The document discusses inequalities and their solutions. It defines absolute and conditional inequalities and explains how to represent solutions using interval, set, and graphical notation. Methods are presented for solving linear, polynomial, rational, and absolute value inequalities. Key steps include determining intervals where an expression is positive or negative, identifying valid intervals based on inequality signs, and expressing the solution in interval notation. Examples are provided throughout to demonstrate these techniques.

1. INEQUALITIES MATH10 ALGEBRAInequalities(Algebra and Trigonometry, Young 2nd Edition, page 136-170)

2. Week 5 Day 1Week 5 Day 1GENERAL OBJECTIVEAt the end of the chapter the students are expected to: Use interval notation.

3. Solve linear and nonlinear inequalities.

4. Solve application problems involving linear inequalities.Week 5 Day 1TODAY’S OBJECTIVEAt the end of the lesson the students are expected to:To identify an inequality.

5. To classify inequalities as absolute or conditional.

6. To use interval and set notation in writing solutions to inequalities.

7. To represent graphically the solution to inequalities.

8. Toapply intersection and union concepts in solving compound inequalities.

9. To solve linear and fractional inequalities.

10. Understand that linear inequalities have one solution, no solution, or an interval solution.DEFINITIONWeek 5 Day 1INEQUALITIESLet a and b denote two real numbers such that thegraph of a on the number line is in the negative direction from the graph of b. Then we say that a is less than b and b is greater than a,or, in symbols: A statement that one quantity is greater than or less than another quantity is called an INEQUALITY.

11. KINDS OF INEQUALITIESWeek 5 Day 1Absolute inequalities are inequalities which is true for all values of x. Example: Conditional inequalities are inequalities which is true for certain values of x. Example:

12. Week 5 Day 1GRAPHING INEQUALITIES and INTERVAL NOTATION

13. Week 5 Day 1FOUR WAYS OF EXPRESSING SOLUTIONS TO INEQUALITIES: inequality notation

14. set notation

15. interval notation

16. graphical representationWeek 5 Day 1EXAMPLE)or[0baa0b a is the left endpoint

17. b is the right endpoint

18. If an inequality is a strict inequality (< or >) parenthesis is used.

19. If an inequality includes an endpoint (> or <) bracket is used.Week 5 Day 1Let x be a real number , x is ….)or(([0ba0ba0baaaa000bbb)or]or

20. Week 5 Day 1Let x be a real number , x is ….]or[baaabab)aor]or

21. Week 5 Day 1Let x be a real number , x is ….or(bbbb[orRR

22. Week 5 Day 1Infinity is not a number. It is a symbol that means continuing indefinitely to the right on the number line.

23. Negative infinity means continuing indefinitely to the left on the number line.

24. In interval notation, the lower number is always written on the left.Week 5 Day 1Example 1(-∞,4)x < 4)○04-404-4

25. Week 5 Day 1Example 2x ≤ 4 (-∞,4]]●04-404-4

26. Week 5 Day 1Example 3x > 4(4, +∞)04-404-4(○

27. Week 5 Day 1Example 4x ≥4 [4, +∞)04-404-4[●

28. Week 5 Day 1EXAMPLE 5)or[4-14-1

29. Week 5 Day 1EXAMPLE 5]or[4040

30. Week 5 Day 1Example 6:Classroom example 1.5.1 page 137Express the following as an inequality and an interval. x is less than -1x is greater than or equal to 3x is greater than -2 and less than or equal to 7.

31. DEFINITIONWeek 5 Day 1UNION AND INTERSECTION

32. Week 5 Day 1DOUBLE OR COMBINED INEQUALITYA statement formed by joining two clauses with the word and is called a conjunction. For a conjunction to be true, both clauses must be true.A statement formed by joining two clauses with the word or is called a disjunction. For a disjunction to be true, at least one of the clauses must be true.

33. ExampleWeek 5 Day 1

34. Week 5 Day 1SOLVING LINEAR INEQUALITIES

35. SOLVING LINEAR INEQUALITIES Week 5 Day 1Linear inequalities are solved using the same procedure as linear equations with the following exception: When you multiply or divide by a negative number, you must reverse the inequality sign. Cross multiplication cannot be used with inequalities. INEQUALITY PROPERTIESWeek 5 Day 11. Simplifying by eliminating parentheses and collecting like terms.2. Adding or subtracting the same quantity on both sides.3.Multiplying or dividing by the same positive number.

36. INEQUALITY PROPERTIESWeek 5 Day 11. Interchanging the two sides of the inequality2.Multiplying or dividing by the same negative number.

37. SOLVING A LINEAR INEQUALITY Week 5 Day 1Example

38. SOLVING A LINEAR INEQUALITIES WITH FRACTION Week 5 Day 1ExampleNote: Common mistake is using cross multiplication to solve fractional inequalities.

39. SOLVING A DOUBLE OR COMPOUND LINEAR INEQUALITY Week 5 Day 1Example

40. SUMMARYWeek 5 Day 1The solution to linear inequalities are solution sets that can be expressed in four ways: Inequality notationSet NotationInterval NotationGraph (number line)Linear inequalities are solved using the same procedures as linear equations with the following exception: when you multiply or divide by a negative number you must reverse the inequality signcross multiplication cannot be used with inequalities.

41. Week 5 Day 2NON LINEAR INEQUALITIES IN ONE VARIABLE

42. TODAY’S OBJECTIVEWeek 5 Day 2At the end of the lesson the students are expected to: Tosolve quadratic inequalities.

43. To solve polynomial inequalities.

44. To solve rational inequalities.

45. To solve absolute value inequalities