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Inequalities mathematics grade nine igcse.ppt
- 2. Learning objectives Students are able to draw a number line to represent the values of inequalities. Students are able to solve inequalities algebraically.
- 3. Inequality Signs An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: < : less than : less than or equal to > : greater than : greater than or equal to
- 4. x < 5 means that whatever value x has, it must be less than 5. What could x be?
- 5. x -2 means that whatever value x has, it must be greater than or equal to -2. What could x be?
- 6. Graphing Rules Symbol Circle Direction of Arrow < Open Left > Open Right Closed Left Closed Right
- 7. Examples: x < 5 x > -2 x -8 x 4
- 8. You Try: x < -6 x > 2 x 0 x -7
- 10. Solving the Inequality w + 5 < 8 Note: We will use the same steps that we did with equations, if a number is added to the variable, we subtract the same number to both sides.
- 11. Answer w + 5 < 8 w + 5 -5 < 8 -5 w + 0 < 3 w < 3 All numbers less than 3 are solutions to this problem!
- 12. Now you try it! 8 + r -2 x - 2 > -2 4 + y 1 7 > x 5 x + 2 3
- 13. Answers to Practice Problems: 8 + r -2 8 -8 + r -2 -8 r + 0 -10 w -10 All numbers from -10 and up (including -10) make this problem true! x - 2 > -2 x - 2 + 2 > -2 + 2 x + 0 > 0 x > 0 All numbers greater than 0 make this problem true! 4 + y 1 4 - 4 + y 1 - 4 y + 0 -3 y -3 All numbers from -3 down (including -3) make this problem true!
- 14. Answers to Practice Problems: x - 5 > 7 x + 2 3 x 5 + 5 > 7 + 5 x + 0 > 12 x > 12 12 0 x + 2 - 2 7 - 2 x + 0 5 x 5 0 5
- 16. Solving Inequalities 10 5 x 5 10 5 5 x 50 x Multiply both sides by the reciprocal of the coefficient
- 17. Solving Inequalities Division Property for Inequalities 20 5 x 5 20 5 5 x 4 x Divide both sides by the coefficient of x
- 18. Solving Inequalities! Solving inequalities is the same as solving equations. There are only 2 things you need to know 1.) If you multiply or divide by a negative number you must switch the sign. -7x < 21 -7 -7 x > -3 2.) You will graph your solutions. Dividing by a negative means switch the sign!!
- 19. Special Case 1: Switching the Signs When solving inequalities, if you multiply or divide by a negative you must switch the signs. Switching the signs: Less than becomes Greater than < switches to > Greater than becomes Less than > switches to < Less than or equal to becomes Greater than or equal to switches to Greater than or equal to becomes Less than or equal to switches to
- 20. Division Property for Inequalities Caution! Dividing by a negative number 20 5 x 5 20 5 5 x 4 x Notice: Sign CHANGED Same if multiplying?
- 21. Multiplication Property for Inequalities Caution! When you multiply by a negative number the sign CHANGES YES ! -x > 2 5 (-5 )-x > 2(-5) 1 5 x < -10
- 22. Solving Inequalities Lets try some on our own ready?
- 23. Solving Inequalities 7 6 x 5 3 x 15 3 x 5 9 x
- 24. 1. x 1 2. 8 x or x 8 Answers for #1 - #4 3. x 5 4. x > 4
- 29. Answers for #5 - #8 5. x -6 6. 21 < x or x > 21 7. x 2 8. x > 15
- 31. Solving Inequalities Follow same steps used to solve equations: 3x + 4 < 13 - 4 -4 3x < 9 3 3 x < 3
- 32. Practice Problem 1 4 5v < 29
- 33. Practice Problem 2 2 + r > 1 9
- 34. Practice Problem 3 Which graph shows the solution to 2x - 10 4?
- 37. Real-Life Application Hint: 90 x 6 90% = 540 pts. You are taking a history course in which your grade is based on six 100 point tests. To earn an A in class, you must have a total of at least 90%. You have scored an 83, 89, 95, 98, and 92 on the first five tests. What is the least amount of points you can earn on the sixth test in order to earn an A in the course? 83+89+95+98+92= 457 457 - 457 + T 540 - 457 T 83
- 38. Example 2 f/3 4 f/3 3 4 3 F 12 To play a board game, there must be at least 4 people on each team. You divide your friends into 3 groups. Write and solve an inequality to represent the number of friends playing the game.
- 39. Example 3: 0.50 x +45 50 0.50 x +45 -45 50 -45 0.50 x 5 0.50 x / 0.50 5 /0.50 x 10 You budget $50 a month for your cell phone plan. You pay $45 for your minutes and 250 text messages. You are charged an extra $0.50 for picture messages. Write and solve an inequality to find the number of picture messages you can send without going over your budget.