Solving Inequalities- Methods for Linear and Quadratic Cases
What Inequalities Actually Are
An inequality is a mathematical statement that shows the relationship between two expressions using symbols like <, >, ≤, or ≥. Unlike equations, which have exact solutions, inequalities describe a range of possible values.
Most students struggle with inequalities because they expect a single answer. You won't get one. You're looking for all values that make the statement true.
This guide covers the two cases you'll encounter most often: linear inequalities and quadratic inequalities. Nothing else.
Solving Linear Inequalities
Linear inequalities involve variables raised to the first power only. The process mirrors solving linear equations, except for one critical rule that trips up most people.
The One Rule That Changes Everything
When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality symbol. That's it. That's the gotcha.
Example: -2x > 6
Divide both sides by -2:
x < -3
Notice the symbol flipped from > to <. Forget this step and every answer you get will be wrong.
Step-by-Step Process
For any linear inequality, follow this sequence:
- Simplify both sides if needed (distribute, combine like terms)
- Collect variable terms on one side
- Isolate the variable using addition/subtraction
- Multiply or divide, flipping the symbol if you use a negative number
- Express your answer in interval notation or as a solution set
Example: Solving 3x - 7 < 2x + 5
Subtract 2x from both sides:
x - 7 < 5
Add 7 to both sides:
x < 12
Answer: x < 12, or in interval notation, (-∞, 12)
Solving Quadratic Inequalities
Quadratic inequalities introduce a second critical concept: sign analysis. The variable is squared, which means you often have two critical points where the expression equals zero.
The goal is to find where the quadratic expression is positive or negative across different intervals.
The Method: Find Roots, Test Intervals
Here's what you actually do:
- Rewrite the inequality as an equation and solve for roots
- Plot the roots on a number line
- Pick a test point from each interval
- Determine which intervals satisfy the original inequality
- Check whether endpoints are included (depends on ≥ or ≤)
Example: Solving x² - 4 > 0
Set equal to zero: x² - 4 = 0
Solve: (x - 2)(x + 2) = 0
Roots: x = 2 and x = -2
Number line has three intervals: (-∞, -2), (-2, 2), (2, ∞)
Test points: x = -3: (-3)² - 4 = 5 > 0 ✓
x = 0: 0² - 4 = -4 < 0 ✗
x = 3: 3² - 4 = 5 > 0 ✓
Solution: x < -2 or x > 2, or in interval notation: (-∞, -2) ∪ (2, ∞)
When the Quadratic Doesn't Factor
Use the quadratic formula when factoring fails:
x = (-b ± √(b² - 4ac)) / 2a
The discriminant (b² - 4ac) tells you what you're dealing with:
- Positive discriminant: two real roots
- Zero discriminant: one repeated root
- Negative discriminant: no real roots
If there's no real root and the quadratic opens upward, the expression is always positive (or always negative depending on the sign).
Linear vs. Quadratic: Quick Comparison
| Feature | Linear Inequality | Quadratic Inequality |
|---|---|---|
| Variable power | First power only | Second power |
| Number of solutions | Half the number line | Can be two intervals |
| Method complexity | Straightforward algebra | Requires sign analysis |
| Critical step | Flip symbol with negatives | Test intervals between roots |
Getting Started: Your Action Checklist
Before you solve any inequality, do this:
- Identify the type — Is it linear or quadratic? This determines your approach.
- Clear denominators — Multiply both sides by the least common denominator if fractions are present. Watch out for sign changes.
- Simplify first — Combine like terms, distribute, and get everything on one side before solving.
- Check your symbol direction — Did you flip when you needed to? Double-check before moving on.
- Verify your answer — Plug a value from your solution back into the original inequality.
Common Mistakes That Will Cost You Points
These errors appear constantly in homework and exams:
- Forgetting to flip the inequality symbol when multiplying or dividing by negatives
- Excluding endpoints when the inequality uses ≤ or ≥
- Including endpoints when the inequality uses strict < or >
- Misidentifying intervals in quadratic problems — test points, don't guess
- Algebra errors when simplifying before solving
Solving Systems of Inequalities
Sometimes you'll get two or more inequalities together. The solution is the overlap where all conditions are satisfied simultaneously.
Graph each inequality on the coordinate plane. Shade the region that satisfies each one. Your answer is where the shaded regions intersect.
For linear systems, this intersection forms a polygon (usually a triangle or quadrilateral). For quadratic systems, you get curved boundary regions that can be tricky to visualize.
When to Use Interval Notation
Interval notation is cleaner than inequality chains. Here's the translation:
- x > a becomes (a, ∞)
- x ≥ a becomes [a, ∞)
- x < b becomes (-∞, b)
- x ≤ b becomes (-∞, b]
- a < x < b becomes (a, b)
Square brackets mean the endpoint is included. Parentheses mean it's not. Always match the notation to the inequality symbol.