More Equations and Inequalities


 Flora Cooper
 2 years ago
 Views:
Transcription
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.
1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationSolving x < a. Section 4.4 Absolute Value Inequalities 391
Section 4.4 Absolute Value Inequalities 391 4.4 Absolute Value Inequalities In the last section, we solved absolute value equations. In this section, we turn our attention to inequalities involving absolute
More informationRational Expressions and Rational Equations
Rational Epressions and Rational Equations 6 6. Rational Epressions and Rational Functions 6. Multiplication and Division of Rational Epressions 6. Addition and Subtraction of Rational Epressions 6.4 Comple
More informationLinear Equations in Two Variables
Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations
More informationPolynomial and Rational Functions
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationLinear Inequality in Two Variables
90 (7) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationSection 7.1 Graphing Linear Inequalities in Two Variables
Section 7.1 Graphing Linear Inequalities in Two Variables Eamples of linear inequalities in two variables include + 6, and 1 A solution of a linear inequalit is an ordered pair that satisfies the
More information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
More informationInequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10,
Chapter 4 Inequalities and Absolute Values Assignment Guide: E = ever other odd,, 5, 9, 3, EP = ever other pair,, 2, 5, 6, 9, 0, Lesson 4. Page 7577 Es. 420. 2328, 2939 odd, 4043, 4952, 5973 odd
More informationSimplification of Rational Expressions and Functions
7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More information2.3 Quadratic Functions
. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More information8.7 Systems of NonLinear Equations and Inequalities
8.7 Sstems of NonLinear Equations and Inequalities 67 8.7 Sstems of NonLinear Equations and Inequalities In this section, we stud sstems of nonlinear equations and inequalities. Unlike the sstems of
More informationLesson 6: Linear Functions and their Slope
Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation
More informationUnit 1 Study Guide Systems of Linear Equations and Inequalities. Part 1: Determine if an ordered pair is a solution to a system
Unit Stud Guide Sstems of Linear Equations and Inequalities 6 Solving Sstems b Graphing Part : Determine if an ordered pair is a solution to a sstem e: (, ) Eercises: substitute in for and  in for in
More information13 Graphs, Equations and Inequalities
13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,
More informationGraphing Linear Inequalities in Two Variables
5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when
More informationPolynomial and Rational Functions
Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationQuadratic Functions and Parabolas
MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens
More informationSection 0.2 Set notation and solving inequalities
Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationRational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza
Rational Functions 7 The ozone laer protects Earth from harmful ultraviolet radiation. Each ear, this laer thins dramaticall over the poles, creating ozone holes which have stretched as far as Australia
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationAnalyzing the Graph of a Function
SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More information2 Analysis of Graphs of
ch.pgs116 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,
More informationGraphing Nonlinear Systems
10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationIdentify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4
Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationGraphing Quadratic Functions
A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which
More informationQ (x 1, y 1 ) m = y 1 y 0
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
More informationIf (a)(b) 5 0, then a 5 0 or b 5 0.
chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic
More informationEssential Question: What are two ways to solve an absolute value inequality? A2.6.F Solve absolute value linear inequalities.
Locker LESSON.3 Solving Absolute Value Inequalities Name Class Date.3 Solving Absolute Value Inequalities Teas Math Standards The student is epected to: A.6.F Essential Question: What are two was to solve
More informationI think that starting
. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
More informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
More informationP1. Plot the following points on the real. P2. Determine which of the following are solutions
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More informationSECTION 25 Combining Functions
2 Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
More informationLinear Inequalities, Systems, and Linear Programming
8.8 Linear Inequalities, Sstems, and Linear Programming 481 8.8 Linear Inequalities, Sstems, and Linear Programming Linear Inequalities in Two Variables Linear inequalities with one variable were graphed
More informationA Summary of Curve Sketching. Analyzing the Graph of a Function
0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph
More informationHigher. Functions and Graphs. Functions and Graphs 18
hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms
More informationSolve the linear programming problem graphically: Minimize w 4. subject to. on the vertical axis.
Do a similar example with checks along the wa to insure student can find each corner point, fill out the table, and pick the optimal value. Example 3 Solve the Linear Programming Problem Graphicall Solve
More informationCoordinate Geometry. Positive gradients: Negative gradients:
8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight
More informationAnytime plan TalkMore plan
CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations
More informationSolving Absolute Value Equations and Inequalities Graphically
4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More information3.4 The PointSlope Form of a Line
Section 3.4 The PointSlope Form of a Line 293 3.4 The PointSlope Form of a Line In the last section, we developed the slopeintercept form of a line ( = m + b). The slopeintercept form of a line is
More information3.1 Graphically Solving Systems of Two Equations
3.1 Graphicall Solving Sstems of Two Equations (Page 1 of 24) 3.1 Graphicall Solving Sstems of Two Equations Definitions The plot of all points that satisf an equation forms the graph of the equation.
More information4.4 Logarithmic Functions
SECTION 4.4 Logarithmic Functions 87 4.4 Logarithmic Functions PREPARING FOR THIS SECTION Before getting started, review the following: Solving Inequalities (Appendi, Section A.8, pp. 04 05) Polnomial
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions  4, use the graph at the right.. Eplain wh the graph
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More information1.6 Graphs of Functions
.6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each coordinate was matched with onl one coordinate. We spent most of our time
More informationExponential and Logarithmic Functions
Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationTranslating Points. Subtract 2 from the ycoordinates
CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that
More informationSummer Review For Students Entering Algebra 2
Summer Review For Students Entering Algebra Board of Education of Howard Count Frank Aquino Chairman Ellen Flnn Giles Vice Chairman Larr Cohen Allen Der Sandra H. French Patricia S. Gordon Janet Siddiqui
More information4.1 PiecewiseDefined Functions
Section 4.1 PiecewiseDefined Functions 335 4.1 PiecewiseDefined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept
More informationWhy should we learn this? One realworld connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the intercept. One realworld connection is to find the rate
More informationSystems of Equations. from Campus to Careers Fashion Designer
Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions 0 0. Algebra and Composition of Functions 0. Inverse Functions 0. Eponential Functions 0. Logarithmic Functions 0. Properties of Logarithms 0. The Irrational Number
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationGraph each function. Compare to the parent graph. State the domain and range. 1. SOLUTION:
 Root Functions Graph each function. Compare to the parent graph. State the domain and range...5.. 5. 6 is multiplied b a value greater than, so the graph is a vertical stretch of. Another wa to identif
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More information3 Rectangular Coordinate System and Graphs
060_CH03_13154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationGRAPHS OF RATIONAL FUNCTIONS
0 (0) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationGRAPHING SYSTEMS OF LINEAR INEQUALITIES
444 (8 5) Chapter 8 Sstems of Linear Equations and Inequalities GETTING MORE INVOLVED 5. Discussion. When asked to graph the inequalit, a student found that (0, 5) and (8, 0) both satisfied. The student
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More informationQuadratic Equations and Inequalities
MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More information11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems
a b Objectives Solve maimum minimum problems involving quadratic functions. Fit a quadratic function to a set of data to form a mathematical model, and solve related applied problems. 11.7 MATHEMATICAL
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationGraphing and transforming functions
Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 0 C)
Sample/Practice Final Eam MAT 09 Beginning Algebra Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the epression. 1) 2  [  ( 38)]
More informationx 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac
Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square
More informationReteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.
Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.
More informationa > 0 parabola opens a < 0 parabola opens
Objective 8 Quadratic Functions The simplest quadratic function is f() = 2. Objective 8b Quadratic Functions in (h, k) form Appling all of Obj 4 (reflections and translations) to the function. f() = a(
More informationSOLVING SYSTEMS OF EQUATIONS
SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,
More informationQuadratic Functions. MathsStart. Topic 3
MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE
More informationAlex and Morgan were asked to graph the equation y = 2x + 1
Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and intercept wa First, I made a table. I chose some values, then plugged
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) 
More informationFunctions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 21 Functions 22 Elementar Functions: Graphs and Transformations 23 Quadratic
More informationReasoning with Equations and Inequalities
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations.
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationMethods to Solve Quadratic Equations
Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a seconddegree
More information