📚 Exercise 3D: Quadratic Inequalities | Exercise 3D：二次不等式
In the AQA A-Level Mathematics curriculum, Chapter 3 focuses on equations and inequalities. Exercise 3D builds on the foundation of solving quadratic equations to introduce one of the most frequently tested skills in Pure Mathematics: solving quadratic inequalities. This exercise bridges algebraic manipulation and graphical reasoning, and mastering it is essential for success in later topics such as optimisation, integration and numerical analysis.
在 AQA 进阶数学课程中，第三章聚焦于方程与不等式。Exercise 3D 在解二次方程的基础上，引入了纯数学中考查频率最高的技能之一：解二次不等式。本练习将代数运算与图像推理相结合，掌握它对于后续的优化问题、积分和数值分析等专题至关重要。
1. What Is a Quadratic Inequality? | 什么是二次不等式？
are constants and a ≠ 0. The solution set of such an inequality is usually an interval, or the union of two intervals, on the real number line — not a single value or a pair of values as with a quadratic equation.
二次不等式是涉及二次表达式的不等式。其标准形式可写作 ax² + bx + c > 0、ax² + bx + c ≥ 0、ax² + bx + c < 0 或 ax² + bx + c ≤ 0，其中 a、b、c 为常数且 a ≠ 0。此类不等式的解集通常是实数轴上的一段区间，或两段区间的并集——而非像二次方程那样是一个值或一对值。
hat makes the quadratic expression positive. The roots 2 and 3 are merely the boundary points that separate positive regions from negative regions.
例如，不等式 x² − 5x + 6 > 0 要求的并不是方程 x² − 5x + 6 = 0 的根，而是使该二次表达式为正的所有实数 x。根 2 和 3 仅仅是分隔正区域与负区域的边界点。
2. Rearranging into Standard Form | 整理为标准形式
Before solving a quadratic inequality, always collect all non-zero terms on one side of the inequality sign. For instance, to solve x² < 2x + 3, subtract 2x + 3 from both sides to obtain x² − 2x − 3 < 0. Only when the inequality is in standard form can you reliably identify the corresponding quadratic equation and its roots.
在求解二次不等式之前，务必将所有非零项移到不等式的一侧。例如，解 x² < 2x + 3 时，两边同时减去 2x + 3，得到 x² − 2x − 3 < 0。只有当不等式处于标准形式时，才能可靠地确定对应的二次方程及其根。
If the coefficient of x² is negative, multiply both sides of the inequality by −1 and reverse the inequality sign. This step is critical: multiplying or dividing an inequality by a negative number flips the direction of the inequality. After this step, the leading coefficient is positive, which makes the graph a U-shaped parabola and allows you to apply the standard sign rules without confusion.
若 x² 的系数为负数，则将不等式两边同时乘以 −1，并反向不等式号。这一步至关重要：不等式两边乘以或除以负数时，不等式方向必须反转。完成此步后，首项系数为正，图像为开口向上的抛物线，此时可以放心应用标准的符号判定规则。
Example: −3x² + 6x < 0 → 3x² − 6x > 0
Here the direction of the inequality has been reversed because both sides were multiplied by −1. The new quadratic 3x² − 6x has a positive leading coefficient and is much easier to analyse graphically.
此处两边同时乘以 −1，因此不等式方向发生了反转。新的二次表达式 3x² − 6x 具有正的首项系数，从图像角度分析要容易得多。
3. Step 1: Solve the Corresponding Equation | 第一步：解对应方程
ts where the graph of y = ax² + bx + c crosses or touches the x-axis, and hence where the sign of the quadratic expression can change.
将不等式号换成等号，并解出相应的二次方程。因式分解、二次求根公式或配方法均可使用。方程的两个根通常记为 x₁ 和 x₂，称为不等式的临界值。这些值标记了 y = ax² + bx + c 的图像穿过或与 x 轴相切的位置，也就是二次表达式符号可能发生变化的点。
Consider the inequality x² − 5x + 6 > 0. Begin by solving:
考虑不等式 x² − 5x + 6 > 0。首先求解：
x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x₁ = 2, x₂ = 3
ntervals, the sign of x² − 5x + 6 remains constant, so testing one value per interval tells you the sign for the whole interval.
临界值为 2 和 3。这两个点将实数轴分成三个互不相交的区间：(−∞, 2)、(2, 3) 和 (3, ∞)。在每个区间内，x² − 5x + 6 的符号保持恒定，因此每段区间内取一个测试值，即可判断整段区间的符号。
4. Step 2: Sketch or Construct a Sign Diagram | 第二步：画图或绘制符号图
d parabola. If the equation has two distinct roots x₁ and x₂ (with x₁ < x₂), the curve lies below the x-axis between the roots and above the x-axis outside the roots. This simple observation gives you the sign pattern directly: negative in the middle, positive on both ends.
当首项系数为正时，y = ax² + bx + c 的图像是开口向上的抛物线。若方程有两个不同的根 x₁ 和 x₂（且 x₁ < x₂），则曲线在两根之间位于 x 轴下方，在两根之外位于 x 轴上方。这一直观观察直接给出了符号模式：中间为负，两端为正。
A sign diagram achieves the same result without drawing a full graph. Mark the critical values in increasing order on a number line, then substitute a convenient test value from each interval into the quadratic expression to determine its sign.
符号图无需画出完整图像即可达到同样效果。在数轴上按递增顺序标出临界值，然后从每个区间选取一个方便的测试值代入二次表达式，判断其符号。
| Interval | (−∞, 2) | (2, 3) | (3, ∞) |
| Test value | x = 0 | x = 2.5 | x = 4 |
| x² − 5x + 6 | 6 | −0.25 | 2 |
| Sign | + | − | + |
2, 3). This table-based method is quick, accurate and easy to reproduce under exam conditions.
符号图清楚地显示，二次表达式在 (−∞, 2) 和 (3, ∞) 上为正，在 (2, 3) 上为负。这种表格化的方法快速、准确，而且在考试条件下易于复现。
5. Step 3: Read Off the Solution Set | 第三步：读出解集
(3, ∞), so the solution is x < 2 or x > 3. In set notation this is written as {x : x < 2} ∪ {x : x > 3}, or equivalently (−∞, 2) ∪ (3, ∞).
一旦知道每个区间内二次函数的符号，选取满足原不等式的区间即可。对于 x² − 5x + 6 > 0，二次函数在 (−∞, 2) 和 (3, ∞) 上为正，所以解为 x < 2 或 x > 3。用集合记号写作 {x : x < 2} ∪ {x : x > 3}，等价地写作 (−∞, 2) ∪ (3, ∞)。
For x² − 5x + 6 < 0, the quadratic is negative on (2, 3), so the solution is 2 < x < 3, or simply (2, 3). Notice that for a positive leading coefficient, the general rule is always the same: "greater than zero" gives two outside intervals, while "less than zero" gives one middle interval between the roots.
对于 x² − 5x + 6 < 0，二次函数在 (2, 3) 上为负，所以解为 2 < x < 3，简写为 (2, 3)。注意，对于正的首项系数，一般规律始终一致："> 0" 给出两段外部区间，而 "< 0" 给出两根本身之间的一段中间区间。
Summary: if a > 0 and x₁ < x₂ are the roots, then ax² + bx + c > 0 ⇔ x < x₁ or x > x₂; and ax² + bx + c < 0 ⇔ x₁ < x < x₂.
This summary can be memorised as a quick reference, but always check the leading coefficient sign first. If a < 0, multiply by −1 before applying the rule.
可将这一总结作为快捷参考加以记忆，但务必先确认首项系数的符号。若 a < 0，先乘以 −1 再套用规则。
6. Strict vs Non-Strict Inequalities | 严格与非严格不等式
uded, so use a closed circle on a number line. In interval notation, round brackets ( ) indicate excluded endpoints and square brackets [ ] indicate included endpoints.
对于严格不等式 > 和 <，临界值不包含在解集之内。在数轴上，排除的端点用空心圆表示。对于非严格不等式 ≥ 和 ≤，临界值包含在解集中，在数轴上用实心圆表示。在区间记号中，圆括号 ( ) 表示端点被排除，方括号 [ ] 表示端点被包含。
Compare the four possibilities for x² − 5x + 6:
比较 x² − 5x + 6 的四种情况：
| Inequality | 广告 / Sponsored ALevel国际高中数学课程 剑桥牛津培生 CAIE Edexcel AQA国际考试局 课本练习册复习指南 英文教材进… 数学 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥69.8
更多咨询请联系16621398022（同微信） 备考资讯每日更新，关注不错过 考纲变化 · 科目分析 · 学习方法  在微信里长按二维码识别关注 或截图后用「扫一扫 → 相册」选择本图 公众号：tutorhao |