Overview
This lecture covers how to solve compound inequalities, graph them on a number line, and express their solutions in interval notation.
Solving Compound Inequalities ('Or' and 'And')
- Compound inequalities can be joined by "or" (union of solutions) or "and" (intersection of solutions).
- To solve, isolate x in each part, performing operations step by step (addition, subtraction, multiplication, division).
- When multiplying or dividing both sides by a negative number, reverse the direction of the inequality.
Graphing Solutions on a Number Line
- Use an open circle for values not included (strict inequality: < or >).
- Use a closed circle for values included (inclusive inequality: ≤ or ≥).
- Shade left for "<" and right for ">", according to the direction of the inequality.
- For "or" inequalities, shade both intervals; for "and", shade the overlapping region.
Interval Notation for Solutions
- Use parentheses "( )" for open intervals (value not included).
- Use brackets "[ ]" for closed intervals (value included).
- Combine intervals using the union symbol "∪" for "or" situations.
- Always use parentheses for infinity: (−∞, a), (b, ∞).
Examples
- 3x + 5 < 8 or –2x + 5 ≤ 1:
- Solution: x < 1 or x ≥ 3; interval notation: (−∞, 1) ∪ [3, ∞)
- 7 ≤ 4x + 3 < 23:
- Solution: 1 ≤ x < 5; interval notation: [1, 5)
- 1/3 x + 2 ≤ 4 or 6 – (2/5)x < 2:
- Solution: x ≤ 6 or x > 10; interval notation: (−∞, 6] ∪ (10, ∞)
- 8 < (2/3)x + 6 ≤ 12:
- Solution: 3 < x ≤ 9; interval notation: (3, 9]
Key Terms & Definitions
- Compound Inequality — An inequality involving two or more inequalities joined by "and" or "or".
- Interval Notation — A way to describe sets of numbers between endpoints using parentheses and brackets.
- Open Circle — Point on a number line not included in the solution (strict inequality).
- Closed Circle — Point on a number line included in the solution (inclusive inequality).
Action Items / Next Steps
- Practice solving compound inequalities and representing their solutions on number lines and in interval notation.