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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

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Chapter 2 Equations, Inequalities, and Problem Solving

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 2.5 Compound Inequalities

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Two inequalities joined by the words and or or are called compound inequalities. x + 5 4 3x ≥ 6 or x + 9 < 8 The solution set of a compound inequality formed by the word and is the intersection of the solution sets of the two inequalities. We use the symbol to represent “intersection.”

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Intersection of Two Sets The intersection of two sets, A and B, is the set of all elements common to both sets. A intersect B is denoted by A B A B

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1 If A = {x | x is an odd number greater than 0 but less than 9} and B = {4, 5, 6, 7, 8}, find A B. Solution List the elements of A. A = {1, 3, 5, 7} The numbers 5 and 7 are in sets A and B. The intersection is {5, 7}.

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2 Solve: x + 4 > 0 and 4x > 0. Solution Solve each inequality separately. x + 4 > 0and4x > 0 x > 4 and x > 0 Graph the two inequalities on two number lines and find their intersection. x > 4 x > 0 x > 4 and x > 0 As we see from the last number line, the solutions are all numbers greater than 0, written as x > 0.

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 3 Solve: 5x > 0 and 3x 4 ≤ 13. Solution Solve each inequality separately. 5x > 0and3x – 4 ≤ 13 x > 0 and 3x ≤ 9 x ≤ 3 Graph the two inequalities and find their intersection. x > 0 x ≤ 3 x > 0 and x ≤ 3 There is no number that is greater than 0 and less than or equal to 3. The answer is no solution.

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall To solve a compound inequality written in compact form, such a 3 < 5 – x < 9, we get x alone in the “middle part.” We must perform the same operations on all three parts of the inequality.

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 4 Solve: 3 < 5 – x < 9. Solution To get x alone, we first subtract 5 from all three parts. Subtract 5 from all three parts. Simplify. Divide all three parts by 1 and reverse the inequality symbols.

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall The solution set of a compound inequality formed by the word or is the union of the solution sets of the two inequalities. We use the symbol to denote “union.” Union of Two Sets The union of two sets, A and B, is the set of elements that belong to either of the sets. A union B is denoted by A A B B

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 6 If A = {x | x is an odd number greater than 0 but less than 9} and B = {4, 5, 6, 7, 8}, find A B. Solution List the elements of A. A = {1, 3, 5, 7} The numbers that are in either set or both sets are {1, 3, 4, 5, 6, 7, 8} This set is the union.

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 7 Solve: 6x – 4 ≤ 12 or x + 2 ≥ 8. Solution Solve each inequality separately. Graph each inequality. The solutions are x ≤ 8/3 or x ≥ 6.

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 8 Solve: 2x – 6 < 2 or 8x < 0. Solution Solve each inequality separately. Graph each inequality. The solutions are all real numbers.

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