This document provides an overview of solving linear inequalities and their applications. It discusses key concepts like the definition of a solution to an inequality, determining if a number is a solution, writing interval notation for graphs of inequalities on number lines, and the addition and multiplication properties of inequalities. Examples are provided to demonstrate how to solve inequalities algebraically and graph their solutions. One example solves and graphs the inequality 126t + 1293 > 3000 to determine the years after 2000 that the average cost of tuition and fees at two-year public institutions will be more than $3000.

2. 1 Solving Linear Equations and Inequalities Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

3. 1.4 Sets, Inequalities, and Interval Notation Slide 3 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

4. 1.4 Sets, Inequalities, and Interval Notation a Determine whether a given number is a solution of an inequality. Inequality Slide 4 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. An inequality is a sentence containing

5. 1.4 Sets, Inequalities, and Interval Notation a Determine whether a given number is a solution of an inequality. Solution of an Inequality Slide 5 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Any replacement or value for the variable that makes an inequality true is called a solution of the inequality. The set of all solutions is called the solution set. When all the solutions of an inequality have been found, we say that we have solved the inequality.

6. 1.4 Sets, Inequalities, and Interval Notation a Determine whether a given number is a solution of an inequality. 1 Determine whether the given number is a solution of the inequality. Slide 6 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

7. 1.4 Sets, Inequalities, and Interval Notation a Determine whether a given number is a solution of an inequality. 1 Slide 7 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. We substitute 5 for x and get 5 + 3 < 6, or 8 < 6, a false sentence. Therefore, 5 is not a solution.

8. 1.4 Sets, Inequalities, and Interval Notation a Determine whether a given number is a solution of an inequality. 3 Determine whether the given number is a solution of the inequality. Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

9. 1.4 Sets, Inequalities, and Interval Notation a Determine whether a given number is a solution of an inequality. 3 Slide 9 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. We substitute –3 for x and get or a true sentence. Therefore, –3 is a solution.

10. 1.4 Sets, Inequalities, and Interval Notation b Write interval notation for the solution set or the graph of an inequality. Slide 10 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. The graph of an inequality is a drawing that represents its solutions.

11. 1.4 Sets, Inequalities, and Interval Notation b Write interval notation for the solution set or the graph of an inequality. 4 Graph on the number line. Slide 11 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

12. 1.4 Sets, Inequalities, and Interval Notation b Write interval notation for the solution set or the graph of an inequality. 4 Slide 12 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. The solutions are all real numbers less than 4, so we shade all numbers less than 4 on the number line. To indicate that 4 is not a solution, we use a right parenthesis “)” at 4.

13. 1.4 Sets, Inequalities, and Interval Notation b Write interval notation for the solution set or the graph of an inequality. Slide 13 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

14. 1.4 Sets, Inequalities, and Interval Notation b Write interval notation for the solution set or the graph of an inequality. Slide 14 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

15. 1.4 Sets, Inequalities, and Interval Notation b Write interval notation for the solution set or the graph of an inequality. Write interval notation for the given set or graph. Slide 15 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

16. 1.4 Sets, Inequalities, and Interval Notation b Write interval notation for the solution set or the graph of an inequality. Slide 16 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

17. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. Slide 17 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Two inequalities are equivalent if they have the same solution set.

18. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. The Addition Principle for Inequalities Slide 18 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

19. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 10 Solve and graph. Slide 19 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

20. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 10 Slide 20 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. We used the addition principle to show that the inequalities x + 5 > 1 and x > –4 are equivalent. The solution set is and consists of an infinite number of solutions. We cannot possibly check them all.

21. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 10 Slide 21 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Instead, we can perform a partial check by substituting one member of the solution set (here we use –1) into the original inequality:

22. Since 4 > 1 is true, we have a partial check. The solution set is or The graph is as follows: 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 10 Slide 22 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

23. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 11 Solve and graph. Slide 23 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

24. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 11 Slide 24 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

25. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 11 Slide 25 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. The inequalities and have the same meaning and the same solutions. The solution set is or more commonly, Using interval notation, we write that the solution set is The graph is as follows:

26. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. The Multiplication Principle for Inequalities Slide 26 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 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

27. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. Slide 27 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. The multiplication principle tells us that when we multiply or divide on both sides of an inequality by a negative number, we must reverse the inequality symbol to obtain an equivalent inequality.

28. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 13 Solve and graph. Slide 28 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

29. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 13 Slide 29 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

30. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 13 Slide 30 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

31. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 15 Solve. Slide 31 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

32. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 15 Slide 32 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

33. 1.4 Sets, Inequalities, and Interval Notation c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. 15 Slide 33 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

34. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. Slide 34 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

35. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. Slide 35 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

36. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. Translating “At Least” and “At Most” Slide 36 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

37. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. 16 Cost of Higher Education. Slide 37 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. The equation C = 126t + 1293 can be used to estimate the average cost of tuition and fees at two-year public institutions of higher education, where t is the number of years after 2000. Determine, in terms of an inequality, the years for which the cost will be more than $3000.

38. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. 16 Slide 38 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Familiarize. We already have a formula. To become more familiar with it, we might make a substitution for t. Suppose we want to know the cost 15 yr after 2000, or in 2015. We substitute 15 for t: C = 126(15) + 1293 = $3183.

39. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. 16 Slide 39 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. We see that in 2015, the cost of tuition and fees at two-year public institutions will be more than $3000. To find all the years in which the cost exceeds $3000, we could make other guesses less than 15, but it is more efficient to proceed to the next step.

40. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. 16 Slide 40 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 2. Translate. The cost C is to be more than $3000. Thus we have C > 3000. We replace C with 126t + 1293 to find the values of t that are solutions of the inequality: 126t + 1293 > 3000.

41. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. 16 Slide 41 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 3. Solve. We solve the inequality:

42. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. 16 Slide 42 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 4. Check. A partial check is to substitute a value for t greater than 13.55. We did that in the Familiarize step and found that the cost was more than $3000. 5. State. The average cost of tuition and fees at two-year public institutions of higher education will be more than $3000 for years more than 13.55 yr after 2000, so we have