Linear Inequalities
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective
Understand, solve, and graphically represent linear inequalities in one and two variables using concrete balances, number lines, and coordinate half-planes.
Interactive Visualizer: Linear Inequalities & Half-Planes
Shaded region represents all coordinate points \((x, y)\) satisfying the inequality.
(a) Concrete Scenario
Consider shopping at a local market stall. If you have a maximum of Ksh 200 in your pocket and each mango costs Ksh 40, the number of mangoes \(m\) you can purchase must satisfy:
\[ 40m \le 200 \]Unlike an equation which gives a single exact number (\(m = 5\)), an inequality describes a full range of possibilities: \(m = 0, 1, 2, 3, 4,\) or \(5\) mangoes.
(b) Geometric Representation
- One Variable (Number Line): A linear inequality represents a continuous ray. We place a circle at the boundary point and shade in the direction of valid values.
- Two Variables (Cartesian Plane): An inequality such as \(y \le 2x + 1\) divides the coordinate plane into two halves separated by the boundary line \(y = 2x + 1\). All coordinate pairs \((x, y)\) on one side satisfy the inequality, creating a shaded half-plane.
(c) Algebraic Rule: The Negative Inversion
Solving linear inequalities follows the same inverse-operation steps as linear equations with one crucial distinction: multiplying or dividing both sides by a negative number reverses the inequality symbol.
Why Does the Sign Flip?
Consider the true statement \(2 < 5\). If we multiply both sides by \(-1\), we get \(-2\) and \(-5\). On the number line, \(-2\) is to the right of \(-5\), which means \(-2 > -5\). The order of values flips when reflected across zero!
Key Formulas
1. Basic Inequality Rules
\[ ax \le b \implies x \le \frac{b}{a} \quad \text{(when } a > 0\text{)} \]\[ -ax \le b \implies x \ge -\frac{b}{a} \quad \text{(when dividing by a negative number)} \]2. Two-Step Linear Inequalities
\[ ax + b > c \implies ax > c - b \implies x > \frac{c - b}{a} \quad (a > 0) \]3. Boundary Lines & Symbols
| Symbol | Meaning | Number Line | 2D Graph Line |
|---|---|---|---|
| \(<, >\) | Strictly less / greater than | Open circle \(\circ\) | Dashed line (\(---\)) |
| \(\le, \ge\) | Less/greater than or equal to | Solid dot \(\bullet\) | Solid line (\(\bf{\text{---}}\)) |
4. Half-Plane Shading Test
For \(y < mx + c\) or \(Ax + By \le C\), test the point \((0,0)\):
- If \((0,0)\) produces a true statement, shade the half-plane containing \((0,0)\).
- If \((0,0)\) produces a false statement, shade the opposite half-plane.
Worked Examples
Example 1 (Easy): Solving a Two-Step Linear Inequality
Solve the inequality \(3x - 5 < 7\) and express the solution set.
- Isolate the variable term: Add 5 to both sides:\[ 3x < 7 + 5 \implies 3x < 12 \]
- Divide by the coefficient: Divide both sides by 3 (positive, so keep sign):\[ x < 4 \]
- Graphical representation: Open circle at \(x = 4\), shade the ray going to the left towards \(-\infty\).
Example 2 (Medium): Inequality Involving Negative Division
Solve and graph the inequality: \(-2(x + 4) \ge 10\).
- Expand the bracket:\[ -2x - 8 \ge 10 \]
- Add 8 to both sides:\[ -2x \ge 18 \]
- Divide by \(-2\) and reverse the inequality sign:\[ x \le \frac{18}{-2} \implies x \le -9 \]
- Graphical representation: Solid closed dot at \(-9\) with arrow extending to the left.
Example 3 (Hard): Graphing a Linear Inequality in Two Variables
Determine the shaded region for the inequality \(2x + 3y < 6\) on the Cartesian plane.
- Find intercepts for the boundary line \(2x + 3y = 6\):
- When \(x = 0\): \(3y = 6 \implies y = 2\) → point \((0, 2)\)
- When \(y = 0\): \(2x = 6 \implies x = 3\) → point \((3, 0)\)
- Draw the boundary line: Since the inequality is strict (\(<\)), connect \((0,2)\) and \((3,0)\) with a dashed line.
- Perform the Test Point Test using \((0,0)\):\[ 2(0) + 3(0) = 0 < 6 \quad \text{(True)} \]
- Shade the correct region: Since \((0,0)\) satisfies the inequality, shade the entire region below/left of the line containing the origin.
Common Mistakes
Misconception 1: Forgetting to Flip the Sign on Negative Division
The Error: Solving \(-3x \le 12\) as \(x \le -4\).
Correction: Dividing or multiplying by a negative number reverses the relational order. \(-3x \le 12 \implies x \ge -4\).
Why it feels right: Standard algebra equation solving does not require any sign inversion, so students apply equation habits without thinking.
Misconception 2: Using Solid vs. Dashed Lines Incorrectly
The Error: Drawing a solid line for \(y > 3x - 1\).
Correction: Strict inequalities (\(<\) and \(>\)) do NOT include the boundary line itself, so the line must be dashed. Inclusive inequalities (\(\le\) and \(\ge\)) use solid lines.
Why it feels right: Learners naturally draw solid axes and lines in coordinate geometry and forget that the line style carries mathematical meaning.
Misconception 3: Guessing the Shading Direction Without a Test Point
The Error: Assuming that "greater than" always means shading directly "to the right" or "above" regardless of whether \(y\) or \(x\) has a negative coefficient.
Correction: In \(-2y > 4x - 6\), dividing by \(-2\) gives \(y < -2x + 3\), meaning the solution is below the line. Always substitute \((0,0)\) into the original inequality to check.
Real World
Practice