2.5 Compound Inequalities

Transcription

1 Section.5 Compound Inequalities 89.5 Compound Inequalities S 1 Find the Intersection of Two Sets. Solve Compound Inequalities Containing and. Find the Union of Two Sets. 4 Solve Compound Inequalities Containing or. Two inequalities joined by the words and or or are called compound inequalities. Compound Inequalities x and x 7 x Ú 5 or -x Finding the Intersection of Two Sets 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. 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 A B EXAMPLE 1 If A = 5x x is an even number greater than 0 and less than 106 and B = 5, 4, 5, 66, find A B. Solution Let s list the elements in set A. A = 5, 4, 6, 86 The numbers 4 and 6 are in sets A and B. The intersection is 54, If A = 5x x is an odd number greater than 0 and less than 106 and B = 51,,, 46, find A B. Solving Compound Inequalities Containing and A value is a solution of a compound inequality formed by the word and if it is a solution of both inequalities. For example, the solution set of the compound inequality x 5 and x Ú contains all values of x that make the inequality x 5 a true statement and the inequality x Ú a true statement. The first graph shown below is the graph of x 5, the second graph is the graph of x Ú, and the third graph shows the intersection of the two graphs. The third graph is the graph of x 5 and x Ú. 5x x 56 5x x Ú 6 5x x 5 and x Ú 6 also 5x x 56 (see below) , 5] [, [, 5] Since x Ú is the same as x, the compound inequality x and x 5 can be written in a more compact form as x 5. The solution set 5x x 56 includes all numbers that are greater than or equal to and at the same time less than or equal to 5. In interval notation, the set 5x x 5 and x Ú 6 or the set 5x x 56 is written as, 54.

2 90 CHAPTER Equations, Inequalities, and Problem Solving Helpful Hint Don t forget that some compound inequalities containing and can be written in a more compact form. Compound Inequality Compact Form Interval Notation x and x 6 x 6, 64 Graph: EXAMPLE Solve: x and x Solution First we solve each inequality separately. x and x x 6 9 and x 6 8 x 6 9 and x 6 4 Now we can graph the two intervals on two number lines and find their intersection. Their intersection is shown on the third number line. 5x x , 9 5x x x 0 x 6 9 and x 6 46 = 5x 0 x 6 46 The solution set is 1 -, , , Solve: x and x Write the solution set in interval notation. EXAMPLE Solve: x Ú 0 and 4x Solution First we solve each inequality separately. x Ú 0 and 4x x Ú 0 and 4x -8 x Ú 0 and x - Now we can graph the two intervals and find their intersection. 5x x Ú 06 5x x -6 [0, , -] x x Ú 0 and x -6 = There is no number that is greater than or equal to 0 and less than or equal to -. The solution set is. Solve: 4x 0 and x Write the solution set in interval notation. Helpful Hint Example shows that some compound inequalities have no solution. Also, some have all real numbers as solutions.

3 Section.5 Compound Inequalities 91 To solve a compound inequality written in a compact form, such as x 6 7, we get x alone in the middle part. Since a compound inequality is really two inequalities in one statement, we must perform the same operations on all three parts of the inequality. For example: x 6 7 means x and 4 - x 6 7, Helpful Hint Don t forget to reverse both inequality symbols. EXAMPLE 4 Solve: x 6 7 Solution To get x alone, we first subtract 4 from all three parts x x Subtract 4 from all three parts x 6 Simplify x x 7 - Divide all three parts by -1 and reverse the inequality symbols. This is equivalent to - 6 x 6. The solution set in interval notation is 1 -,, and its graph is shown Solve: x 6 9. Write the solution set in interval notation. EXAMPLE 5 Solve: -1 x + 5. Solution First, clear the inequality of fractions by multiplying all three parts by the LCD. -1 x a x + 5b 1 Multiply all three parts by the LCD. - x Use the distributive property and multiply x Subtract 15 from all three parts. -18 x -9 Simplify. -18 x -9 Divide all three parts by. -9 x - 9 The graph of the solution is shown. Simplify. t The solution set in interval notation is c -9, - 9 d. 5 Solve: -4 x - 1. Write the solution set in interval notation.

4 9 CHAPTER Equations, Inequalities, and Problem Solving Finding the Union of Two Sets 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. Helpful Hint The word either in this definition means one or the other or both. 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 B. A B A B EXAMPLE 6 If A = 5x x is an even number greater than 0 and less than 106 and B = 5, 4, 5, 66, find A B. Solution Recall from Example 1 that A = 5, 4, 6, 86. The numbers that are in either set or both sets are 5,, 4, 5, 6, 86. This set is the union. 6 If A = 5x x is an odd number greater than 0 and less than 106 and B = 5,, 4, 5, 66, find A B. 4 Solving Compound Inequalities Containing or A value is a solution of a compound inequality formed by the word or if it is a solution of either inequality. For example, the solution set of the compound inequality x 1 or x Ú contains all numbers that make the inequality x 1 a true statement or the inequality x Ú a true statement. 5x x 16 5x x Ú 6 5x x 1 or x Ú 6 1 -, 1] [, 1 -, 1] [, In interval notation, the set 5x x 1 or x Ú 6 is written as 1 -, 1] [,. EXAMPLE 7 Solve: 5x - 10 or x + 1 Ú 5. Solution First we solve each inequality separately. 5x - 10 or x + 1 Ú 5 5x 1 or x Ú 4 x 1 or x Ú 4 5 Now we can graph each interval and find their union. e x ` x 1 5 f { a -, 1 5 d 5x x Ú 46 [4, e x ` x 1 5 or x Ú 4 f { a -, 1 d [4, 5

5 Section.5 Compound Inequalities 9 The solution set is a -, 1 5 d [4,. 7 Solve: 8x or x - 1 Ú. Write the solution set in interval notation. EXAMPLE 8 Solve: -x or 6x 6 0. Solution First we solve each inequality separately. -x or 6x 6 0 -x 6 or x 6 0 x 7-1 or x 6 0 Now we can graph each interval and find their union. 5x x x x x x 7-1 or x 6 06 = all real numbers The solution set is 1 -,. 1-1, , , Solve: -x or 5x 7 0. Write the solution set in interval notation. Answer to Concept Check: b is not correct CONCEPT CHECK Which of the following is not a correct way to represent the set of all numbers between - and 5? a. 5x - 6 x 6 56 b. - 6 x or x 6 5 c. 1 -, 5 d. x 7 - and x 6 5 Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. or and compound 1. Two inequalities joined by the words and or or are called inequalities.. The word means intersection.. The word means union. 4. The symbol represents intersection. 5. The symbol represents union. 6. The symbol is the empty set.

6 94 CHAPTER Equations, Inequalities, and Problem Solving Martin-Gay Interactive Videos See Video.5 Watch the section lecture video and answer the following questions Based on Example 1 and the lecture before, complete the following statement. For an element to be in the intersection of sets A and B, the element must be in set A in set B. 8. In Example, how can using three number lines help us find the solution to this and compound inequality? 9. Based on Example 4 and the lecture before, complete the following statement. For an element to be in the union of sets A and B, the element must be in set A in set B. 10. In Example 5, how can using three number lines help us find the solution to this or compound inequality?.5 Exercise Set MIXED If A = 5x 0 x is an even integer6, B = 5x 0 x is an odd integer6, C = 5,, 4, 56, and D = 54, 5, 6, 76, list the elements of each set. See Examples 1 and C D. C D. A D 4. A D 5. A B 6. A B 7. B D 8. B D 9. B C 10. B C 11. A C 1. A C Solve each compound inequality. Graph the solution set and write it in interval notation. See Examples and. 1. x 6 1 and x x 0 and x Ú x - and x Ú x 6 and x x 6-1 and x x Ú -4 and x 7 1 Solve each compound inequality. Write solutions in interval notation. See Examples and. 19. x + 1 Ú 7 and x - 1 Ú 5 0. x + Ú and 5x - 1 Ú x and x 0. x and 4x x 6-8 and x x -1 and x Solve each compound inequality. See Examples 4 and x x x x x x x x Solve each compound inequality. Graph the solution set and write it in interval notation. See Examples 7 and 8.. x 6 4 or x x Ú - or x 5. x -4 or x Ú 1 6. x 6 0 or x x 7 0 or x 6 8. x Ú - or x -4 Solve each compound inequality. Write solutions in interval notation. See Examples 7 and x -4 or 5x - 0 Ú x 10 or x - 5 Ú x or 6x x or 4x x or x x - 1 Ú -5 or 5 + x 11 MIXED Solve each compound inequality. Write solutions in interval notation. See Examples 1 through x 6 and x x and x x 6 or x x or x x x

7 Section.5 Compound Inequalities x x x + Ú and x x - 1 Ú and -x x Ú 5 or x x or -x x x x x x and 1 + x x 0 and -x x + 5 or 7x x 6 7 or x x 7 7 and x + Ú x 6-6 or 1 - x x x x x x x REVIEW AND PREVIEW Evaluate the following. See Sections 1. and Find by inspection all values for x that make each equation true. 77. x = x = x = x = - CONCEPT EXTENSIONS Use the graph to answer Exercises 81 and 8. Number of Single-Family Homes (in thousands) Year Source: U.S. Census Bureau United States Single-Family Homes Housing Starts vs. Housing Completions Started Completed 81. For which years were the number of single-family housing starts greater than 1500 and the number of single-family home completions greater than 1500? 8. For which years were the number of single-family housing starts less than 1000 or the number of single-family housing completions greater than 1500? 8. In your own words, describe how to find the union of two sets. 84. In your own words, describe how to find the intersection of two sets. Solve each compound inequality for x. See the example below. To solve x x 6 x + 5, notice that this inequality contains a variable not only in the middle but also on the left and the right. When this occurs, we solve by rewriting the inequality using the word and. x x and x 6 x x and x x x 7 - and x x 7-4 x x - 6 x x x + 6 x x x - - x 10 - x 88. 7x x 11 + x - 6 x 6 5 or 1 -, 5

8 96 CHAPTER Equations, Inequalities, and Problem Solving 89. 5x x x x x x The formula for converting Fahrenheit temperatures to Celsius temperatures is C = 5 1F -. Use this formula for Exercises 9 91 and During a recent year, the temperatures in Chicago ranged from -9 C to 5 C. Use a compound inequality to convert these temperatures to Fahrenheit temperatures. 9. In Oslo, the average temperature ranges from -10 to 18 Celsius. Use a compound inequality to convert these temperatures to the Fahrenheit scale. Solve. 9. Christian D Angelo has scores of 68, 65, 75, and 78 on his algebra tests. Use a compound inequality to find the scores he can make on his final exam to receive a C in the course. The final exam counts as two tests, and a C is received if the final course average is from 70 to Wendy Wood has scores of 80, 90, 8, and 75 on her chemistry tests. Use a compound inequality to find the range of scores she can make on her final exam to receive a B in the course. The final exam counts as two tests, and a B is received if the final course average is from 80 to Absolute Value Equations 1 Solve Absolute Value Equations. 1 Solving Absolute Value Equations In Chapter 1, we defined the absolute value of a number as its distance from 0 on a number line. units units = and = In this section, we concentrate on solving equations containing the absolute value of a variable or a variable expression. Examples of absolute value equations are 0 x 0 = -5 = 0 y z = 0 z Since distance and absolute value are so closely related, absolute value equations and inequalities (see Section.7) are extremely useful in solving distance-type problems such as calculating the possible error in a measurement. For the absolute value equation 0 x 0 =, its solution set will contain all numbers whose distance from 0 is units. Two numbers are units away from 0 on the number line: and -. units units Thus, the solution set of the equation 0 x 0 = is 5, -6. This suggests the following: Solving Equations of the Form 0 X 0 a If a is a positive number, then 0 X 0 = a is equivalent to X = a or X = -a. EXAMPLE 1 Solve: 0 p 0 =. 0 0 Solution Since is positive, p = is equivalent to p = or p = -. To check, let p = and then p = - in the original equation. p = Original equation p = Original equation = Let p =. - = Let p = -. = True = True The solutions are and - or the solution set is 5, Solve: q =.