Solving Compound Linear Inequalities - Complete Guide

1. Compound Linear Inequalities

2. Identify the Symbol • Less Than • Less Than or Equal To • Greater Than • Greater Than or Equal To

3. Identify the Symbol • Less Than • Less Than or Equal To • Greater Than • Greater Than or Equal To

4. Identify the Symbol • Less Than or Equal To • Greater Than or Equal To • All Real Numbers • Not Equal To

5. Identify the Symbol • Less Than or Equal To • Greater Than or Equal To • All Real Numbers • Not Equal To

6. Conjunction • Mathematical sentences joined by “and” • Meaning: an intersection

7. Disjunction • Mathematical sentences joined by “or” • Meaning: a union

8. Which is it? • All students who have red hair and are boys. • All students who have brown hair or wear glasses.

9. x < -3 and x > 1 Where on the number line are both of these statements true?

10. Solving Conjunctions • Graph both inequalities. • Find the intersection. (overlapping portions) • Write the answer as an inequality.

11. -4 -3 0 -2 1 3 -6 -5 -1 2 Conjunction – Case 1 • x < -3 and x > 1

12. Conjunction – Case 1 • No overlap and arrows going in the opposite direction • No solutions

13. -4 -3 0 -2 1 3 -6 -5 -1 2 Conjunction – Case 2 • x > -3 and x < 1 • Both must be true. • -3 < x < 1

14. Conjunction – Case 2 • Overlapping and arrows going in the opposite direction • The solution is between the two numbers.

15. -4 -3 0 -2 1 3 -6 -5 -1 2 Conjunction – Case 3 • x < -3 and x < 1 • Both must be true. • x < -3

16. Conjunction – Case 3 • Overlapping and arrows going in the same direction. • The solution will be a single greater than/less than inequality.

17. -4 -3 0 -2 1 3 -6 -5 -1 2 Conjunction – Case 3 • x > -3 and x > 1 • Both must be true. • x > 1

18. Solving Disjunctions • Graph both inequalities. • Find the union. (Join the two graphs) • Write the answer as an inequality.

19. -4 -3 0 -2 1 3 -6 -5 -1 2 Disjunction – Case 1 • x < -3 or x > 1

20. Disjunction – Case 1 • No overlap and arrows going in the opposite direction • The solution is the original inequalities.

21. -4 -3 0 -2 1 3 -6 -5 -1 2 Disjunction – Case 2 • x > -3 or x < 1 • Either can be true. • All Real Numbers

22. Disjunction – Case 2 • Overlapping and arrows going in the opposite direction • If every part of the number line is covered at least once, then the solution is all real numbers.

23. -4 -3 0 -2 1 3 -6 -5 -1 2 Disjunction – Case 3 • x < -3 or x < 1 • Either can be true. • x < 1

24. Disjunction – Case 3 • Overlapping and arrows going in the same direction. • The solution will be a single greater than/less than inequality.

25. -4 -3 0 -2 1 3 -6 -5 -1 2 Disjunction – Case 3 • x > -3 or x > 1 • Either can be true. • x > -3

26. x > 3 or x > 1 • x > 1 • x > 3 • All real numbers • None of these

27. x > 3 and x > 1 • x > 1 • x > 3 • All real numbers • None of these

28. -1 0 3 1 4 6 -3 -2 2 5 Disjunction • x > 3 or x  0

29. -1 0 3 1 4 6 -3 -2 2 5 Conjunction • x > 3 and x  0

30. -7 -6 -3 -5 -2 0 -9 -8 -4 -1 -2x > 4 or x + 8 < 1 • Solve each inequality first! • x < -2 or x < -7 • x < -2

31. 3x – 5 < 1 and x – 5 > -3 • x = 2 • x > 2 • -2 < x < 2 • The empty set • None of these

32. -2 -1 2 0 3 5 -4 -3 1 4 3x – 5 < 1 or x – 5 > -3 • x < 2 or x > 2 • x ≠ 2

33. -2 < x + 1 < 5 • “x + 1 lies between -2 and 5.” • Always a conjunction. • Write as two separate inequalities, then solve as usual. • x + 1 > -2 and x + 1 < 5

34. -15 < 3(x – 1) < 12 • x < -4 • -4 < x < 5 • x < 5 • The empty set • None of these

35. 4x > -12 or x + 6 < 5 • x > -3 • -3 < x < -1 • All real numbers • The empty set • None of these

36. Section 2.7 • p. 74

37. -2 -1 2 0 3 5 -4 -3 1 4 Page 74 • 6.

38. Page 74 • 8. 1 < x  7 • 10. All real numbers