Linear Equations and Inequalities
A Complete Lesson for Pre-University Algebra
1. Introduction & Motivation
Linear equations and inequalities are among the most fundamental objects in all of mathematics. Whether you are computing a budget, analyzing a scientific relationship, or designing an engineering system, the ability to set up and solve linear equations and inequalities is indispensable.
- Taxi fare: A flat fee of \$3.00 plus \$2.50 per mile → $C = 2.50m + 3$
- Gym membership: \$30/month plus \$5 per class → $C = 5n + 30$
- Budget constraint: You can spend at most \$200 → $5n + 30 \leq 200$
- Production planning: A factory must produce at least 500 units to break even → $x \geq 500$
This lesson covers definitions, solving techniques, special cases, inequalities, interval notation, compound inequalities, and applications.
2. Linear Equations in One Variable
2.1 Definition
The key feature is that the variable $x$ appears only to the first power — there are no $x^2$, $\sqrt{x}$, or $\frac{1}{x}$ terms. A linear equation describes a straight line when graphed, and the variable appears with no products, quotients, or nonlinear terms such as squares or roots.
✓ Linear Equations
- $3x - 7 = 0$
- $2(x+4) = 5x - 1$
- $\frac{x}{3} + 2 = \frac{x-1}{6}$
✗ NOT Linear
- $x^2 + 3x = 5$ (quadratic)
- $\sqrt{x} = 4$ (radical)
- $\frac{1}{x} + 2 = 0$ (rational)
2.2 What Is a Solution?
Verification example: Is $x = 4$ a solution of $3x - 12 = 0$?
$3(4) - 12 = 12 - 12 = 0 \;\checkmark$ True!
2.3 General Solving Strategy
The goal is to isolate the variable on one side of the equation using the properties of equality: you may add, subtract, multiply, or divide both sides by the same nonzero number without changing the solution.
| Step | Action | Purpose |
|---|---|---|
| 1 | Clear fractions/decimals (multiply by LCD) | Simplify arithmetic |
| 2 | Distribute and combine like terms on each side | Simplify each side |
| 3 | Collect variable terms on one side, constants on the other | Separate knowns from unknowns |
| 4 | Divide both sides by the coefficient of the variable | Isolate the variable |
| 5 | Substitute back into the original equation | Verify correctness |
2.4 Visualizing a Linear Equation
A linear equation $ax + b = 0$ can be visualized as finding where the line $y = ax + b$ crosses the $x$-axis. The solution is the $x$-intercept.
Drag the sliders to change $a$ and $b$. The red point shows the solution of $ax + b = 0$.