Linear Inequalities

Equations say "these two things are equal." Inequalities say "these two things have a relationship" — maybe one is bigger, maybe one is at least as big, maybe one is at most another. Inequalities describe whole ranges of valid values, not just single solutions. They're the language of real-world constraints: budgets, speeds, distances, capacities.

- **>** — strictly greater than. "x > 5" means x is more than 5.\n- **<** — strictly less than. "x < 5" means x is less than 5.\n- **≥** — greater than or equal to. "x ≥ 5" includes 5.\n- **≤** — less than or equal to. "x ≤ 5" includes 5.\n\nThe difference between strict (>, <) and non-strict (≥, ≤) matters. Whether a boundary value is included or excluded changes the solution.

Solving inequalities is almost exactly like solving equations — with one crucial exception.\n\n**Regular rules**: add, subtract, multiply by positives, divide by positives — all fine, inequality stays the same.\n\n**The one exception**: when you **multiply or divide by a NEGATIVE number, flip the inequality**.\n\nExample: solve −2x + 3 > 9.\n\n- −2x > 6 (subtracted 3)\n- x < −3 (divided by −2, FLIPPED the inequality)\n\nWhy? Because multiplying by a negative reverses order. (5 > 3, but −5 < −3.) Forgetting this flip is the #1 mistake in this topic.

- **x > 5**: open circle at 5, arrow pointing right. (Open because 5 is not included.)\n- **x ≥ 5**: closed (filled) circle at 5, arrow pointing right. (Closed because 5 IS included.)\n- **x < −2**: open circle at −2, arrow pointing left.\n- **x ≤ −2**: closed circle at −2, arrow pointing left.\n\nCompound inequalities like "−3 < x ≤ 7" become line segments — open on the left, closed on the right.

In two variables (x and y), the graph of a linear inequality is a **shaded half-plane**.\n\nTo graph y < 2x + 3:\n\n1. Graph y = 2x + 3 as you would an equation. This is the **boundary**.\n2. Make the line **dashed** (because < is strict). Use **solid** if ≤.\n3. Test a point not on the line (like (0, 0)): is 0 < 2(0) + 3? Yes, 0 < 3. So shade the side containing (0, 0).\n4. Shade the correct half.\n\nAny (x, y) in the shaded region is a solution. Any outside is not.

A **system** of inequalities has multiple inequalities that must ALL be true at once. Graphically, the solution is the **overlap** of all shaded regions.\n\nExample: y ≥ x − 1 AND y ≤ −x + 3.\n\nGraph both, shade both. The overlap — usually a bounded region — contains every (x, y) that satisfies BOTH.\n\nSystems of inequalities are the foundation of **linear programming** — a real-world optimization technique used in scheduling, supply chains, finance, and machine learning. If you know the constraints as inequalities, you can find the best solution inside the feasible region.

Linear inequalities are more than a step up from equations — they're the bridge to optimization, linear programming, and the real-world math of making decisions under constraints. Every business decision, every scheduling problem, every resource allocation issue involves inequalities. Learning them well now pays dividends for the rest of math, science, economics, and computer science.