Inequalities
An inequality is like an equation but uses <, >, ≤, or ≥ instead of =. Solving is similar — isolate the variable — with one crucial difference: multiplying or dividing by a negative number flips the inequality sign.
Key Concepts
Solving Inequalities
2x − 3 > 7 → add 3 → 2x > 10 → divide by 2 → x > 5. The solution is all values greater than 5. Graph it on a number line: open circle at 5 (not included), arrow pointing right.
Flipping the Sign
−3x < 12 → divide both sides by −3 → flip the sign → x > −4. Test: pick x = 0. Does −3(0) < 12? Yes ✓. The flip rule is the most common mistake in inequalities.
Compound Inequalities
−2 < x + 1 ≤ 5. Subtract 1 from all three parts: −3 < x ≤ 4. The solution is all x between −3 (exclusive) and 4 (inclusive). Graph: open circle at −3, closed circle at 4, shaded between.
🆕 Inequality Number Line
Solve an inequality and see it graphed on a number line!
✅ Check Your Understanding
1. When does the inequality sign flip?
2. Solve: x + 3 > 10. Solution is...
3. An open circle on a number line means the value is...