Graphing Inequalities: Solving with Graphical Methods | 不等式图像：图形法求解

📚 Graphing Inequalities: Solving with Graphical Methods | 不等式图像：图形法求解

Graphical methods turn abstract inequalities into visual regions on a coordinate plane. Instead of manipulating algebra alone, you plot the boundary curve and then decide which side of the curve satisfies the inequality.

图形法将抽象的不等式转化为坐标平面上的可视区域。你不再仅仅依赖代数变形，而是画出边界曲线，再判断曲线的哪一侧满足不等式。

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1. The Core Idea: Boundary Lines and Half-Planes | 核心思想：边界直线与半平面

For a linear inequality such as y > 2x + 1, the boundary is the straight line y = 2x + 1. This line splits the plane into two half-planes: one where y > 2x + 1 and one where y < 2x + 1.

对于像 y > 2x + 1 这样的线性不等式，边界是直线 y = 2x + 1。这条线把平面分成两个半平面：一个满足 y > 2x + 1，另一个满足 y < 2x + 1。

The boundary itself is only included when the inequality sign is ≥ or ≤. If the sign is > or <, the boundary is drawn dashed to show that points on the line are not solutions.

只有当不等号是 ≥ 或 ≤ 时，边界本身才被包含。如果符号是 > 或 <，边界应画成虚线，表示直线上的点不是解。

• Solid line for ≥ or ≤ | 实线表示 ≥ 或 ≤
• Dashed line for > or < | 虚线表示 > 或 <
• Shade the half-plane satisfying the inequality | 对满足不等式的半平面进行阴影标记

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2. Drawing a Linear Boundary Accurately | 准确绘制线性边界

To draw a straight line, find two or three points on it. For y = 2x + 1, choose x = 0 to get y = 1, and x = 2 to get y = 5. Plot these points and join them with a straight line.

要画一条直线，找到它上面的两个或三个点。对于 y = 2x + 1，取 x = 0 得到 y = 1，取 x = 2 得到 y = 5。描出这些点并用直线连接。

If the inequality is given in the form ax + by > c, it can be easier to find the intercepts. For example, 2x + 3y = 6 crosses the x-axis at (3, 0) and the y-axis at (0, 2). Connecting these two points gives the boundary.

如果不等式以 ax + by > c 的形式给出，求截距会更方便。例如，2x + 3y = 6 与 x 轴交于 (3, 0)，与 y 轴交于 (0, 2)。连接这两点就得到边界。

Intercept method: set x = 0 to find the y-intercept, then set y = 0 to find the x-intercept.

截距法：令 x = 0 求 y 截距，再令 y = 0 求 x 截距。

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3. Choosing the Correct Side: Test Points | 选择正确的一侧：测试点

Once the boundary line is drawn, select a test point that is not on the line. The origin (0, 0) is usually the easiest, provided the line does not pass through it. Substitute the coordinates into the original inequality.

画出边界线后，选择一个不在线上的测试点。原点 (0, 0) 通常最方便，前提是直线不经过它。将坐标代入原不等式。

If the inequality is satisfied, shade the side containing the test point. If it is not satisfied, shade the opposite side.

如果不等式成立，就阴影标出测试点所在的一侧；如果不成立，就阴影标出另一侧。

For example, for y > 2x + 1, test (0, 0): 0 > 1 is false, so the shaded region is the side not containing the origin.

例如，对于 y > 2x + 1，测试 (0, 0)：0 > 1 不成立，因此阴影区域是不含原点的那一侧。

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4. Shading Rules and Conventions | 阴影规则与约定

In Edexcel A-Level mathematics, you are usually asked to shade the region that satisfies all the inequalities. This is often referred to as the feasible region or the solution region.

在爱德思 A-Level 数学中，通常要求你标出满足所有不等式的区域。这个区域常被称为可行域或解区域。

There are two common conventions: shading the wanted region, or shading the unwanted regions. Always read the question carefully to see which convention is required.

有两种常见约定：阴影标出想要的区域，或者阴影标出不想要的区域。务必仔细读题，确认题目要求哪种约定。

Convention | 约定	What to shade | 阴影标出什么
Wanted region | 所求区域	Shade the region that satisfies all inequalities | 标出满足所有不等式的区域
Unwanted region | 非所求区域	Shade every region that fails at least one inequality | 标出至少违反一个不等式的所有区域

When using the “unwanted region” convention, the unshaded part left on your diagram is the final answer. This is common in linear programming questions.

使用“非所求区域”约定时，图中未标阴影的部分就是最终答案。这在线性规划问题中很常见。

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5. Systems of Linear Inequalities | 线性不等式组

Real problems usually involve several inequalities at once. For example, a set of constraints might be y ≤ 2x + 1, y ≥ 3, and x ≥ 0.

实际问题通常同时涉及多个不等式。例如，一组约束条件可能是 y ≤ 2x + 1、y ≥ 3 和 x ≥ 0。

Plot each boundary line on the same axes. Then identify, for each inequality, which side is valid. The feasible region is the overlap of all the valid half-planes.

在同一坐标平面上画出每条边界线。然后确定每个不等式的有效侧。可行域就是所有有效半平面的交集。

A point in the feasible region must satisfy every inequality simultaneously.

可行域中的点必须同时满足每一个不等式。

Use different shading directions or a single final shade to make the overlap clear. If shading unwanted regions, the answer is the unshaded polygon that remains.

使用不同的阴影方向或最终统一的阴影来清楚显示交集。如果标出非所求区域，那么答案就是剩下的未标阴影的多边形。

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6. Quadratic Inequalities: Curved Boundaries | 二次不等式：曲线边界

For inequalities involving quadratics, such as y > x² − 4x + 3, the boundary is a parabola. Draw the parabola first, then decide whether to shade above or below it.

对于涉及二次函数的不等式，如 y > x² − 4x + 3，边界是抛物线。先画出抛物线，然后判断在它上方还是下方标阴影。

Since the inequality has y on the left and the quadratic on the right, y > f(x) means shade above the curve. For y < f(x), shade below the curve.

由于不等式左侧是 y、右侧是二次函数，y > f(x) 表示在曲线上方标阴影。对于 y < f(x)，则在曲线下方标阴影。

If the quadratic is written with x on both sides, for example x² − 3x < 10, rearrange it to make one side zero first: x² − 3x − 10 < 0. Then factorise or use the graph of y = x² − 3x − 10 to find where the curve is below the x-axis.

如果二次不等式两边都含 x，例如 x² − 3x < 10，先移项使一边为零：x² − 3x − 10 < 0。然后因式分解或用 y = x² − 3x − 10 的图像找出曲线位于 x 轴下方的区间。

Factorise: x² − 3x − 10 = (x − 5)(x + 2) < 0, so −2 < x < 5.

因式分解：x² − 3x − 10 = (x − 5)(x + 2) < 0，因此 −2 < x < 5。

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7. Solving Quadratic Inequalities Using Sign Diagrams | 用符号图解二次不等式

For a quadratic inequality in one variable, a sign diagram is a quick graphical tool. Sketch the x-axis and mark the roots. Then test one x-value in each interval to determine the sign of the quadratic expression.

对于一元二次不等式，符号图是一种快捷的图形工具。在 x 轴上标出根，然后在每个区间内测试一个 x 值，判断二次表达式的符号。

For y = (x − 5)(x + 2), the roots are −2 and 5. For x < −2, both factors are negative, so the product is positive. For −2 < x < 5, one factor is negative, so the product is negative. For x > 5, both factors are positive, so the product is positive.

对于 y = (x − 5)(x + 2)，根为 −2 和 5。当 x < −2 时，两个因子都为负，乘积为正。当 −2 < x < 5 时，一个因子为负，乘积为负。当 x > 5 时，两个因子都为正，乘积为正。

The graph of a quadratic with a positive coefficient of x² is a U-shaped parabola. This means the region between the roots lies below the x-axis, and the regions outside the roots lie above it.

当 x² 系数为正时，二次函数图像是开口向上的抛物线。这意味着两根之间的区域位于 x 轴下方，两根之外位于 x 轴上方。

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8. Intersection Points and Critical Values | 交点与临界值

When solving systems of inequalities, the critical points are where the boundary lines or curves intersect. These points define the vertices of the feasible region.

解不等式组时，临界点是边界直线或曲线的交点。这些点决定可行域的顶点。

To find an intersection point, solve the two boundary equations simultaneously. For example, the lines y = 2x + 1 and y = 3 intersect when 3 = 2x + 1, so x = 1 and the point is (1, 3).

要求交点，需要联立求解两条边界方程。例如，直线 y = 2x + 1 和 y = 3 相交时，3 = 2x + 1，得到 x = 1，交点为 (1, 3)。

For a parabola and a line, set the equations equal to each other. The resulting quadratic may give two, one, or zero intersection points.

对于抛物线与直线，令两个方程相等。得到的二次方程可能有两个、一个或零个交点。

Example: y = x² − 1 and y = x + 1 ⇒ x² − 1 = x + 1 ⇒ x² − x − 2 = 0 ⇒ x = 2 or x = −1.

例：y = x² − 1 与 y = x + 1 ⇒ x² − 1 = x + 1 ⇒ x² − x − 2 = 0 ⇒ x = 2 或 x = −1。

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9. Common Pitfalls and Exam Tips | 常见错误与考试提示

One of the most common mistakes is using a solid line for a strict inequality. Remember that > and < mean the boundary is not included, so the line must be dashed.

最常见的错误之一是对严格不等式使用实线。请记住，> 和 < 表示边界不包含在内，因此必须使用虚线。

Another mistake is shading the wrong side. Always test a point, and never assume that “above” is always correct. For horizontal lines, y > k means above the line, but for slanted or curved boundaries, a test point is safer.

另一个错误是标错阴影侧。一定要测试一个点，不要假设“上方”总是正确。对于水平线，y > k 表示线上方，但对于倾斜或曲线边界，使用测试点更稳妥。

• Always label the boundary equation on your graph | 始终在图上标出边界方程
• Use a ruler for straight lines | 画直线时使用直尺
• Show your test point working clearly | 清晰展示测试点的计算过程
• Check whether the question asks for integer solutions, maximum values, or just the region | 检查题目要求的是整数解、最大值，还是仅需画出区域

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10. Worked Example: Feasible Region | 综合例题：可行域

Question: Shade the region satisfying y ≥ x − 1, y ≤ 4, and y ≥ 0. State the coordinates of all vertices.

题目：标出满足 y ≥ x − 1、y ≤ 4 和 y ≥ 0 的区域，并写出所有顶点的坐标。

Step 1: Draw the line y = x − 1 as a solid line because the inequality is ≥. Test (0, 0): 0 ≥ −1 is true, so shade the side containing the origin.

第 1 步：因为不等式是 ≥，将 y = x − 1 画成实线。测试 (0, 0)：0 ≥ −1 成立，因此标出含原点的一侧。

Step 2: Draw y = 4 as a solid horizontal line. The inequality y ≤ 4 means the region below this line.

第 2 步：画水平实线 y = 4。不等式 y ≤ 4 表示这条线以下的区域。

Step 3: Draw y = 0 as the x-axis. The inequality y ≥ 0 means the region above the x-axis.

第 3 步：画 x 轴即 y = 0。不等式 y ≥ 0 表示 x 轴上方的区域。

Step 4: Find the intersections. The line y = x − 1 meets y = 0 when x = 1, giving (1, 0). The line y = x − 1 meets y = 4 when x = 5, giving (5, 4). The vertical boundary at x = 0 gives (0, 4) and (0, 0) from the axes.

第 4 步：求交点。y = x − 1 与 y = 0 相交时 x = 1，得到 (1, 0)。y = x − 1 与 y = 4 相交时 x = 5，得到 (5, 4)。x = 0 与两轴给出 (0, 4) 和 (0, 0)。

Therefore the feasible region is a quadrilateral with vertices (0, 0), (1, 0), (5, 4), and (0, 4).

因此可行域是一个四边形，顶点为 (0, 0)、(1, 0)、(5, 4) 和 (0, 4)。

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11. Linear Programming Context | 线性规划背景

Graphical inequality solving is central to linear programming. You may be given an objective such as maximise P = 3x + 2y subject to a set of constraints expressed as inequalities.

图形法解不等式是线性规划的核心。题目可能给出目标，例如在满足一组用不等式表示的约束条件下，最大化 P = 3x + 2y。

After drawing the feasible region, evaluate the objective function at each vertex. The optimal value occurs at one of the vertices, unless the objective line is parallel to a boundary, in which case the whole edge is optimal.

画出可行域后，在每个顶点处计算目标函数的值。最优点出现在某个顶点上，除非目标直线平行于某条边界，此时整条边都是最优点。

Key rule: For a linear objective over a convex polygon, the optimum is at a vertex.

关键规则：在线性目标与凸多边形约束下，最优点位于顶点。

Always check whether the question asks for integer coordinates. Sometimes the optimum vertex has non-integer coordinates, and you must test nearby integer points inside the feasible region.

始终检查题目是否要求整数坐标。有时最优点不是整数坐标，你需要测试可行域内附近的整数点。

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12. Final Checklist | 最终检查清单

Before moving on, quickly check: Is the boundary line dashed or solid? Have I tested a point? Is my shading visibly clear? Have I labelled every boundary? Have I stated all required intersection points?

继续之前，快速检查：边界线是虚线还是实线？我是否测试了一个点？我的阴影是否清晰可见？是否标出了所有边界？是否写出了所有要求的交点？

Inequality | 不等式	Boundary style | 边界样式	Shading side | 阴影侧
y > f(x)	Dashed | 虚线	Above curve | 曲线上方
y ≥ f(x)	Solid | 实线	Above curve | 曲线上方
y < f(x)	Dashed | 虚线	Below curve | 曲线下方
y ≤ f(x)	Solid | 实线	Below curve | 曲线下方

Graphical methods will help you develop spatial intuition and solve inequality problems efficiently. Practise with past paper questions until shading and vertex-finding become automatic.

图形法能帮助你建立空间直觉，并高效解决不等式问题。通过练习历年真题，直到画阴影和求顶点变得熟练自然。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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