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Ultimate guide to linear inequalities

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Ultimate guide to linear inequalities

1. 1. Solving Linear Inequalities 1.4 Sets, Inequalities, and Interval Notation 1.5 Intersections, Unions, and Compound Inequalities 1.6 Absolute-Value Equations and Inequalities
2. 2. 1.4 Sets, Inequalities, and Interval Notation OBJECTIVES a Determine whether a given number is a solution of an inequality. b Write interval notation for the solution set or the graph of an inequality. c Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. d Solve applied problems by translating to inequalities.
3. 3. 1.4 Sets, Inequalities, and Interval Notation Inequality An inequality is a sentence containing
4. 4. 1.4 Sets, Inequalities, and Interval Notation Solution of an Inequality 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.
5. 5. 1.4 Sets, Inequalities, and Interval Notation Determine whether a given number is a solution of an a inequality. EXAMPLE 1 Determine whether the given number is a solution of the inequality.
6. 6. 1.4 Sets, Inequalities, and Interval Notation Determine whether a given number is a solution of an a inequality. EXAMPLE 1 Solution We substitute 5 for x and get 5 + 3 < 6, or 8 < 6, a false sentence. Therefore, 5 is not a solution.
7. 7. 1.4 Sets, Inequalities, and Interval Notation Determine whether a given number is a solution of an a inequality. EXAMPLE 3 Determine whether the given number is a solution of the inequality.
8. 8. 1.4 Sets, Inequalities, and Interval Notation Determine whether a given number is a solution of an a inequality. EXAMPLE 3 Solution We substitute –3 for x and get or a true sentence. Therefore, –3 is a solution.
9. 9. 1.4 b Sets, Inequalities, and Interval Notation Write interval notation for the solution set or the graph of an inequality. The graph of an inequality is a drawing that represents its solutions.
10. 10. 1.4 Sets, Inequalities, and Interval Notation Write interval notation for the solution set or the b graph of an inequality. EXAMPLE 4 Graph on the number line.
11. 11. 1.4 Sets, Inequalities, and Interval Notation Write interval notation for the solution set or the b graph of an inequality. EXAMPLE 4 Solution 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.
12. 12. 1.4 b Sets, Inequalities, and Interval Notation Write interval notation for the solution set or the graph of an inequality.
13. 13. 1.4 b Sets, Inequalities, and Interval Notation Write interval notation for the solution set or the graph of an inequality.
14. 14. 1.4 Sets, Inequalities, and Interval Notation Write interval notation for the solution set or the b graph of an inequality. EXAMPLE Write interval notation for the given set or graph.
15. 15. 1.4 Sets, Inequalities, and Interval Notation Write interval notation for the solution set or the b graph of an inequality. Solution EXAMPLE
16. 16. 1.4 c Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the multiplication principle and then graph the inequality. Two inequalities are equivalent if they have the same solution set.
17. 17. 1.4 Sets, Inequalities, and Interval Notation The Addition Principle for Inequalities
18. 18. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 10 Solve and graph.
19. 19. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 10 Solution 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.
20. 20. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 10 Solution Instead, we can perform a partial check by substituting one member of the solution set (here we use –1) into the original inequality:
21. 21. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 10 Solution Since 4 > 1 is true, we have a partial check. The solution set is or The graph is as follows:
22. 22. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 11 Solve and graph.
23. 23. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 11 Solution
24. 24. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 11 Solution 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:
25. 25. 1.4 Sets, Inequalities, and Interval Notation 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
26. 26. 1.4 Sets, Inequalities, and Interval Notation 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.
27. 27. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 13 Solve and graph.
28. 28. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 13 Solution
29. 29. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 13 Solution
30. 30. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 15 Solve.
31. 31. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 15 Solution
32. 32. 1.4 Sets, Inequalities, and Interval Notation Solve an inequality using the addition principle and the c multiplication principle and then graph the inequality. EXAMPLE 15 Solution
33. 33. 1.4 d Sets, Inequalities, and Interval Notation Solve applied problems by translating to inequalities.
34. 34. 1.4 d Sets, Inequalities, and Interval Notation Solve applied problems by translating to inequalities.
35. 35. 1.4 Sets, Inequalities, and Interval Notation Translating “At Least” and “At Most”
36. 36. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. EXAMPLE 16 Cost of Higher Education. 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.
37. 37. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. EXAMPLE 16 Solution 1. 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.
38. 38. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. EXAMPLE 16 Solution We see that in 2015, the cost of tuition and fees at twoyear 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.
39. 39. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. EXAMPLE 16 Solution 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.
40. 40. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. EXAMPLE 16 Solution 3. Solve. We solve the inequality:
41. 41. 1.4 Sets, Inequalities, and Interval Notation d Solve applied problems by translating to inequalities. EXAMPLE 16 Solution 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