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# 4. solving inequalities

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### 4. solving inequalities

1. 1. 1.4 Solving Inequalities OBJECTIVES a Determine whether a given number is a solution of an inequality. b Graph an inequality on the number line. c Solve inequalities using the addition principle. d Solve inequalities using the multiplication principle. e Solve inequalities using the addition principle and the multiplication principle together. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 1
2. 2. 1.4 Solving Inequalities a Determine whether a given number is a solution of an inequality. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 2
3. 3. 1.4 Solving Inequalities SOLUTION A replacement that makes an inequality true is called a solution. The set of all solutions is called the solution set. When we have found the set of all solutions of an inequality, we say that we have solved the inequality. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 3
4. 4. 1.4 a Solving Inequalities Determine whether a given number is a solution of an inequality. EXAMPLE Determine whether 2 is a solution of x < 2. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 4
5. 5. 1.4 Solving Inequalities Determine whether a given number is a solution of an a inequality. EXAMPLE Determine whether 6 is a solution of Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 5
6. 6. 1.4 b Solving Inequalities Graph an inequality on the number line. A graph of an inequality is a drawing that represents its solutions. An inequality in one variable can be graphed on the number line. An inequality in two variables can be graphed on the coordinate plane. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 6
7. 7. 1.4 Solving Inequalities b Graph an inequality on the number line. EXAMPLE The solutions are all those numbers less than 2. They are shown on the number line by shading all points to the left of 2. The open circle at 2 indicates that 2 is not part of the graph. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 7
8. 8. 1.4 Solving Inequalities b Graph an inequality on the number line. EXAMPLE The solutions are shown on the number line by shading the point for –3 and all points to the right of –3. The closed circle at –3 indicates that –3 is part of the graph. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 8
9. 9. 1.4 Solving Inequalities b Graph an inequality on the number line. EXAMPLE The inequality is read “–3 is less than or equal to x and x is less than 2,” or “x is greater than or equal to –3 and x is less than 2.” In order to be a solution of this inequality, a number must be a solution of both and x < 2. We can see from the graphs that the solution set consists of the numbers that overlap in the two solution sets in Examples 5 and 6. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 9
10. 10. 1.4 Solving Inequalities b Graph an inequality on the number line. EXAMPLE The open circle at 2 means that 2 is not part of the graph. The closed circle at –3 means that is part of the graph. The other solutions are shaded. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 10
11. 11. 1.4 Solving Inequalities c Solve inequalities using the addition principle. Any solution of one inequality is a solution of the other—they are equivalent. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 11
12. 12. 1.4 Solving Inequalities THE ADDITION PRINCIPLE FOR INEQUALITIES For any real numbers a, b, and c: In other words, when we add or subtract the same number on both sides of an inequality, the direction of the inequality symbol is not changed. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 12
13. 13. 1.4 Solving Inequalities c Solve inequalities using the addition principle. As with equation solving, when solving inequalities, our goal is to isolate the variable on one side. Then it is easier to determine the solution set. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 13
14. 14. 1.4 c Solving Inequalities Solve inequalities using the addition principle. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 14
15. 15. 1.4 c Solving Inequalities Solve inequalities using the addition principle. A shorter notation for sets is called set-builder notation. is read Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 15
16. 16. 1.4 Solving Inequalities THE MULTIPLICATION PRINCIPLE FOR INEQUALITIES For any real numbers a and b, and any positive number c: For any real numbers a and b, and any negative number c: Similar statements hold for Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 16
17. 17. 1.4 Solving Inequalities THE MULTIPLICATION PRINCIPLE FOR INEQUALITIES In other words, when we multiply or divide by a positive number on both sides of an inequality, the direction of the inequality symbol stays the same. When we multiply or divide by a negative number on both sides of an inequality, the direction of the inequality symbol is reversed. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 17
18. 18. 1.4 Solving Inequalities d Solve inequalities using the multiplication principle. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 18
19. 19. 1.4 Solving Inequalities d Solve inequalities using the multiplication principle. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 19
20. 20. 1.4 Solving Inequalities e Solve inequalities using the addition principle and the multiplication principle together. Remember to reverse the inequality symbol when multiplying or dividing on both sides by a negative number. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 20
21. 21. 1.4 e Solving Inequalities Solve inequalities using the addition principle and the multiplication principle together. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 21
22. 22. 1.4 Solving Inequalities Solve inequalities using the addition principle and the e multiplication principle together. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 22
23. 23. 1.4 e Solving Inequalities Solve inequalities using the addition principle and the multiplication principle together. EXAMPLE First, we use the distributive law to remove parentheses. Next, we collect like terms and then use the addition and multiplication principles for inequalities to get an equivalent inequality with x alone on one side. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 23
24. 24. 1.4 Solving Inequalities Solve inequalities using the addition principle and the e multiplication principle together. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 24
25. 25. 1.4 Solving Inequalities Solve inequalities using the addition principle and the e multiplication principle together. EXAMPLE The greatest number of decimal places in any one number is two. Multiplying by 100, which has two 0’s, will clear decimals. Then we proceed as before. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 25
26. 26. 1.4 Solving Inequalities Solve inequalities using the addition principle and the e multiplication principle together. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 26
27. 27. 1.4 Solving Inequalities Solve inequalities using the addition principle and the e multiplication principle together. EXAMPLE The number 6 is the least common multiple of all the denominators. Thus we first multiply by 6 on both sides to clear the fractions. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 27
28. 28. 1.4 Solving Inequalities Solve inequalities using the addition principle and the e multiplication principle together. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 28
29. 29. 1.4 Solving Inequalities Solve inequalities using the addition principle and the e multiplication principle together. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 29