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Chapter 2 Equations, Inequalities, and Problem Solving

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

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.”

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

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}.

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.

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.

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.

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.

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

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.

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.

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.