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- 1. INEQUALITIES <br />MATH10 <br />ALGEBRA<br />Inequalities(Algebra and Trigonometry, Young 2nd Edition, page 136-170) <br />
- 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. Solve linear and nonlinear inequalities.
- 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. 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.</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. 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. Week 5 Day 1<br />GRAPHING INEQUALITIES and INTERVAL NOTATION<br />
- 13. Week 5 Day 1<br />FOUR WAYS OF EXPRESSING SOLUTIONS TO INEQUALITIES:<br /><ul><li> inequality notation
- 14. set notation
- 15. interval notation
- 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. 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.</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. 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. 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. 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. 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.</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. 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. 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. 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. Week 5 Day 1<br />EXAMPLE 5<br />)<br />or<br />[<br />4<br />-1<br />4<br />-1<br />
- 29. Week 5 Day 1<br />EXAMPLE 5<br />]<br />or<br />[<br />4<br />0<br />4<br />0<br />
- 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. DEFINITION<br />Week 5 Day 1<br />UNION AND INTERSECTION<br />
- 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. Example<br />Week 5 Day 1<br />
- 34. Week 5 Day 1<br />SOLVING LINEAR INEQUALITIES <br />
- 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. 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. SOLVING A LINEAR INEQUALITY <br />Week 5 Day 1<br />Example<br />
- 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. SOLVING A DOUBLE OR COMPOUND LINEAR INEQUALITY <br />Week 5 Day 1<br />Example<br />
- 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. Week 5 Day 2<br />NON LINEAR INEQUALITIES IN ONE VARIABLE<br />
- 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. To solve polynomial inequalities.
- 44. To solve rational inequalities.
- 45. To solve absolute value inequalities
- 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. Week 5 Day 2<br />SOLVING QUADRATIC INEQUALITY<br /><ul><li>The square root method cannot be used for quadratic inequalities.
- 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. 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. Week 5 Day 2<br />SOLVING A POLYNOMIAL INEQUALITY<br />Solve each inequality:<br />
- 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. 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. Determine values that make the numerator or denominator equal to zero
- 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. Week 5 Day 2<br />SOLVING A RATIONAL INEQUALITY<br />Solve each inequality:<br />
- 56. Week 5 Day 2<br />ABSOLUTE VALUE INEQUALITIES<br />PROPERTIES OF ABSOLUTE VALUE INEQUALITIES<br />
- 57. Week 5 Day 2<br />SOLVING AN ABSOLUTE VALUE INEQUALITY<br />Solve each inequality:<br />
- 58. APPLICATIONS INVOLVING LINEAR INEQUALITY <br />Example<br />
- 59. APPLICATIONS INVOLVING LINEAR INEQUALITY <br />
- 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. 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 />

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