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# Inequalities lesson 4

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• Week 5 Day 1
• Week 5 Day 1
• Week 5 Day 1
• Week 5 Day 1
• ### Inequalities lesson 4

1. 1. INEQUALITIES <br />MATH10 <br />ALGEBRA<br />Inequalities(Algebra and Trigonometry, Young 2nd Edition, page 136-170) <br />
2. 2. Week 5 Day 1<br />Week 5 Day 1<br />GENERAL OBJECTIVE<br />At the end of the chapter the students are expected to:<br /><ul><li> Use interval notation.
3. 3. Solve linear and nonlinear inequalities.
4. 4. Solve application problems involving linear inequalities.</li></li></ul><li>Week 5 Day 1<br />TODAY’S OBJECTIVE<br />At the end of the lesson the students are expected to:<br /><ul><li>To identify an inequality.
5. 5. To classify inequalities as absolute or conditional.
6. 6. To use interval and set notation in writing solutions to inequalities.
7. 7. To represent graphically the solution to inequalities.
8. 8. Toapply intersection and union concepts in solving compound inequalities.
9. 9. To solve linear and fractional inequalities.
10. 10. Understand that linear inequalities have one solution, no solution, or an interval solution.</li></li></ul><li>DEFINITION<br />Week 5 Day 1<br />INEQUALITIES<br />Let 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: <br />A statement that one quantity is greater than or less than another quantity is called an INEQUALITY. <br />
11. 11. KINDS OF INEQUALITIES<br />Week 5 Day 1<br />Absolute inequalities are inequalities which is true for all values of x.<br /> Example: <br /><ul><li> Conditional inequalities are inequalities which is true for certain values of x.</li></ul> Example: <br />
12. 12. Week 5 Day 1<br />GRAPHING INEQUALITIES and INTERVAL NOTATION<br />
13. 13. Week 5 Day 1<br />FOUR WAYS OF EXPRESSING SOLUTIONS TO INEQUALITIES:<br /><ul><li> inequality notation
14. 14. set notation
15. 15. interval notation
16. 16. graphical representation</li></li></ul><li>Week 5 Day 1<br />EXAMPLE<br />)<br />or<br />[<br />0<br />b<br />a<br />a<br />0<br />b<br /><ul><li> a is the left endpoint
17. 17. b is the right endpoint
18. 18. If an inequality is a strict inequality (< or >) parenthesis is used.
19. 19. If an inequality includes an endpoint (> or <) bracket is used.</li></li></ul><li>Week 5 Day 1<br />Let x be a real number , x is ….<br />)<br />or<br />(<br />(<br />[<br />0<br />b<br />a<br />0<br />b<br />a<br />0<br />b<br />a<br />a<br />a<br />a<br />0<br />0<br />0<br />b<br />b<br />b<br />)<br />or<br />]<br />or<br />
20. 20. Week 5 Day 1<br />Let x be a real number , x is ….<br />]<br />or<br />[<br />b<br />a<br />a<br />a<br />b<br />a<br />b<br />)<br />a<br />or<br />]<br />or<br />
21. 21. Week 5 Day 1<br />Let x be a real number , x is ….<br />or<br />(<br />b<br />b<br />b<br />b<br />[<br />or<br />R<br />R<br />
22. 22. Week 5 Day 1<br /><ul><li>Infinity is not a number. It is a symbol that means continuing indefinitely to the right on the number line.
23. 23. Negative infinity means continuing indefinitely to the left on the number line.
24. 24. In interval notation, the lower number is always written on the left.</li></li></ul><li>Week 5 Day 1<br />Example 1<br />(-∞,4)<br />x < 4<br />)<br />○<br />0<br />4<br />-4<br />0<br />4<br />-4<br />
25. 25. Week 5 Day 1<br />Example 2<br />x ≤ 4<br /> (-∞,4]<br />]<br />●<br />0<br />4<br />-4<br />0<br />4<br />-4<br />
26. 26. Week 5 Day 1<br />Example 3<br />x > 4<br />(4, +∞)<br />0<br />4<br />-4<br />0<br />4<br />-4<br />(<br />○<br />
27. 27. Week 5 Day 1<br />Example 4<br />x ≥4<br /> [4, +∞)<br />0<br />4<br />-4<br />0<br />4<br />-4<br />[<br />●<br />
28. 28. Week 5 Day 1<br />EXAMPLE 5<br />)<br />or<br />[<br />4<br />-1<br />4<br />-1<br />
29. 29. Week 5 Day 1<br />EXAMPLE 5<br />]<br />or<br />[<br />4<br />0<br />4<br />0<br />
30. 30. Week 5 Day 1<br />Example 6:<br />Classroom example 1.5.1 page 137<br />Express the following as an inequality and an interval. <br />x is less than -1<br />x is greater than or equal to 3<br />x is greater than -2 and less than or equal to 7.<br />
31. 31. DEFINITION<br />Week 5 Day 1<br />UNION AND INTERSECTION<br />
32. 32. Week 5 Day 1<br />DOUBLE OR COMBINED INEQUALITY<br />A 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.<br />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.<br />
33. 33. Example<br />Week 5 Day 1<br />
34. 34. Week 5 Day 1<br />SOLVING LINEAR INEQUALITIES <br />
35. 35. SOLVING LINEAR INEQUALITIES <br />Week 5 Day 1<br />Linear inequalities are solved using the same procedure as linear <br />equations with the following exception:<br /><ul><li> When you multiply or divide by a negative number, you</li></ul> must reverse the inequality sign.<br /><ul><li> Cross multiplication cannot be used with inequalities. </li></li></ul><li>INEQUALITY PROPERTIES<br />Week 5 Day 1<br />1. Simplifying by eliminating parentheses and collecting like terms.<br />2. Adding or subtracting the same quantity on both sides.<br />3.Multiplying or dividing by the same positive number.<br />
36. 36. INEQUALITY PROPERTIES<br />Week 5 Day 1<br />1. Interchanging the two sides of the inequality<br />2.Multiplying or dividing by the same negative number.<br />
37. 37. SOLVING A LINEAR INEQUALITY <br />Week 5 Day 1<br />Example<br />
38. 38. SOLVING A LINEAR INEQUALITIES WITH FRACTION <br />Week 5 Day 1<br />Example<br />Note: Common mistake is using cross multiplication to solve fractional <br /> inequalities. <br />
39. 39. SOLVING A DOUBLE OR COMPOUND LINEAR INEQUALITY <br />Week 5 Day 1<br />Example<br />
40. 40. SUMMARY<br />Week 5 Day 1<br /><ul><li>The solution to linear inequalities are solution sets that can be </li></ul>expressed in four ways:<br /> Inequality notation<br />Set Notation<br />Interval Notation<br />Graph (number line)<br /><ul><li>Linear inequalities are solved using the same procedures as </li></ul> linear equations with the following exception: <br /> when you multiply or divide by a negative number you <br /> must reverse the inequality sign<br />cross multiplication cannot be used with inequalities.<br />
41. 41. Week 5 Day 2<br />NON LINEAR INEQUALITIES IN ONE VARIABLE<br />
42. 42. TODAY’S OBJECTIVE<br />Week 5 Day 2<br />At the end of the lesson the students are expected to:<br /><ul><li> Tosolve quadratic inequalities.
43. 43. To solve polynomial inequalities.
44. 44. To solve rational inequalities.
45. 45. To solve absolute value inequalities
46. 46. To solve application problems involving inequalities .</li></li></ul><li>Week 5 Day 2<br />POLYNOMIAL INEQUALITIES<br /><ul><li>Zeros of a polynomial are the values of x that make the polynomial </li></ul> equal to zero.<br /><ul><li>These zeros divide the real number line into test intervals where the</li></ul> the value of the polynomial is either positive or negative.<br />STEPS:<br />Write inequality in standard form (zero on one side).<br />Identify zeros (factor if possible otherwise use quadratic formula)<br />Draw the number line with zeros labeled.<br />Determine the sign of the polynomial in each interval.<br />Identify which interval(s) make the inequality true.<br />Write the solution in interval notation.<br />
47. 47. Week 5 Day 2<br />SOLVING QUADRATIC INEQUALITY<br /><ul><li>The square root method cannot be used for quadratic inequalities.
48. 48. Dividing both sides by the variable (x) cannot be used for quadratic inequalities</li></ul>Common mistakes:<br /><ul><li>Taking the square root of both sides.
49. 49. Dividing both sides by the variable (x).</li></li></ul><li>Week 5 Day 2<br />SOLVING QUADRATIC INEQUALITY<br />Solve each quadratic inequality:<br />
50. 50. Week 5 Day 2<br />SOLVING A POLYNOMIAL INEQUALITY<br />Solve each inequality:<br />
51. 51. Week 5 Day 2<br />SOLVING A RATIONAL INEQUALITY<br /><ul><li> A rational expression have numerators and denominators , thus the</li></ul> we have the following possible combinations:<br /><ul><li>To solve rational inequalities we use a similar procedure for solving </li></ul> polynomial inequalities, with one exception. You must eliminate <br /> values for x that make the denominator equal to zero.<br /><ul><li> Once expressions are combined into a single fraction the value that </li></ul> make either the numerator or the denominator equal to zero divide <br /> the number line into intervals. <br />
52. 52. Week 5 Day 2<br />SOLVING A RATIONAL INEQUALITY<br />STEPS:<br />Write inequality in standard form (zero on one side).<br />Identify zeros .<br /><ul><li> Write as a single fraction
53. 53. Determine values that make the numerator or denominator equal to zero
54. 54. Always exclude values that make the denominator = 0. </li></ul>Draw the number line with zeros labeled.<br />Determine the sign of the polynomial in each interval.<br />Identify which interval(s) make the inequality true.<br />Write the solution in interval notation.<br />
55. 55. Week 5 Day 2<br />SOLVING A RATIONAL INEQUALITY<br />Solve each inequality:<br />
56. 56. Week 5 Day 2<br />ABSOLUTE VALUE INEQUALITIES<br />PROPERTIES OF ABSOLUTE VALUE INEQUALITIES<br />
57. 57. Week 5 Day 2<br />SOLVING AN ABSOLUTE VALUE INEQUALITY<br />Solve each inequality:<br />
58. 58. APPLICATIONS INVOLVING LINEAR INEQUALITY <br />Example<br />
59. 59. APPLICATIONS INVOLVING LINEAR INEQUALITY <br />
60. 60. Week 5 Day 2<br />SUMMARY<br />The following procedure can be used for solving polynomial and <br />rational inequalities.<br />Write inequality in standard form (zero on one side).<br />Determine the zeros; if it is a rational function, note the domain restrictions. <br /><ul><li> Polynomial Inequality</li></ul> - Factor if possible, otherwise, use quadratic formula<br /><ul><li>Rational Inequality</li></ul> - Write as a single fraction<br /> - Determine values that make the numerator or denominator equal to zero<br /> -Always exclude values that make the denominator = 0. <br />Draw the number line with zeros labeled.<br />Determine the sign of the polynomial in each interval.<br />Identify which interval(s) make the inequality true.<br />Write the solution in interval notation.<br />
61. 61. Week 5 Day 2<br />CLASSWORK <br />#s 85 page 144 <br />#s 25 page 154 <br />#s 43,49 page 162<br />HOMEWORK <br />#s 69,83,86,89,91,99 pages 143-144<br />#s 10,13,19,28,38,44,55 page 154<br />#s 53,57,62 page 162<br />