving absolute values:
**对于正实数 a, 有 (For positive real number a, we have):**
* `|x| ≤ a ⇔ -a ≤ x ≤ a`
* `|x| > a ⇔ x < -a 或 x > a`
**说明 (Explanation):** When solving inequalities (systems) involving absolute values, special attention should be paid to the distinction and connection between "且" (and) and "或" (or). The solution set for inequalities (systems) is usually represented using intervals, which is more concise.