CHAPTER 2 Solving Equations and Inequalities
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1 CHAPTER Solving Equations and Inequalities Section. Linear Equations and Problem Solving Section. Solving Equations Graphically Section. Comple Numbers Section. Solving Quadratic Equations Algebraically Section. Solving Other Types of Equations Algebraically Section. Solving Inequalities Algebraically and Graphically.... Section.7 Linear Models and Scatter Plots Review Eercises Practice Test
2 CHAPTER Solving Equations and Inequalities Section. Linear Equations and Problem Solving You should know how to solve linear equations: a b 0. An identity is an equation whose solution consists of every real number in its domain. To solve an equation you can: (a) Add or subtract the same quantity from both sides. Multiply or divide both sides by the same nonzero quantity. To solve an equation that can be simplified to a linear equation: (a) Remove all symbols of grouping and all fractions. Combine like terms. (c) Solve by algebra. (d) Check the answer. A solution that does not satisfy the original equation is called an etraneous solution. You should be able to set up mathematical models to solve problems. You should be able to translate key words and phrases. (a) Equality: Addition: Equals, Equal to, is, are, was, Sum, plus, greater, increased by, more than, will be, represents eceeds, total of (c) Subtraction: (d) Multiplication: Difference, minus, less than, Product, multiplied by, twice, times, percent of decreased by, subtracted from, reduced by, the remainder (e) Division: (f) Consecutive: Quotient, divided by, ratio, per Net, subsequent You should know the following formulas: (a) Perimeter: Area:. Square: P s. Square: A s. Rectangle: A LW. Rectangle: P L W. Circle: A r. Circle: C r. Triangle: (c) Volume. Cube: V s. Rectangular solid: V LW H. Cylinder: V r h. Sphere: V r (e) Compound Interest: A P r n nt A bh (d) Simple Interest: I Prt (f) Distance: D r t (g) Temperature: F 9 C You should be able to solve word problems. Study the eamples in the tet carefully. 8
3 8 Chapter Solving Equations and Inequalities Vocabulary Check. equation. solve. identities, conditional equations. a b 0. etraneous. Mathematical modeling 7. formulas. (a)? is a solution. (c) is undefined is not a solution. (d)? 8 is not a solution.? is not a solution.. (a)? 0 is undefined. is not a solution. is a solution. (c) 0? (d)? is not a solution. is not a solution.. (a)? 0? 9 0 is not a solution. 0 is not a solution. (c)? (d)? is not a solution. is a solution.
4 Section. Linear Equations and Problem Solving 8 7. is an identity by the Distributive Property. It is true for all real values of is an identity since is conditional. There are real values of for which the equation is not true.. Method : Method : Graph y and y in the 8 same viewing window. These lines intersect at Method : Method : Graph y and y in the same viewing window. These lines intersect at y y 7 y. y y 7 y 7 y y y y y 7 8 y y y 9 y z z 0 z z 0 z z 0 z 0 z 0 z z 7. z 0z u u z 8 0z 00 u u z 7 0z 00 u u 7 7 8z 00 u u 8 7 z 7 8 u 0 u 0
5 8 Chapter Solving Equations and Inequalities A check reveals that is an etraneous solution, so there is no solution.. A bh. A bh A b h A P r n nt P A r n nt P A r n nt. S rl a r Sr S rl a Sr rl S a rs L S a 7. V a b V a b 9. P l w w P l w P l P l r S a S L. V r h. h V r S rh. r S h PV nrt R PV nt inches
6 Section. Linear Equations and Problem Solving (a) l l.w P l w.w w w (c) w w meters and l. 7. meters w test test test test. (a) Test average Four tests at 00 points each 00 points points needed for an A. So far, points points needed on fourth test.. Rate Distance Time Total time Total distance Rate 0 kilometers hours 00 kilometershour 00 kilometers 00 kilometershour hours. Model: (Distance) (rate)(time time Labels: Distance miles, rate r, Equation: time 00 r r r distance rate 00 hours, r time distance rate 00 0 hours The average speed for the round trip was approimately. miles per hour. ) 7. Let h height of the building in feet. (a) 80 ft h ft ft Not drawn to scale h feet feet 80 feet. feet h 80..h 0 h 9. feet 9. I Prt $90
7 88 Chapter Solving Equations and Inequalities 7. Let amount in % fund. Then,000 amount in % fund , You must invest $8000 in the % fund, and, in the % fund. 7. Let amount invested in DVD players, and 7. Let number of pounds of $.9 nuts. Then y amount invested in VCRs. 00 number of pounds of $.89 nuts. y 0,000 y 0, y 0.90,000, ,000, $0,000 y $0, Use 0 pounds of each kind lbs of $.9 nuts 00 0 lbs of $.89 nuts 77. A bh h A b feet 79. (a) l w l w, h w V lwh ww w 0 h 9 w 0 8. Te mperature ( C) 0 0:00 A.M. :00 A.M. :00 P.M. :00 P.M. :00 P.M. :00 P.M. :00 P.M. :00 P.M. :00 P.M. w Time w 8 inches Dimensions 8 inches 8. W W L 8. False. 0 is a quadratic equation child feet from the 0-pound 0 or 0 0
8 Section. Solving Equations Graphically You need a b c or b c a a c. One answer is a, c and b and another is a, c, and b. 89. Equivalent equations are derived from the substitution principle and simplification techniques. They have the same solution(s). 8 and are equivalent equations. 9. c 0 c, c 0 c c c 8c c 8 9. y 8 9. y y y y y f g f g 8 0. fg8 f8g f g fg f9 7 Section. Solving Equations Graphically You should be able to find the intercepts of the graph of an equation. You should be able to find the zeros of a function y f by solving the equation f 0. You should be able to find the solutions of an equation graphically using a graphing utility. You should be able to use the zoom and trace features to find solutions to any desired accuracy. You should be able to find the points of intersection of two graphs. Vocabulary Check. -intercept, y-intercept. zero. point of intersection. y Let Let y 0: 0, 0 -intercept 0: y 0 y 0, y-intercept. y Let Let y 0: 0,, 0,, 0 0: y 0 0 0, y-intercept -intercepts
9 90 Chapter Solving Equations and Inequalities. y Let y 0: 0 0, 0, 0,, 0 Let 0: y , 0 y-intercept -intercepts 7. y If 0, then 0y 0, which is impossible. Similarly, y 0 is impossible. Hence there are no intercepts. 9. y y 0 Let y 0: 0, 0 Let 0: y 0 y 0, -intercept y-intercept., 0, 0, 8 y 0 0 y. 0 0, 0, 0, y y 0. f f ,, 0,,
10 Section. Solving Equations Graphically 9 9. f f f f f f f f 0
11 9 Chapter Solving Equations and Inequalities , , 0., , , , , (a) Because of the sign change, < <. CONTINUED
12 Section. Solving Equations Graphically 9. CONTINUED Because of the sign change,.8 < <.9. To improve accuracy, evaluate the epression for values in this interval and determine where the sign changes. (c) Let y..8. The graph of crosses the -ais at.8. y 7. y 9. y y y y 0 y, y, y, y,, y,. y 0 y 0. y y 0 0 8, y 8 0, y 8, y y y, y,. y 9 7. y, y = 9 (, ) 8 y y, y.9,.898,.9, y = (.9,.898) y = 0 (.9, 7.899) 9 y = 9. y y, y 0, 0,, 8,, 8 y = (, 8) (, 8) (0, 0) y =
13 9 Chapter Solving Equations and Inequalities 7. (a) The second method decreases the accuracy. 7. (a) t 80 Domain: (c) If the time was hours and minutes, then t and. miles. 7. (a) A 0. Domain: (a) Area A (c) If the final miture is 0% concentrate, then A 0. and. gallons. (c) A 00.7 units 79. (a) T I S 0,000 0,000 If S 00 0, $800. (c) If T,800 0, $700. (d) If T,00 0,000 then 000. Thus, S 0,000 $ (a) Intersection:.7, 88.7 (c) The slopes indicate the change in population per year. Arizona s population is growing faster..t t 8t t.7 A S 88.7 The point.7, 88.7 indicates the year, 98, in which the two populations were the same, about 88.7 thousand. (d) For 00, t 0 and S thousand and A 79 thousand. Answers will vary. 8. True 8. False. Two linear equations could have an infinite number of points of intersection. For eample, y and y. 87. From the table, f 0 for. 89. From the table, g ffor. In this 9. case, f and g.
14 Section. Comple Numbers Section. Comple Numbers You should know how to work with comple numbers. Operations on comple numbers (a) Addition: a bi c di a c b di Subtraction: a bi c di a c b di (c) Multiplication: a bic di ac bd ad bci (d) Division: The comple conjugate of a bi is a bi: a bia bi a b The additive inverse of a bi is a bi. The multiplicative inverse of a bi is a bi a b. a a i for a > 0. a bi a bi c di c di c di c di ac bd bc ad c d c d i Vocabulary Check. (a) ii iii (c) i.,. comple, a bi. real, imaginary. Mandelbrot Set. a bi 9 i. a b i 8i. a 9 a a i b b 8 b 7. 0i 9. i i i i i 0.i. i 7 i 7 i i i i 7 i 9. i 7i i 7i 0i. i i i 9 0 i 9 7 i
15 9 Chapter Solving Equations and Inequalities...i.8.i. 7.i. ii i i 0i 0 9. i i i i i i i. i8 i i 0i. i 0 0 i 0 i 0 i 0i 0. i i i i i i 80i 80i 7. i is the comple conjugate of i. 9. i is the comple conjugate of i. i i 9 i i. 0i is the comple conjugate of. i is the comple conjugate of 0 0i. i. 0i0i 0 i i 9. i i i i i i i i 7. i i 8 0i i i 8 0 i 9.. i i i i i i i i i i i i i i i i i i i i.. i i i 0i i 9 0i 9 0i 9 0i 0 9i i i i i 8i i 8i i i i 8i 9i 9 8i 9i 8i 8i 8i 00 7i i 8 97i i
16 Section. Comple Numbers i i i i i 9. i i i 7 i i i 7i. i i i i i i i i. 8 i i i i i 9i i i 9 i 8 i i i i i 9i i i 9 i 8 The three numbers are cube roots of 8.. i 7. i i 7. i 7. Imaginary ais Imaginary ais Imaginary ais 7 7 Real ais Real ais Real ais 77. The comple number is in the Mandelbrot Set since for c i, i, the corresponding Mandelbrot sequence is i, i, i, 7 i, which is bounded. Or in decimal form 0,77 97, 09 i, 0.i, 0. 0.i, i, i, i, i. 8,,0,07,8,9,97,9,7,78 i 79. z i z i z z z i i i 8 z 8 i 8 i i 8 i i i i i 8 i i
17 98 Chapter Solving Equations and Inequalities 8. False. A real number a 0i a is equal to 8. False. For eample, i i, which its conjugate. is not an imaginary number. 8. True. Let z a b i and z a b i. Then 87. z z a b ia b i 0 0 a a b b a b b a i a a b b a b b a i a b ia b i a b i a b i z z. 89. Section. Solving Quadratic Equations Algebraically You should be able to solve a quadratic equation by factoring, if possible. You should be able to solve a quadratic equation of the form u d by etracting square roots. You should be able to solve a quadratic equation by completing the square. You should know and be able to use the Quadratic Formula: For a b c 0, a 0, b ± b ac. a You should be able to determine the types of solutions of a quadratic equation by checking the discriminant b ac. (a) If b ac > 0, there are two distinct real solutions. If b ac 0, there is one repeated real solution. (c) If b ac < 0, there is no real solution. You should be able to solve certain types of nonlinear or nonquadratic equations. For equations involving radicals or fractional powers, raise both sides to the same power. For equations that are of the quadratic type, au bu c 0, a 0, use either factoring or the quadratic equation. For equations with fractions, multiply both sides by the least common denominator to clear the fractions. For equations involving absolute value, remember that the epression inside the absolute value can be positive or negative. Always check for etraneous solutions. Vocabulary Check. quadratic equation. factoring, etracting square roots, completing the square, Quadratic Formula. discriminant. position, t v 0 t s 0, initial velocity, initial height
18 .. Standard form: 0 Section. Solving Quadratic Equations Algebraically Standard form: or 0 0 or 0 0 or 0 0 or or or or 0 or a b 0 a b a b 0 a b 0 a b a b 0 a b ±9 ±7 0. ± ±i ± i 0. ± 0.8i ± ± ±, ± 7, impossible 7.. ± ± 8, ±7 ±7 ± ± ±
19 00 Chapter Solving Equations and Inequalities ± ±i ± i ± 89 ± 89. y. y (a) (a) 9 9 The -intercepts are, 0 and, 0. The - intercepts are and, 0., 0 (c) 0 (c) 0 ± ± or ± ± or 7. y 0 (a) (c) The - intercept is, 0. 0
20 Section. Solving Quadratic Equations Algebraically The graph does not have any -intercepts and thus The graph has one -intercept 7, 0 and hence the the equation has no real solution. equation has one real solution The graph does not have any -intercepts and hence the equation has no real solution. 8 0 b ± b ac a ± ± ± ± ± b ± b ac a ± ± ± 7 ± i ± i ± 8 ± i 8 ± i ± ±9 9 or 9 or ± ± 9 ± ± i b ± b ac a 7 ± 7 7 ± 7 7 ±
21 0 Chapter Solving Equations and Inequalities b ± b ac a ± ± other answers possible (other answers possible) ± (other answers possible) (other answers possible) 7. i i (a) i i 0 w 0 0 (other answers possible) w + ww w w 0 (c) w 8w 0 w, length w 8 width feet, length 8 feet 79. Let be the length of the square base. Then Thus, the original piece of material is of length cm. Size: centimeters 8. (a) s 0 8 and v 0 0 s t 8 t s (c) The object reaches the ground between 0 and seconds, 0,. In fact, t 0. seconds. 8. (a) s t v 0 t s 0 Distance 00 mileshour0 seconds00 secondshour 0 t miles.7 miles 9,77. feet t 8000 t 00 t 0. seconds
22 Section. Solving Quadratic Equations Algebraically 0 8. (a) P 0.0t.9t t.9 ± t.9t ±.7 0. t 8., 0.89 Taking the positive root, t 8. or 998. Similarly, P 0 yields 0.9, or 000. Answers will vary. (c) (d) P 7 when t., or 00. (e) Answers will vary. 87. (a) C , If C 0, then.797 degrees. (c) If the temperature is increased 0 to 0, then C increases from 79. to 97.7, a factor of Let u be the speed of the eastbound plane. Then u 0 speed of northbound plane. u u 0 0 9u 9u 0 0 8u 900u,00 0 8u 900u,9,00 0 Using a graphing utility, u 9.7 mph and u mph. 9. False. The solutions are comple numbers. 9. False. The solutions are either both imaginary or both real (a) u u 0, u u u 0 u 0 u u 0 u u 0 u , (c) Answers will vary. 97. Add the two solutions and the radicals cancel. S b b ac a b b ac a 99. Answers will vary. b a 0. The parabola opens upward and its verte is at,. Matches (e)
23 0 Chapter Solving Equations and Inequalities Yes, y is a function of. y No, y is not a function of. y ±0. Yes, y is a function of. y. Answers will vary. Section. Solving Other Types of Equations Algebraically You should know the properties of inequalities. (a) Transitive: a < b and b < c implies a < c. Addition: a < b and c < d implies a c < b d. (c) Adding or Subtracting a Constant: a ± c < b ± c if a < b. (d) Multiplying or Dividing by a Constant: For a < b, a. If c > 0, then ac < bc and. If c < 0, then ac > bc and c < b c. You should know that if 0 if < 0. a c > b c. You should be able to solve absolute value inequalities. (a) < a if and only if a < < a. > a if and only if < a or > a. You should be able to solve polynomial inequalities. (a) Find the critical numbers.. Values that make the epression zero. Values that make the epression undefined Test one value in each interval on the real number line resulting from the critical numbers. (c) Determine the solution intervals. You should be able to solve rational and other types of inequalities. Vocabulary Check. n. etraneous. quadratic type , ±
24 Section. Solving Other Types of Equations Algebraically ,, t 8 0 t 8t t 0 t t 0 t 0 t t 0 t. s s s Let u s s. u u 0 0 s. y (a) 9 9 u u u 0 u u 0 u s s s s s s y 0 9 (a) 0 0 -intercepts: ±, 0, ±, 0 (c) (c) 0 9 -intercepts:, 0, 0, 0,, (d) The -intercepts are the same as the solutions. (d) The -intercepts are the same as the solutions.
25 0 Chapter Solving Equations and Inequalities , , etraneous is not possible. Note: You can see graphically that there is only one solution
26 Section. Solving Other Types of Equations Algebraically 07.. ± ± 9, ± 8 ± 7, , which is etraneous. 9. y 0. y 7 (a) (a) 0. (c) -intercepts:, 0,, (d) The -intercepts and the solutions are the same. (c) 0. -intercepts: 0, 0,, (d) The -intercepts and the solutions are the same.
27 08 Chapter Solving Equations and Inequalities a, b, c ± ± a, b, c 0 ± ± ± ±.. OR 0 0 ± Only, and are solutions to the original equation. and are etraneous. Note that the graph of has two -intercepts. y
28 Section. Solving Other Types of Equations Algebraically ( is etraneous.) OR 7, ± 7 7 is etraneous. 7. y (a) 9. (a) y (c) -intercept:, (d) The -intercepts and the solutions are the same. (c) -intercepts:, 0,, 0 0 or or (d) The -intercepts and the solutions are the same. 7. Let original number of students. The original 7. Let v the average speed of the plane. The time cost per student is 700 and the new cost per for the -mile trip is v hours. By increasing student is 700. Hence, the speed by 0 mph, the time is v 0 hours. Hence, , , students. v 0 vv 0 v 7v 9,000 v 0v 7v v 0 ± 0 9,000 v 0v 9, ± 7,00. v v 0 Taking the positive square root, v. mph, and v 0 9. mph.
29 0 Chapter Solving Equations and Inequalities 7. A P r n nt, n, t r r r.00 r 0.00 r 0.0, or % 77. (a) Nt 0 8.7t, 0 t Year N Year N in 99; 000 in 000 (c) 0 8.7t t 9 t.8 t., or t t 79 t.9 t 0., or 000 (d) t 0.,. (e) For 00, t 7., or 07. For 000, t.0, or 0. Answers will vary C , or,0 passengers p ,900 units
30 Section. Solving Other Types of Equations Algebraically 8. (a) 00 (c) F a., b., c.8. ± Because is restricted to 0, choose.7 pounds per square inch. 8. False. An equation can have any number of etraneous solutions. For eample, see Eample other answers possible (other answers possible) i i other answers possible 9. The distance between, and, 0 is a 9 9 a b b, 9 One solution is a 9, b 9. Another solution is a 0, b a b, a b One solution is a 9 and b. Another solution is a b
31 Chapter Solving Equations and Inequalities 0. z z z z z z z z zz z z z z zz z z zz Section. Solving Inequalities Algebraically and Graphically You should be able to solve an inequality algebraically using the Properties of Inequalities. You should be able to solve inequalities involving absolute values. You should be able to solve polynomial inequalities using critical numbers and test intervals. You should be able to solve rational inequalities. You should be able to solve inequalities using a graphing utility. Vocabulary Check. negative. double. a a. a, a. zeros, undefined values. <. <. Matches (f ). Matches (d). Matches (e). 7. (a) (c) >? 0 Yes, is a solution. > 0 >? 0 > 0 Yes, is a solution. (d) >? 0 7 > 0 No, is not a solution. >? 0 9 > 0 No, is not a solution.
32 Section. Solving Inequalities Algebraically and Graphically 9. < (a) 0 (c) <? 0 <?? No, 0 is not a solution. <? <???? Yes, is a solution. (d)? < <? 0.8? Yes, is a solution. <? <???? No, is not a solution.. 0 < > 00 > < < <. 7 < < 9. 8 < 8 7 < < > < 0 < < < 9 < < 9 < < < < < 7 < 9 9 <. 8 Using the graph, (a) y for and y 0 for. Algebraically: (a) y y 0 0
33 Chapter Solving Equations and Inequalities > 0 < 0 or > 0 < or > y 0 Using the graph, (a) 0 y for and y 0 for. Algebraically: (a) 0 y <. < 7 < < < > 7 > < or > < or > 0. 0 < 7. < < < y 7 < < > > < < 0 9 Graphically, (a) y for and y for or 7. Algebraically: (a) y y or or 7 9. The midpoint of the interval, is 0. The interval represents all real numbers no more than three units from The midpoint of the interval, is 0. The two intervals represent all numbers more than three units from 0. 0 > >
34 Section. Solving Inequalities Algebraically and Graphically. All real numbers within 0 units of All real numbers at least five units from 7. > 0 9. > 0 Critical numbers:, Testing the intervals,,, and,, we have > 0 on, and,. Similarly, < 0 on,. 0 Entirely negative: ± Entirely positive: ± 0 0 0, 0.8,.8, 0 0,. > 0 for all. There are no critical numbers because 0. The only test interval is,.. < 8 < < < 0 Critical numbers:, Critical numbers: 7, Test intervals:,,,,, Test intervals:, 7, 7,,, Test: Is 0? Test: Is 7 < 0? Solution set:,, Solution set: 7, Critical number: 0, ± Test intervals:,,, 0, 0,,, Test: Is 0? Solution set:, 0, 0 Since b ac 7 < 0, there are no real solutions to 0. In fact, > 0 for all. No solution. 9 > 0 > 0 Critical numbers:,, Testing the four intervals, we see that 9 > 0 on, and,.. (a) f g when. f g when. (c) f > g when >. 0
35 Chapter Solving Equations and Inequalities. y Algebraically, Critical numbers:, 0 (a) y 0 when or. y when 0. Testing the intervals,,,, and,, you obtain or. Critical numbers: 0, Testing the intervals, 0, 0,, and,, you obtain > 0 9. < 0 0 > 0 < 0 Critical numbers: 0, ± Test intervals:,,, 0, 0,,, Test: Is > 0? Critical numbers: Test intervals: < 0,,,,,, Solution set: 0, 0, Test: Is < 0? Solution set:,, 7. y Need: 0 Domain: all real Domain:, (a) y 0 when 0 <. y when <. 77. Need: 0 0 or Domain:,, 79. (a) Pt 000 This occurs at the point of intersection, t, or 99. Less than one million: Pt < 000 This occurs for t <, or before 99. Greater than one million: Pt > 000 This occurs for t >, or after 99.
36 Section. Solving Inequalities Algebraically and Graphically 7 8. (a) s t v 0 t s 0 s t 0t > 8 s t 0t t 0t 8 < 0 s t0 t t t < 0 s 0 when t 0 seconds. s > 8 when < t <. 8. (a) < D < 0 for.8 < t < 0.09, or between 99 and 000 (c) < D < 0 < 0.0t 0.7t.0 < 0 To solve these inequalities, find the critical numbers. 0.0t 0.7t t 0.7 ± ± Because 0 < t <, select the negative sign, t.8. Hence, < D for.8 < t. Similarly, D < 0 for t < (d) No. Dt < 0 for all t. Vt 87..7t 7.9.7t 7. t. The number of hours playing video games eceeded in 00. Vt Nt.7t 7.9.t t.7 t 0.7 According to these models, the number of hours reading daily newspapers and playing video games will be the same in When t, v vibrations per second. 9. When 00 v 00,. < t <.. 9. (a) Option A: Option B: 00 At 0.t 9. False. If 0 8, then 0 and Bt 0.0t 8. A B (c) At Bt when t 0. Bt is the better choice if you use less than 0 minutes. At is the better choice if you use more than 0 minutes. (d) Answers will vary.
37 8 Chapter Solving Equations and Inequalities 97. The polynomial f a b is zero at a and b. 99. (iv) a < b 0. f 0. f (ii) a < b (iii) a < a b < b (i) a < a b < b y y 0. y 07. y Answers will vary. y y 7 y 7 y 7 y f f 7 Section.7 Linear Models and Scatter Plots You should know how to construct a scatter plot for a set of data. You should recognize if a set of data has a positive correlation, negative correlation, or neither. You should be able to fit a line to data using the point-slope formula. You should be able to use the regression feature of a graphing utility to find a linear model for a set of data. You should be able to find and interpret the correlation coefficient of a linear model. Vocabulary Check. positive. negative. fitting a line to data.,. (a) Monthly sales (in thousands of dollars) y Years of eperience Yes, the data appears somewhat linear. The more eperience,, corresponds to higher sales, y.. Negative correlation y decreases as increases.. No correlation
38 Section.7 Linear Models and Scatter Plots 9 7. (a) (0, ) (, ) (, 0) y y = + (, ) (, ) 9. (a) (0, ) y (, ) (, ) (, ) (, ) y = y 0.. y Correlation coefficient: Correlation coefficient: (c) (c) 7 8 (d) Yes, the model appears valid. (d) Yes, the model appears valid.. (a) d d 0.07F 0. Elongation Force F (c) d 0.0F or F.d 0.09 (d) If F, d 0.0. cm.. (a) (c) 00 (d) For 00, t and y., or $,,00. 0 y.t 8 00 For 00, t 0 and y 97, or $,97,000. Yes, the answers seem reasonable. (e) The slope is.. It says that the mean salary increases by $,00 per year. 0 Yes, the model is a good fit.
39 0 Chapter Solving Equations and Inequalities. (a) 0 7. (a) C.t.70 P 0.t Correlation coefficient: 0.99 (c) 700 (c) (d) The model is a good fit. (e) For 00, t, y $8.98. For 00, t 0, y $.7. (f) Answers will vary. The model is not a good fit. (d) For 00, t 0 and P, or,000 people. Answers will vary. 9. (a) T.7t 9 Correlation coefficient: (c) The slope indicated the number of new stores opened per year. (d) T.7t 9 > 800.7t > 87 t >.8 The number of stores will eceed 800 near the end of 0. (e) Year Data Model The model is not a good fit, especially around t.. True. To have positive correlation, the y-values tend to increase as increases.. Answers will vary.. 7. h, 0, > 0 (a) h h f (a) f fw w w w w , ± 9 7 ± 7
40 Review Eercises for Chapter Review Eercises for Chapter. (a) 0? 0? (c) No, is not a solution.? 9?? (d) No, 0 is not a solution.?.??. No, is not a solution. Yes, is a solution September s profit October s profit 89,000 Let September s profit. Then 9 0. October s profit , ,000 7, ,000 September: $,000; October: $,000
41 Chapter Solving Equations and Inequalities. (a) h F 9 C h h meters high.7 9 C. 9 C 8 m m 7 cm C.. Celsius 9. y. Let 0: y, y-intercept: 0, Let y 0:, -intercept:, 0 y Let 0: y 8, y-intercept: 0, 8 Let y 0:, 8, -intercepts:, 0, 8, Solution:. Solution:.0 Solutions:.07, y 7. y From second equation, y. Then: y y 7 y 9 7 y and Intersection point:, y y From equation, y. Then: y 9 8 y Intersection points:,, 9, 8. i. i 7i 7i 7. 7 i i 7 i i 7i 9. i 8i i 0i 0 i. i i. 9 i i 0 8i i 9i 7i. 0 8i i 0 0i i i i
42 Review Eercises for Chapter i 7i 9 i 9 9 i 9 i i i i i 80 i i i. i i i i i i 0i i i 7 7 i. i. i 7. i 9. Imaginary ais Imaginary ais Imaginary ais Real ais 7 Real ais Real ais ,, ,,, ±8 0 0 ± 0 0 ± ± , ± ± ± 0 ± ± i
43 Chapter Solving Equations and Inequalities 8. 0 ± 8 ± ± i 87. (a) C 0 when t., or 00. (c) t.9t 8 0 t.9t 8 0 t.9 ± ± 88.., 0.99 (etraneous) (d) C 00 when t, or 00. C 000 when t 8, or 008. (e) Answers will vary or 0 ±, ±i 0,, 8 0 or ±, ± ± ± or 0 0, etraneous or, etraneous No solution. (You can verify that the graph of y lies above the -ais.)
44 Review Eercises for Chapter or ± ± 9 0 ± t , t 0 t t ± ± or t. 0 0 or 0 or. or or or The only solutions to the original equation are or. and are etraneous. 7. Let number of farmers. 9. 8,000 8,000 8, , ,000 farmers 000 A P r n nt r.99 r 7 r.997 r 0.0, or %
45 Chapter Solving Equations and Inequalities. (a) Year P (millions) (c) P 8. when t 0.9, or late 000. (d) t t 0.9 t 0.9 t 0.9 (e) P 9 when t, or 0. (f) Answers will vary.. 8 <. < 8 < > 7. > 9 > > >,, 0 < < 9 > < <. or < < < < 0 or which can be written as,. which can be written as, 0, Test intervals: or,,,,,,, Critical numbers: 0,, Critical numbers:, Testing the three intervals,we obtain. Testing the four intervals, we obtain 0 or
46 Review Eercises for Chapter 7. < 0. Critical numbers:, Test intervals:,,,,, Test: Is < 0? Solution set:,, Critical numbers: , Testing the three intervals, we obtain 0 or < is defined for all $0.,,. (a) Grade-point average y Eam score Yes, the relationship is approimately linear. Higher entrance eam scores,, are associated with higher grade-point averages, y.. (a) (c) Speed (in meters per second) s s 0t Time (in seconds) (Approimations will vary.) s 9.7t 0.; (d) For t., S.7 m/sec. t. False. A function can have only one y-intercept. (Vertical Line Test) 7. False. The slope can be positive, negative, or They are the same. A point a, 0 is an -intercept if it is a solution point of the equation. In other words, a is a zero of the function.. In fact, i i.. (a) i 0 i 0 0 i ii i i (c) i 0 i i 8 (d) i 7 i i i i
47 8 Chapter Solving Equations and Inequalities Chapter Practice Test. Solve the equation 0.. Solve the equation and verify your answer with a graphing utility. Verify your answer with a graphing utility.. Solve A a bh for a.. 0 is what percent of 00?. Cindy has $.0 in quarters and nickels. How many of each coin does she have if there are coins in all?. Ed has $,000 invested in two funds paying 9 % and % simple interest, respectively. How much is invested in each if the yearly interest is $8.0? 7. Use a graphing utility to approimate any points of intersection of y and y. 8. Use a graphing utility to approimate any points of intersection of y and y. 9. Write in standard form. i i 0. Write i in standard form.. Solve 8 0 by factoring.. Solve by taking the square root of both sides.. Solve 9 0 by completing the square.. Solve 0 by the Quadratic Formula.. Solve 0 by the Quadratic Formula.. The perimeter of a rectangle is 00 feet. Find the dimension so that the enclosed area will be 0,000 square feet. 7. Find two consecutive even positive integers whose product is. 8. Solve 0 0 by factoring. 9. Solve. 0. Solve 8.. Solve 0.. Solve >.. Solve <.. Solve <.. Solve 9.. Use a graphing utility to find the least squares regression line for the points, 0, 0,,, and,. Graph the points and the line.
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