More Equations and Inequalities


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1 Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities Problem Recognition Eercises Equations and Inequalities 9. Linear Inequalities in Two Variables As ou stud Chapter 9 ou will be able to recognize and solve a variet of equations and inequalities. As ou work through the chapter, write the solution set for each inequalit here. Then use the letter net to each answer to complete the puzzle. Solve each inequalit.. and i.. or s.. 7 and d.. 7 or v.. i.. o. 7. n i.,,,,,,,,,, He wears glasses during his math class because it improves
2 Chapter 9 More Equations and Inequalities Section 9. Concepts. Union and Intersection. Solving Compound Inequalities: And. Solving Compound Inequalities: Or. Applications of Compound Inequalities Compound Inequalities. Union and Intersection In Chapter we graphed simple inequalities and epressed the solution set in interval notation and in setbuilder notation. In this chapter, we will solve compound inequalities that involve the union or intersection of two or more inequalities. A Union B and A Intersection B The union of sets A and B, denoted A B, is the set of elements that belong to set A or to set B or to both sets A and B. The intersection of two sets A and B, denoted A B, is the set of elements common to both A and B. The concepts of the union and intersection of two sets are illustrated in Figures 9 and 9. A B A B A B A union B The elements in A or B or both A B A intersection B The elements in A and B Figure 9 Figure 9 Eample Finding the Union and Intersection of Two Intervals Find the union or intersection as indicated. a. a, b, b.,, Solution: a. a, b, To find the intersection, graph each interval separatel. Then find the real numbers common to both intervals. ) a, b ), The intersection is ),. The intersection is the overlap of the two intervals:, b.,, To find the union, graph each interval separatel. The union is the collection of real numbers that lie in the first interval, the second interval, or both intervals.
3 Section 9. Compound Inequalities ), ), The union is,. ) The union consists of all real numbers in the red interval along with the real numbers in the blue interval:, Skill Practice. Find the intersection,,. Write the answer in interval notation.. Find the union,,. Write the answer in interval notation.. Solving Compound Inequalities: And The solution to two inequalities joined b the word and is the intersection of their solution sets. The solution to two inequalities joined b the word or is the union of their solution sets. Steps to Solve a Compound Inequalit. Solve and graph each inequalit separatel.. If the inequalities are joined b the word and, find the intersection of the two solution sets. If the inequalities are joined b the word or, find the union of the two solution sets.. Epress the solution set in interval notation or in setbuilder notation. As ou work through the eamples in this section, remember that multipling or dividing an inequalit b a negative factor reverses the direction of the inequalit sign. Eample Solving Compound Inequalities: And Solve the compound inequalities. a. and 7 b..a.. and.8a 9.. c. and Solution: a. and 7 Solve each inequalit separatel. 7 7 and and Reverse the first inequalit sign. Skill Practice Answers.,.,
4 Chapter 9 More Equations and Inequalities 7 Take the intersection of the solution sets: The solution is or equivalentl, in interval notation,,. b..a.. and.a..a... a. and and and.8a 9...8a..8a a 7. a a. Reverse the second inequalit sign. a a 7. The intersection of the solution sets is the empt set: { } There are no real numbers that are simultaneousl less than. and greater than.. Hence, there is no solution. c. and a b and a b 7 9 and Solve each inequalit separatel ) ) Take the intersection of the solution sets: 7 The solution is 7 or, in interval notation,,. Skill Practice Solve the compound inequalities.. 8 and and.. 7. and.. Skill Practice Answers. ;,. No solution. ;, In Section.7, we learned that the inequalit a b is the intersection of two simultaneous conditions implied on. a b is equivalent to a and b
5 Section 9. Compound Inequalities Eample Solving Compound Inequalities: And Solve the inequalit. Solution: and and and 7 Set up the intersection of two inequalities. Solve each inequalit. Reverse the direction of the inequalit signs. and 7 and 7 Rewrite the inequalities. Take the intersection of the solution sets. The solution is or, equivalentl in interval notation,,. Skill Practice Solve the inequalit.. TIP: As an alternative approach to Eample, we can isolate the variable in the middle portion of the inequalit. Recall that the operations performed on the middle part of the inequalit must also be performed on the left and righthand sides. 7 Subtract from all three parts of the inequalit. Simplif. Divide b in all three parts of the inequalit. Remember to reverse inequalit signs.) 7 Simplif. Rewrite the inequalit. Skill Practice Answers. e ƒ f ; c, b
6 Chapter 9 More Equations and Inequalities. Solving Compound Inequalities: Or Eample Solving Compound Inequalities: Or Solve the compound inequalities. a. 7 or b. or c. or 7 Solution: a or or 9 or or Solve each inequalit separatel. Reverse the inequalit signs. Take the union of the solution sets or. or, equivalentl in interval nota The solution is or tion,,,. b. or or 7 Solve each inequalit separatel. or 7 7 Take the union of the solution sets. The union of the solution sets is { is an real number} or equivalentl,.
7 Section 9. Compound Inequalities c. or 7 a b or 7 Solve each inequalit separatel. or a b or ) 7 ) 7 Take the union of the solution sets:. The union of the solution sets is or, in interval notation,,. Skill Practice ) Solve the compound inequalities. 7. t 8 or t or or.7 7. Applications of Compound Inequalities Compound inequalities are used in man applications, as shown in Eamples and. Eample Translating Compound Inequalities The normal level of throidstimulating hormone TSH) for adults ranges from. to.8 microunits per milliliter mu/ml. Let represent the amount of TSH measured in microunits per milliliter. a. Write an inequalit representing the normal range of TSH. b. Write a compound inequalit representing abnormal TSH levels. Solution: a...8 b.. or 7.8 Skill Practice. The length of a normal human pregnanc is from 7 to weeks, inclusive. a. Write an inequalit representing the normal length of a pregnanc. b. Write a compound inequalit representing an abnormal length for a pregnanc. Skill Practice Answers 7. t t or t 7 ;,, 8. All real numbers;, 9. 7 ;, a. 7 w b. w 7 or w 7
8 Chapter 9 More Equations and Inequalities Eample Translating and Solving a Compound Inequalit The sum of a number and is between and. Find all such numbers. Solution: Let represent a number. 9 8 Translate the inequalit. Subtract from all three parts of the inequalit. The number ma be an real number between 9 and 8: 9 8. Skill Practice Answers. An real number between and : n n Skill Practice. The sum of twice a number and is between and. Find all such numbers. Section 9. Boost our GRADE at mathzone.com! Stud Skills Eercise. Define the ke terms. Practice Eercises Practice Problems SelfTests NetTutor eprofessors Videos a. Compound inequalit b. Intersection c. Unionction 9. Review Eercises For Eercises 8, review solving linear inequalities from Section.7. Write the answers in interval notation.. u 7. z.. p q t 8. Concept : Union and Intersection For Eercises 9, find the intersection and union of sets as indicated. Write the answers in interval notation. 9. a.,,. a.,,. a. a, b a, 9 b b.,, b.,, b. a, b a, 9 b. a..,..,.. a.,,. a.,, b..,..,. b.,, b.,,
9 Section 9. Compound Inequalities 7 Concept : Solving Compound Inequalities: And For Eercises, solve the inequalit and graph the solution. Write the answer in interval notation and. a 7 and a 7. t 7 9 and t p p and 9p p 9. k k 9 and. w 7 w and k k 7 w 7 8w. p and p. a 9 and a. and. 8 and 7 7. Write t as two separate inequalities.. Write.8 as two separate inequalities. 7. Eplain wh has no solution. 8. Eplain wh t has no solution. 9. Eplain wh 7 7 has no solution.. Eplain wh 7 w 7 has no solution. For Eercises, solve the inequalit and graph the solution set. Write the answer in interval notation.. b 9. k 9. a... 7 t Concept : Solving Compound Inequalities: Or For Eercises, solve the inequalit and graph the solution set. Write the answer in interval notation.. h or h 7. 7 or. or. or z 9. or...z z or z
10 8 Chapter 9 More Equations and Inequalities 7. or 8. p 7 or p 9. or. or v v 8 u 7 u t. 7 t or. 7 or..w.w or.w.w...a.a or.a 9.a Mied Eercises For Eercises, solve the inequalit. Write the answer in interval notation.. a. 9 and. a and 7. b. 9 or b or a. 8. or.. 8. a. r 8 or r 8 b. 8. and.. b. r 8 and r t t or. 7 w w or t 8 t w w. 7 9 and. 7 and or 7. or 7 Concept : Applications of Compound Inequalities 7. The normal number of white blood cells for human blood is between 8 and,8 cells per cubic millimeter, inclusive. Let represent the number of white blood cells per cubic millimeter. a. Write an inequalit representing the normal range of white blood cells per cubic millimeter. b. Write a compound inequalit representing abnormal levels of white blood cells per cubic millimeter.
11 Section 9. Polnomial and Rational Inequalities 9 8. Normal hemoglobin levels in human blood for adult males are between and grams per deciliter g/dl), inclusive. Let represent the level of hemoglobin measured in grams per deciliter. a. Write an inequalit representing normal hemoglobin levels for adult males. b. Write a compound inequalit representing abnormal levels of hemoglobin for adult males. 9. The normal number of platelets in human blood is between. and. platelets per cubic millimeter, inclusive. Let represent the number of platelets per cubic millimeter. a. Write an inequalit representing a normal platelet count per cubic millimeter. b. Write a compound inequalit representing abnormal platelet counts per cubic millimeter. 7. Normal hemoglobin levels in human blood for adult females are between and g/dl, inclusive. Let represent the level of hemoglobin measured in grams per deciliter. a. Write an inequalit representing normal hemoglobin levels for adult females. b. Write a compound inequalit representing abnormal levels of hemoglobin for adult females. 7. Twice a number is between and. Find all such numbers. 7. The difference of a number and is between and 8. Find all such numbers. 7. One plus twice a number is either greater than or less than. Find all such numbers. 7. Onethird of a number is either less than or greater than. Find all such numbers. Polnomial and Rational Inequalities. Solving Inequalities Graphicall In Sections.7 and 9., we solved simple and compound linear inequalities. In this section we will solve polnomial and rational inequalities. We begin b defining a quadratic inequalit. Quadratic inequalities are inequalities that can be written in an of the following forms: a b c a b c 7 a b c a b c where a Recall from Section 8. that the graph of a quadratic function defined b f a b c a is a parabola that opens upward or downward. The quadratic inequalit f 7 or equivalentl a b c 7 is asking the question, For what values of is the value of the function positive above the ais)? The inequalit f or equivalentl a b c is asking, For what values of is the value of the function negative below the ais)? The graph of a quadratic function can be used to answer these questions. Section 9. Concepts. Solving Inequalities Graphicall. Solving Polnomial Inequalities b Using the Test Point Method. Solving Rational Inequalities b Using the Test Point Method. Inequalities with Special Case Solution Sets Eample Using a Graph to Solve a Quadratic Inequalit Use the graph of f 8 in Figure 9 to solve the inequalities. a. 8 b. 8 7
12 Chapter 9 More Equations and Inequalities Solution: From Figure 9, we see that the graph of f 8 is a parabola opening upward. The function factors as f. The intercepts are at and, and the intercept is, 8). 8 8 f) 8 f) Figure 9 a. The solution to 8 is the set of real numbers for which f. Graphicall, this is the set of all values corresponding to the points where the parabola is below the ais shown in red). Hence for or equivalentl,, ) b. The solution to 8 7 is the set of values for which f 7. This is the set of values where the parabola is above the ais shown in blue). Hence 8 7 Skill Practice 8 for or 7 or,,. Refer to the graph of f to solve the inequalities. a. 7 b. 8 8 Notice that and define the boundaries of the solution sets to the inequalities in Eample. These values are the solutions to the related equation 8. Skill Practice Answers a. ƒ or 7 ;,, b. ƒ ;, TIP: The inequalities in Eample are strict inequalities. Therefore, and where f ) are not included in the solution set. However, the corresponding inequalities using the smbols and do include the values where f. Hence, The solution to 8 is or, The solution to 8 is or or,,
13 Section 9. Polnomial and Rational Inequalities Eample Using a Graph to Solve a Rational Inequalit Use the graph of a. b. Solution: g 7 in Figure 9 to solve the inequalities. g) g) Figure 9 a. Figure 9 indicates that g is below the ais for shown in red). Therefore, the solution to is ƒ or, equivalentl,,. b. Figure 9 indicates that g is above the ais for 7 shown in blue). Therefore, the solution to is ƒ 7 or, equivalentl,,. 7 Skill Practice. Refer to the graph of g to solve the inequalities. a. b. 7 TIP: Notice that defines the boundar of the solution sets to the inequalities in Eample. At the inequalit is undefined. Skill Practice Answers a. 7 ;, b. ;,
14 Chapter 9 More Equations and Inequalities. Solving Polnomial Inequalities b Using the Test Point Method Eamples and demonstrate that the boundar points of an inequalit provide the boundaries of the solution set. Boundar Points The boundar points of an inequalit consist of the real solutions to the related equation and the points where the inequalit is undefined. Testing points in regions bounded b these points is the basis of the test point method to solve inequalities. Solving Inequalities b Using the Test Point Method. Find the boundar points of the inequalit.. Plot the boundar points on the number line. This divides the number line into regions.. Select a test point from each region and substitute it into the original inequalit. If a test point makes the original inequalit true, then that region is part of the solution set.. Test the boundar points in the original inequalit. If a boundar point makes the original inequalit true, then that point is part of the solution set. Eample Solving Polnomial Inequalities b Using the Test Point Method Solve the inequalities b using the test point method. a. b. 7 Solution: a. Step : Find the boundar points. Because polnomials are defined for all values of, the onl boundar points are the real solutions to the related equation. Solve the related equation. The boundar points are and.
15 Section 9. Polnomial and Rational Inequalities Region I Region II Region III Step : Plot the boundar points. Step : Select a test point from each region. Test?? Test?? Test? 8?? False Test?? True Test a b a b? 8? False Step : Test the boundar points.?? False a 9 b? 9?? False Neither boundar point makes the inequalit true. Therefore, the boundar points are not included in the solution set. False True False TIP: The strict inequalit,, ecludes values of for which. This implies that the boundar points are not included in the solution set. The solution is or equivalentl in interval notation,. Calculator Connections Graph Y and Y. Notice that Y for Y. Y Y, ), ) b. 7 Step : Find the boundar points.
16 Chapter 9 More Equations and Inequalities I II III IV V Step : Plot the boundar points. Step : Select a test point from each region. Test : 7? Test : 7? Test : 7? Test : 7? 7? False 7? False 7 7? True 7 7? False Test : 7? False False True False True 7? True Step : The boundar points are not included because the inequalit, is strict. The solution is or 7 or, equivalentl in interval notation,,,. Calculator Connections 7, Graph Y. Y is positive above the ais) for or 7 or equivalentl,,. Y ) ) ) Skill Practice Solve the inequalities b using the test point method. Write the answers in interval notation.. 7. t t t 7 Eample Solving a Polnomial Inequalit Using the Test Point Method Solve the inequalit b using the test point method. Skill Practice Answers.,,.,,, Solution: ; Step : Find the boundar points in the related equation. Since this equation is not factorable, use the quadratic formula to find the solutions. b ; b ac a
17 Section 9. Polnomial and Rational Inequalities ; 7 7. and 7. Region I Region II Region III Step : Plot the boundar points. Step : Select a test point from each region. Test? True The solution is in interval notation: a, Test?? True? False? True False True 7 7 e or f 7 d c Step : Test the boundar points. Both boundar points make the inequalit true. Therefore, both boundar points are included in the solution set. 7, b. Test? or equivalentl Skill Practice Solve the inequalit using the test point method. Write the answer in interval notation... Solving Rational Inequalities b Using the Test Point Method The test point method can be used to solve rational inequalities. A rational inequalit is an inequalit in which one or more terms is a rational epression. The solution set to a rational inequalit must eclude all values of the variable that make the inequalit undefined. That is, eclude all values that make the denominator equal to zero for an rational epression in the inequalit. Eample Solving a Rational Inequalit b Using the Test Point Method Solve the inequalit b using the test point method. Skill Practice Answers. a, b
18 Chapter 9 More Equations and Inequalities Solution: a b Clear fractions. Solve for. 7 The solution to the related equation is 7, and the inequalit is undefined for. Therefore, the boundar points are and 7. Region I Region II Region III Step : Plot boundar points. Step : Select test points. 7 8 Test?? True Test :?? Undefined Step : Find the boundar points. Note that the inequalit is undefined for. Hence is automaticall a boundar point. To find an other boundar points, solve the related equation. Test? 7? False Test 7: 7 7? 9? True Test 8 8 8??? True Step : Test the boundar points. The boundar point cannot be included in the solution set, because it is undefined in the inequalit. The boundar point 7 makes the original inequalit true and must be included in the solution set. 7 8 The solution is, 7,. or 7 or, equivalentl in interval notation,
19 Section 9. Polnomial and Rational Inequalities 7 Calculator Connections Graph Y and Y. Y has a vertical asmptote at. Furthermore, Y Y at 7. Y Y that is, Y is below ) for and for 7. Y Y 7, ) 7 Y Skill Practice Solve the inequalit b using the test point method. Write the answer in interval notation.. Eample Solve the inequalit. Solution: a b The onl boundar point is. Solving a Rational Inequalit b Using the Test Point Method 7 7 Step : Find the boundar points. Note that the inequalit is undefined for, so is a boundar point. To find an other boundar points, solve the related equation. Clear fractions. There is no solution to the related equation. Region I Region II Step : Plot boundar points. Test : 7? 7? False Test : Step : Select test points. 7? 7 True Step : The boundar point cannot be included in the solution set because it is undefined in the original inequalit. False True Skill Practice Answers The solution is 7 or equivalentl in interval notation,,.. a, d
20 8 Chapter 9 More Equations and Inequalities Skill Practice 7. Solve the inequalit.. Inequalities with Special Case Solution Sets The solution to an inequalit is often one or more regions on the real number line. Sometimes, however, the solution to an inequalit ma be a single point on the number line, the empt set, or the set of all real numbers. Eample 7 Solving Inequalities Solve the inequalities. a. 9 b. c. 9 d Solution: a. 9 Notice that 9 is a perfect square trinomial. Factor 9. The quantit is a perfect square and is greater than or equal to zero for all real numbers,. The solution is all real numbers,,. True True True TIP: The graph of f 9, or equivalentl f, is equal to zero at and positive above the ais) for all other values of in its domain. 8 8 f) 9 f) 8 8 b. c This is the same inequalit as in part a) with the eception that the inequalit is strict. The solution set does not include the point where 9. Therefore, the boundar point is not included in the solution set. False True True The solution set is or 7 or equivalentl,,. 9 A perfect square cannot be less than zero. However, is equal to zero at. Therefore, the solution set is. False False Skill Practice Answers 7., True
21 Section 9. Polnomial and Rational Inequalities 9 d. 9 A perfect square cannot be negative; therefore, there are no real numbers such that. There is no solution. Skill Practice Solve the inequalities Skill Practice Answers 8. All real numbers;, 9.,,.. No solution Section 9. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems SelfTests NetTutor. Define the ke terms. a. Quadratic inequalit b. Boundar points c. Test point method d. Rational inequalit eprofessors Videos Review Eercises For Eercises 8, solve the compound inequalities. Write the solutions in interval notation or 8. a 7 or a 7 a. k 7 and 7 k and Concept : Solving Inequalities Graphicall For Eercises 9, estimate from the graph the intervals for which the inequalit is true f) 8 a. f 7 b. f a. g b. g 7 c. f d. f c. g d. g g)
22 Chapter 9 More Equations and Inequalities.. a. h b. h a. k b. c. h d. h 7 c. k 7 d. k k Concept : Solving Polnomial Inequalities b Using the Test Point Method For Eercises 8, solve the equation and related inequalities.. a.. a. b. b. c. 7 c. 7. a. 7. a. q q b. 7 b. q q c. 7 7 c. q q 7. a. p p p 8. a. ww w b. p p p b. ww w c. p p p c. ww 7 w For Eercises 9 8, solve the polnomial inequalit. Write the answer in interval notation. 9. t 7t. p p t t. mm m.. a a. w w b. c.. tt t.. w ww 7 8 p p p w w 7 w 8. p p p
23 Section 9. Polnomial and Rational Inequalities Concept : Solving Rational Inequalities b Using the Test Point Method For Eercises 9, solve the equation and related inequalities. 8 z 9. a.. a.. a.. a. a z 8 z b. b. b. b. a 7 z 8 z c. c. c. c. 7 a z w 8 w w 8 w w 8 w For Eercises, solve the rational inequalities. Write the answer in interval notation. b.... b 7 a a Concept : Inequalities with Special Case Solution Sets For Eercises 7, solve the inequalities Mied Eercises For Eercises 7 9, identif the inequalit as one of the following tpes: linear, quadratic, rational, or polnomial degree 7. Then solve the inequalit and write the answer in interval notation p p 7 p t w w t 7t
24 Chapter 9 More Equations and Inequalities a t a t 9. p 8 p t Graphing Calculator Eercises 9. To solve the inequalit 7 enter Y as and determine where the graph is above the ais. Write the solution in interval notation. 9. To solve the inequalit enter Y as and determine where the graph is below the ais. Write the solution in interval notation. 97. To solve the inequalit, enter Y as and determine where the graph is below the ais. Write the solution in interval notation. 98. To solve the inequalit 7, enter Y as and determine where the graph is above the ais. Write the solution in interval notation. For Eercises 99, determine the solution b graphing the inequalities Absolute Value Equations. Solving Absolute Value Equations An equation of the form a is called an absolute value equation. The solution includes all real numbers whose absolute value equals a. For eample, the solutions to the equation are as well as, because and. In Chapter, we introduced a geometric interpretation of. The absolute value of a number is its distance from zero on the number line Figure 9). Therefore, the solutions to the equation are the values of that are units awa from zero. Section 9. Concepts. Solving Absolute Value Equations. Solving Equations Having Two Absolute Values
25 Section 9. Absolute Value Equations units units or Figure 9 Absolute Value Equations of the Form If a is a real number, then a. If a, the equation a is equivalent to a or a.. If a, there is no solution to the equation a. Eample Solving Absolute Value Equations Solve the absolute value equations. a. b. w c. p d. Solution: a. The equation is in the form a, where a. or Rewrite the equation as a or a. b. w c. w w or w p Isolate the absolute value to write the equation in the form w a. Rewrite the equation as w a or w a. TIP: The absolute value must be isolated on one side of the equal sign before a can be rewritten as a or a. p or p Rewrite as two equations. Notice that the second equation p is the same as the first equation. Intuitivel, p is the onl number whose absolute value equals. d. This equation is of the form a, but a is negative. There is no number whose No solution absolute value is negative. Skill Practice Solve the absolute value equations.. 7. v. w. z Skill Practice Answers.. 7 or 7 v or v. w. No solution
26 Chapter 9 More Equations and Inequalities We have solved absolute value equations of the form a. Notice that can represent an algebraic quantit. For eample, to solve the equation w, we still rewrite the absolute value equation as two equations. In this case, we set the quantit w equal to and to, respectivel. w w or w Steps to Solve an Absolute Value Equation. Isolate the absolute value. That is, write the equation in the form a, where a is a constant real number.. If a, there is no solution.. Otherwise, if a, rewrite the absolute value equation as a or a.. Solve the individual equations from step.. Check the answers in the original absolute value equation. Eample Solving Absolute Value Equations Solve the absolute value equations. a. w b. c Solution: a. w w or w The equation is alread in the form a, where w. Rewrite as two equations. w 8 or w Solve each equation. w or w Check: w Check: w Check the solutions in the original w w equation. 8 Calculator Connections To confirm the answers to Eample a), graph Y abs and Y. The solutions to the equation w are the coordinates of the points of intersection, ) and,., ), ) Y Y
27 Section 9. Absolute Value Equations b. c c No solution Isolate the absolute value. The equation is in the form a, where c and a. Because a, there is no solution. There are no numbers c that will make an absolute value equal to a negative number. Avoiding Mistakes: Alwas isolate the absolute value first. Otherwise ou will get answers that do not check. Calculator Connections The graphs of Y abs and Y do not intersect. Therefore, there is no solution to the equation c. Y Y Skill Practice Solve the absolute value equations.. 9. z Eample Solving Absolute Value Equations Solve the absolute value equations. a. ` b. p ` 7 9. p.9.9 Solution: a. ` p ` 7 9 ` p ` ` p ` p or p p ` p ` p or p p or p or p Isolate the absolute value. Rewrite as two equations. Multipl all terms b to clear fractions. Both solutions check in the original equation. Skill Practice Answers. or. No solution
28 Chapter 9 More Equations and Inequalities b.. p.9.9. p. p or. p p. p. The solution is.. Isolate the absolute value. Rewrite as two equations. Notice that the equations are the same. Subtract. from both sides. Check: p.. p Skill Practice Solve the absolute value equations. 7. ` a `. Solving Equations Having Two Absolute Values Some equations have two absolute values. The solutions to the equation are or. That is, if two quantities have the same absolute value, then the quantities are equal or the quantities are opposites. Equalit of Absolute Values implies that or. Eample Solve the equations. a. w w b. Solving an Equation Having Two Absolute Values 8 Solution: a. w w Avoiding Mistakes: To take the opposite of the quantit w, use parentheses and appl the distributive propert. Skill Practice Answers 7. a or a 8.. w w or w w w w or w w w or 7w w or 7w w or w 7 The solutions are and 7. Rewrite as two equations, or. Solve for w. Both values check in
29 Section 9. Absolute Value Equations 7 b. 8 8 or 8 contradiction The onl solution is. Skill Practice 8 or 8 Solve the equations t t Rewrite as two equations, or. Solve for. checks in the original equation. Skill Practice Answers 9. or. t Section 9. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises. Define the ke term absolute value equation. Practice Problems SelfTests NetTutor eprofessors Videos Review Eercises For Eercises 7, solve the inequalities. Write the answers in interval notation.. a 7 and a 7. 7 or t Concept : Solving Absolute Value Equations For Eercises 8 9, solve the absolute value equations. 8. p 7 9. q..... w. 8 w 8. q 7. p 8. ` 9. `
30 8 Chapter 9 More Equations and Inequalities. ` 7z w. `. ` `. ` ` 7 t ` ` ` 7. w ` b k 7. ` k ` 9 7. ` h ` 7 Concept : Solving Equations Having Two Absolute Values For Eercises, solve the absolute value equations ` w w p ` ` 7. ` ` ` ` p ` n 7 n..m. 8.m. w w 9a 9a h h.n 9 7.n... 8 Epanding Your Skills. Write an absolute value equation whose solution is the set of real numbers units from zero on the number line.. Write an absolute value equation whose solution is the set of real numbers units from zero on the number line.. Write an absolute value equation whose solution is the set of real numbers units from zero on the number line. 7. Write an absolute value equation whose solution is the set of real numbers 9 units from zero on the number line. 7 For Eercises 8, solve the absolute value equations w 8. w w. ` `
31 Section 9. Absolute Value Inequalities 9 Graphing Calculator Eercises For Eercises 7, enter the left side of the equation as Y and enter the right side of the equation as Y. Then use the Intersect feature or Zoom and Trace to approimate the values where the two graphs intersect if the intersect) Solving Absolute Value Inequalities b Definition In Section 9., we studied absolute value equations in the form a. In this section we will solve absolute value inequalities. An inequalit in an of the forms a, a, 7 a, or a is called an absolute value inequalit. Recall that an absolute value represents distance from zero on the real number line. Consider the following absolute value equation and inequalities.. Absolute Value Inequalities or Solution: The set of all points units from zero on the number line units units Section 9. Concepts. Solving Absolute Value Inequalities b Definition. Solving Absolute Value Inequalities b the Test Point. Translating to an Absolute Value Epression
32 Chapter 9 More Equations and Inequalities. 7 or 7 Solution: The set of all points more than units from zero units units. Solution: The set of all points less than units from zero units units Absolute Value Equations and Inequalities Let a be a real number such that a 7. Then Equation/ Solution Inequalit Equivalent Form) Graph a a or a a a 7 a a or 7 a a a a a a a a To solve an absolute value inequalit, first isolate the absolute value and then rewrite the absolute value inequalit in its equivalent form. Eample Solving Absolute Value Inequalities Solve the inequalities. a. w 7 b. ` t ` Solution: a. w 7 w w w w The solution is w w,. Isolate the absolute value first. The inequalit is in the form a, where w. Rewrite in the equivalent form a a. Solve for w. or, equivalentl in interval notation,
33 Section 9. Absolute Value Inequalities Calculator Connections Graph Y abs and Y 7. On the given displa window, Y Y Y is below ) for Y., 7), 7) Y Y 7 b. ` t ` ` t ` ` t ` t or t t or t 7 a tb or a tb 7 Write the inequalit with the absolute value on the left. Isolate the absolute value. The inequalit is in the form a, where t. Rewrite in the equivalent form a or a. Solve the compound inequalit. Clear fractions. t or t The solution is t t or t,,. or, equivalentl in interval notation, Calculator Connections Graph Y abs and Y. On the given displa window, Y Y for or. Y, ), ) Y Skill Practice Solve the inequalities. Write the solutions in interval notation.. ƒ t ƒ. ` c ` B definition, the absolute value of a real number will alwas be nonnegative. Therefore, the absolute value of an epression will alwas be greater than Skill Practice Answers. 7,., 9 h,
34 Chapter 9 More Equations and Inequalities a negative number. Similarl, an absolute value can never be less than a negative number. Let a represent a positive real number. Then The solution to the inequalit 7 a is all real numbers,,. There is no solution to the inequalit a. Eample Solving Absolute Value Inequalities Solve the inequalities. a. d 7 b. d 7 7 Solution: a. d 7 d Isolate the absolute value. An absolute value epression cannot be less than a negative number. Therefore, there is no solution. No solution b. d 7 7 d 7 All real numbers,, Isolate the absolute value. The inequalit is in the form 7 a, where a is negative. An absolute value of an real number is greater than a negative number. Therefore, the solution is all real numbers. Calculator Connections B graphing Y abs 7 and Y 7 Y, we see that Y 7 Y Y is above Y for all real numbers on the given displa window. Y Skill Practice Solve the inequalities.. ƒ p ƒ. ƒ p ƒ 7 Eample Solving Absolute Value Inequalities Solve the inequalities. a. b. 7 Skill Practice Answers. No solution. All real numbers;, Solution: a. The absolute value is alread isolated. The absolute value of an real number is nonnegative. Therefore, the solution is all real numbers,,. b. 7 An absolute value will be greater than zero at all points ecept where it is equal to zero. That is, the points) for which must be ecluded from the solution set.
35 Section 9. Absolute Value Inequalities or The second equation is the same as the first. Therefore, eclude from the solution. The solution is,,. or equivalentl in interval notation, Calculator Connections Graph Y abs. From the graph, Y at the intercept). On the given displa window, Y 7 for or 7. Y Skill Practice Solve the inequalities.. ƒ ƒ. ƒ ƒ 7. Solving Absolute Value Inequalities b the Test Point Method For each problem in Eample, the absolute value inequalit was converted to an equivalent compound inequalit. However, sometimes students have difficult setting up the appropriate compound inequalit. To avoid this problem, ou ma want to use the test point method to solve absolute value inequalities. Solving Inequalities b Using the Test Point Method. Find the boundar points of the inequalit. Boundar points are the real solutions to the related equation and points where the inequalit is undefined.). Plot the boundar points on the number line. This divides the number line into regions.. Select a test point from each region and substitute it into the original inequalit. If a test point makes the original inequalit true, then that region is part of the solution set.. Test the boundar points in the original inequalit. If a boundar point makes the original inequalit true, then that point is part of the solution set. Skill Practice Answers.,. e a, f ; b h a, b
36 Chapter 9 More Equations and Inequalities To demonstrate the use of the test point method, we will repeat the absolute value inequalities from Eample. Notice that regardless of the method used, the absolute value is alwas isolated first before an further action is taken. Eample Solving Absolute Value Inequalities b the Test Point Method Solve the inequalities b using the test point method. a. w 7 b. ` t ` Solution: a. w 7 w w w or w w or w or w w Isolate the absolute value. Step : Solve the related equation. Write as an equivalent sstem of two equations. These are the onl boundar points. Region I Region II Region III Step : Plot the boundar points. Step : Select a test point from each region. Test w :? 7 Test w : Test w :? 7? 7? 7? 7? 7 False? 7? 7 True? 7? 7 9? 7 False False True False Step : Because the original inequalit is a strict inequalit, the boundar points where equalit occurs) are not included. The solution is w w or, equivalentl in interval notation,,.
37 Section 9. Absolute Value Inequalities b. ` t ` ` t ` ` t ` ` t ` t or t t 7 or t t or t Region I Region II Region III Write the inequalit with the absolute value on the left. Isolate the absolute value. Step : Solve the related equation. Write the union of two equations. These are the boundar points. Step : Plot the boundar points. Step : Select a test point from each region. Test t : Test t : Test t :?? `? ` ` ` ` `????? 8? True? False True Step : The original inequalit uses the sign. Therefore, the boundar points where equalit occurs) must be part of the solution set. True False True The solution is or,,. or, equivalentl in interval notation, Skill Practice Solve the inequalities b using the test point method. 7. t 8. ` c ` 7 Skill Practice Answers 7. c, 8 d 8., 8 h,
38 Chapter 9 More Equations and Inequalities Eample Solving Absolute Value Inequalities Solve the inequalities. a. ` b. ` ` ` Solution: a. ` The absolute value is alread isolated. ` No solution Because the absolute value of an real number is nonnegative, an absolute value cannot be strictl less than zero. Therefore, there is no solution to this inequalit. b. ` The absolute value is alread isolated. ` An absolute value will never be less than zero. However, an absolute value ma be equal to zero. Therefore, the onl solutions to this inequalit are the solutions to the related equation. ` ` The solution set is. Set up the related equation. This is the onl boundar point. False True False Calculator Connections Graph Y abs. Notice that on the given viewing window the graph of Y does not etend below the ais. Therefore, there is no solution to the inequalit Y. Because Y at, the inequalit Y has a solution at. Y Skill Practice Solve the inequalities. 9. `. ` ` ` Skill Practice Answers 9. No solution.
39 Section 9. Absolute Value Inequalities 7. Translating to an Absolute Value Epression Absolute value epressions can be used to describe distances. The distance between c and d is given b c d. For eample, the distance between and on the number line is as epected. Eample Epressing Distances with Absolute Value Write an absolute value inequalit to represent the following phrases. a. All real numbers, whose distance from zero is greater than units b. All real numbers, whose distance from 7 is less than units Solution: a. All real numbers, whose distance from zero is greater than units 7 or simpl 7 b. All real numbers, whose distance from 7 is less than units 7 or simpl 7 Skill Practice units units Write an absolute value inequalit to represent the following phrases.. All real numbers whose distance from zero is greater than units. All real numbers whose distance from is less than units units Absolute value epressions can also be used to describe boundaries for measurement error. Eample 7 Epressing Measurement Error with Absolute Value Latoa measured a certain compound on a scale in the chemistr lab at school. She measured 8 g of the compound, but the scale is onl accurate to. g. Write an absolute value inequalit to epress an interval for the true mass,, of the compound she measured. Solution: Because the scale is onl accurate to. g, the true mass,, of the compound ma deviate b as much as. g above or below 8 g. This ma be epressed as an absolute value inequalit: 8.. or equivalentl Skill Practice Answers. ƒ ƒ 7. ƒ ƒ
40 8 Chapter 9 More Equations and Inequalities Skill Practice Skill Practice Answers. ƒ t ƒ.. Vonzell molded a piece of metal in her machine shop. She measured the thickness at mm. Her machine is accurate to. mm. Write an absolute value inequalit to epress an interval for the true measurement of the thickness, t, of the metal. Section 9. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises. Define the ke term absolute value inequalit. Practice Problems SelfTests NetTutor eprofessors Videos Review Eercises For Eercises, solve the equations... 7 For Eercises 7, solve the inequalities and graph the solution set. Write the solution in interval notation.. w 9. and. m 7 or m 7 7. b 7 or b 7 Concepts and : Solving Absolute Value Inequalities For Eercises 8 8, solve the equations and inequalities. For each inequalit, graph the solution set and epress the solution in interval notation. 8. a. 9. a. a. a. 7 b. 7 b. a 7 b. 7 7 c. c. a c. 7. a. w. a. p. a. b. w 7 b. p 7 b. 7 c. w c. p c.. a.. a. z. a. b. 7 b. z 7 b. 7 c. c. z c.
41 Section 9. Absolute Value Inequalities 9 7. a. p 8. a. b. p 7 b. k 7 k 7 7 c. p c. k 7 For Eercises 9 8, solve the absolute value inequalities b using either the definition or the test point method. Graph the solution set and write the solution in interval notation t. p 7.. 7n k 7 w 8. h 9 9. ` `. ` ` m 7. ` `. ` ` m p.. z 7. c t Concept : Translating to an Absolute Value Epression For Eercises 9, write an absolute value inequalit equivalent to the epression given. 9. All real numbers whose distance from is greater than 7. All real numbers whose distance from is less than. All real numbers whose distance from is at most. All real numbers whose distance from is at least
42 Chapter 9 More Equations and Inequalities. A oz jug of orange juice ma not contain eactl oz of juice. The possibilit of measurement error eists when the jug is filled in the factor. If the maimum measurement error is. oz, write an absolute value inequalit representing the range of volumes,, in which the orange juice jug ma be filled.. The length of a board is measured to be. in. The maimum measurement error is. in. Write an absolute value inequalit that represents the range for the length of the board,.. A bag of potato chips states that its weight is oz. The maimum measurement error is ; 8 oz. Write an absolute value inequalit that represents the range for the weight,, of the bag of chips. 7. A 8in. bolt varies in length b at most in. Write an absolute value inequalit that represents the range for the length,, of the bolt. 7. The width, w, of a bolt is supposed to be cm but ma have a.cm margin of error. Solve w., and interpret the solution to the inequalit in the contet of this problem. 8. In the election, Senator Barak Obama was projected to receive 7% of the votes with a margin of error of %. Solve p.7., and interpret the solution to the inequalit in the contet of this problem. Epanding Your Skills For Eercises 9, match the graph with the inequalit: a. b. 7 c. d. 7 Graphing Calculator Eercises To solve an absolute value inequalit b using a graphing calculator, let Y equal the left side of the inequalit and let Y equal the right side of the inequalit. Graph both Y and Y on a standard viewing window and use an Intersect feature or Zoom and Trace to approimate the intersection of the graphs. To solve Y 7 Y, determine all values where the graph of Y is above the graph of Y. To solve Y Y, determine all values where the graph of Y is below the graph of Y. For Eercises 7, solve the inequalities using a graphing calculator
43 Problem Recognition Eercises Equations and Inequalities. ` ` ` ` Chapter 9 Problem Recognition Eercises Equations and Inequalities For Eercises, identif the categor for each equation or inequalit choose from the list here). Then solve the equation or inequalit. Linear Quadratic Polnomial degree greater than ) Rational Absolute value. z z 9 7. a a t t p p p p
44 Chapter 9 More Equations and Inequalities. 7. ` ` 7. b b b Section 9. Concepts. Graphing Linear Inequalities in Two Variables. Compound Linear Inequalities in Two Variables. Graphing a Feasible Region Linear Inequalities in Two Variables. Graphing Linear Inequalities in Two Variables A linear inequalit in two variables and is an inequalit that can be written in one of the following forms: a b c, a b 7 c, a b c, or a b c, provided a and b are not both zero. A solution to a linear inequalit in two variables is an ordered pair that makes the inequalit true. For eample, solutions to the inequalit are ordered pairs, ) such that the sum of the  and coordinates is less than. This inequalit has an infinite number of solutions, and therefore it is convenient to epress the solution set as a graph. To graph a linear inequalit in two variables, we will follow these steps. Graphing a Linear Inequalit in Two Variables. Solve for, if possible.. Graph the related equation. Draw a dashed line if the inequalit is strict, or. Otherwise, draw a solid line.. Shade above or below the line as follows: Shade above the line if the inequalit is of the form 7 a b or a b. Shade below the line if the inequalit is of the form a b or a b. This process is demonstrated in Eample. Eample Graphing a Linear Inequalit in Two Variables Graph the solution set.
45 Section 9. Linear Inequalities in Two Variables Solution: Solve for. Net graph the line defined b the related equation. Because the inequalit is of the form a b, the solution to the inequalit is the region below the line. See Figure 9. Figure 9 Skill Practice Graph the solution set.. After graphing the solution to a linear inequalit, we can verif that we have shaded the correct side of the line b using test points. In Eample, we can pick an arbitrar ordered pair within the shaded region. Then substitute the  and coordinates in the original inequalit. If the result is a true statement, then that ordered pair is a solution to the inequalit and suggests that other points from the same region are also solutions. For eample, the point, ) lies within the shaded region Figure 97).? True Substitute, ) in the original inequalit. The point, ) from the shaded region is a solution. In Eample, we will graph the solution set to a strict inequalit. A strict inequalit uses the smbol or. In such a case, the boundar line will be drawn as a dashed line. This indicates that the boundar itself is not part of the solution set. Test point, ) Figure 97 Eample Graphing a Linear Inequalit in Two Variables Graph the solution set. Solution: 7 7 Solve for. Reverse the inequalit sign. Skill Practice Answers.
46 Chapter 9 More Equations and Inequalities Figure 98 Graph the line defined b the related equation,. The boundar line is drawn as a dashed line because the inequalit is strict. Also note that the line passes through the origin. Because the inequalit is of the form 7 a b, the solution to the inequalit is the region above the line. See Figure 98. Skill Practice. Graph the solution set. Test point, ) Figure 99 In Eample, we cannot use the origin as a test point, because the point, ) is on the boundar line. Be sure to select a test point strictl within the shaded region. In this case, we choose, ). See Figure 99.? True Substitute, ) in the original inequalit. The point, ) from the shaded region is a solution to the original inequalit. In Eample, we encounter a situation in which we cannot solve for the variable. Eample Graphing a Linear Inequalit in Two Variables Graph the solution set. Solution: Skill Practice Answers.. Figure 9 Skill Practice. Test point, ) Graph the solution set. In this inequalit, there is no variable. However, we can simplif the inequalit b solving for. Graph the related equation. This is a vertical line. The boundar is drawn as a solid line, because the inequalit is not strict,. To shade the appropriate region, refer to the inequalit,. The points for which is greater than are to the right of. Therefore, shade the region to the right of the line Figure 9). Selecting a test point such as, ) from the shaded region indicates that we have shaded the correct side of the line. Substitute. True
47 Section 9. Linear Inequalities in Two Variables. Compound Linear Inequalities in Two Variables Some applications require us to find the union or intersection of two or more linear inequalities. Eample Graphing a Compound Linear Inequalit Graph the solution set of the compound inequalit. Solution: Solve each inequalit for. First inequalit 7 7 and Second inequalit The inequalit is of the form The inequalit is of the form 7 a b. Graph above the a b. Graph below the boundar line. See Figure 9). boundar line. See Figure 9). Figure 9 Figure 9 The region bounded b the inequalities is the region above the line and below the line. This is the intersection or overlap of the two regions shown in purple in Figure 9). Figure 9 The intersection is the solution set to the sstem of inequalities. See Figure 9. Figure 9
48 Chapter 9 More Equations and Inequalities Skill Practice Graph the solution set.. 7 Eample demonstrates the union of the solution sets of two linear inequalities. Eample Graphing a Compound Linear Inequalit Graph the solution set of the compound inequalit. or Solution: First inequalit Second inequalit The graph of is the region The inequalit is of the form a b. on and below the horizontal Graph a solid line and the region below line. See Figure 9.) the line. See Figure 9.) Figure 9 Figure 9 Skill Practice Answers.. The solution to the compound inequalit or is the union of these regions, Figure 97. Skill Practice. or Figure 97 Graph the solution set.
49 Section 9. Linear Inequalities in Two Variables 7 Eample Graphing Compound Linear Inequalities Describe the region of the plane defined b the following sstems of inequalities. and Solution: on the ais and in the second and third quadrants. on the ais and in the first and second quadrants. The intersection of these regions is the set of points in the second quadrant with the boundar included). Skill Practice. and Graph the region defined b the sstem of inequalities.. Graphing a Feasible Region When two variables are related under certain constraints, a sstem of linear inequalities can be used to show a region of feasible values for the variables. Eample 7 Graphing a Feasible Region Susan has two tests on Frida: one in chemistr and one in pscholog. Because the two classes meet in consecutive hours, she has no stud time between tests. Susan estimates that she has a maimum of hr of stud time before the tests, and she must divide her time between chemistr and pscholog. Let represent the number of hours Susan spends studing chemistr. Let represent the number of hours Susan spends studing pscholog. a. Find a set of inequalities to describe the constraints on Susan s stud time. b. Graph the constraints to find the feasible region defining Susan s stud time. Solution: a. Because Susan cannot stud chemistr or pscholog for a negative period of time, we have and. Furthermore, her total time studing cannot eceed hr:. A sstem of inequalities that defines the constraints on Susan s stud time is Skill Practice Answers.
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