Chapter 1 Equations and Inequalities

Transcription

1 Chapter Equations and Inequalities Section ( ) ( ) +. (r ) + (r ) r + r r r r ().( + ) ( + )( ) (+ )( ) + ( ) [ ( )] ( + ) ( + ) ( ) + Conditional equation. + + ( ) (+ ) ( + ) + + Contradiction. ( ) Identity. [ ( )] + [ + ] + [ + ] Identity. + Conditional equation. or. or

2 Chapter : Equations and Inequalities. or. + + or +. or ( ) (). + + Contradiction. There is no solution.. a b, b > a b or a b a+ b a b a+ b a b or Revenue The revenue first eceeded $ billion in.. d t. + Cost + t or t t t t t. Rounded to the nearest yard, sq yards can be carpeted for. hours hours minutes $. Ruben will be eactly miles from Barstow after hours and after hours and minutes.., t Percent remaining,, t, t t t. seconds min. sec min sec. ma.( a) min.( a).( ).( ).()..(). The maimum eercise heart rate for a person who is years of age is beats per minute (to the nearest beat). The minimum eercise heart rate for a person who is years of age is beats per minute (to the nearest beat).

3 Section.. a + b c a c b c b, a a. + + if + if + < < ( + ) an identity +. + ( + ) if + + ( + ) + { } if + < < + + an identity The case + < has no solution since there is no real number such that < and. { }. + + if if + < < + ( + ) an identity + The case + < has no solution since there is no real number such that < and.... Prepare for Section... l + w (l + w) The distributive property bh. b h bh The associative property of multiplication. ab + b + a b + a b+ a a+ b a b a b b a ab ab ab

4 Chapter : Equations and Inequalities Section.. V π r h V π r h V h π r. I Pr t I t Pr. Gm m F d Fd Gm m Fd m Gm. an a + ( n ) d. an a ( n ) d an a d n a S r S( r) a S Sr a S a Sr S a r S. qb rating [.(. ) +.(. ) +.(.) + (..(.))].. SMOG +.. W. GFI.(.+ )... Let the number W P L + W (W ) + W W + W W W ft L W () ft + + () The shortest side is cm. The longer sides are each cm.

5 Section.. d t d ( t) Let t the time to run to the end of the track. Let t the time in seconds to jog back. t ( t) t t t t d () meters. d (t + ) d t Let t the time (in hours) of the second plane. Let t + the time (in hours) of the first plane. d ( t + ) d t ( t + ) t t + t t t t hours. Let + y + z the distance to Jon s house, where the. Let the score on the net test. distance uphill, y the distance down hill, and z the distance on level ground. Note that, the distance uphill on the way to Jon s house equals the distance down hill on the way home, + and that z, the distance down hill on the way to Jon s house equals the distance uphill on the way home. Also note that y is the distance on level ground on the way to Jon s house and on + the way home. rt d t d r A score of will produce an average of. rate time distance Up To Jon s y Level house y Down z z Up Back home Level y y Down z z y z y z y z y z () + y + z + + y + z + y + z ( + y + z) + y + z The distance to Jon s house is miles.

6 Chapter : Equations and Inequalities. Let the number of sunglasses. Profit Revenue Cost,..,.. Let cost last year.... The cost of a computer last year was $.. The manufacturer must sell sunglasses to make a profit of $,.. Let amount invested at %. (, ) amount invested at.%.. +.(, ). +.., $ was invested at %. $ was invested at.% ( ).() ( ), +,,, tickets at $ each tickets at $ each g pure silver.% % % +.() +..( + ) $ additional investment.....().( ).... liters of water. $ $ $ + ( ) () + lb of $ coffee lb of $ coffee

7 Section... or or + + () ( + ) + () ( ) + + () ( + ) + +. Let t the time it takes both electricians working together to wire the house. The first electrician does of the job every hour. The second electrician does of the job every hour. t + t t + t t + t t t. hours g of pure gold. Let price of book. price of bookmark. + (. )..... The price of the book is $.. The price of the bookmark is $..... Connecting Concepts... () + () +.. ft (.) A -lb force is needed to lift lbs. t ( t) t t t t. d t d. The distance to the target is feet (to the nearest foot).

8 Chapter : Equations and Inequalities Diophantus was years old when he died.... Prepare for Section.. ( + )( ). (+ )( ). + + i. ( ) ( ) ( )() ( ). ( ) + ( ) ()() + (). ( i) ( i) + i+ i + i+ Section.. ( + )( ) + or. ( + )( ) + or. + ( )( + ) or +. ( ) or. ( ) [( ) ] [( ) + ] ( )( ) or. ± ±

9 Section.. ± ±. + ± ± i. ( ) ± ± ± + or. ( ) + ( ) ± ± i ± i ( + ) + ± ± + or. + + ( ) ± ± + or ( + ) + ± + ± i ± i i or +i ( + ) +± ± ± ± ± ± +, or ± ± +, or ± ± + or

10 Chapter : Equations and Inequalities., a, b, c b± b ac a ( ) ± ( ) ()( ) () ± + ± ± + or or. + ± ()( ) () ± + ± + or ± ()() () ( ) ± ( ) ()() () ± ± ± ± +, or ± ± i + i, or ± i. + + () + ± ()( ) () ± ± +, or. ± (.) (.)(.) (.).±.+..±..±., or. ( ) ± ( ) ()( ) () ± + ± ±, or

11 Section.. b ac ( ) ()( ) + Two distinct real numbers. +. b ac ( ) ()() Two distinct nonreal comple numbers + b ac ( ) ()() One real number. + b ac () ()( ) + Two distinct real numbers. + + b ac () ()() Two distinct nonreal comple numbers. a + a d d. in.. a b a b a + b c b + b b + b b b. b. (.) a. The TV is. in. high and. in. wide.., t + t+, t + t a, b, c ± ()( ) t () ± ±. t. or t. (not t ). years after is..,. +,. +, a., b, c, ± (.)(, ) (.) ± ±.. + or...., or, racquets must be sold.

12 Chapter : Equations and Inequalities. Let w width of region Then w length. Area length(width) w w w w w w+ ( w w+ ) w w+ ( w )( w ) w w or w w w () w () The region is either feet wide and feet long, or feet wide and feet long.. Solve D + + for with D a, b, c ± ( )( ) ( ) ± ±. +.. or... mile. miles. Solve h t +.t+ for t where h. t +.t+ t +.t+. ± (.) ( )() t ( ). ±. t. or t. He was in the air for. s.. Solve h t + t for t where h. t + t t + t a, b, c ± ( )( ) t ( ) ± ±. +.. t or t... seconds. seconds. Solve s.t for t where s to find the time it takes the ball to reach the fence..t t. seconds Net, evaluate h t + t+. where t. to determine if the ball is at least feet in the air when it reaches the fence. h (.) + (.) +. h. No, the ball will not clear the fence.. Solve h n( n ) for n where h. nn ( ) nn ( ) n n n n ( n )( n+) n or n + n people n (no)

13 Section.. Solve P.t +.t+. for t where P..t +.t+..t +.t. a., b., c.. ± (.) (.)(.) t (.). ±.. ± t or t.... The percent of U.S. citizens that are divorced will first reach %. years from, which is in.. a. If t represents the year, then the year is represented by t. Evaluate A.t +.t+ for t. A.() +.() +.() +.() million pounds b. Solve A.t +.t+ for t where A..t +.t+.t +.t a., b., c. ±. (.)( ) t (.). ± ±.. ± t t. or... (not t ). years from will be in.... Connecting Concepts. a. l l+ w w l l w( l+ w) l lw+ w l lw w Use quadratic formula solving for l in terms of w. ( w) ± ( w) ()( w ) l (). + ; b a ( ) c ( ) ( ) ; a Yes, and are roots of. w± w w± w + w b. l + (). ft c. Answers will vary. positive solution only

14 Chapter : Equations and Inequalities. b ( ) ( + i ) + ( i) ; a c () ( + i )( i) i + ; a Yes, + i and i are roots of Prepare for Section... ( ) ( ) ( + )( ) ( + )( ). / ( ). / ( ). (+ ) + + ( ) ( + ) () + + ( + ) ( ) ( )( + ),, or Section.. + ( ) ( ) ( )( ) ( )( )( + ),, or. ( ) ( )( + ),, or. ( ) ( + )( ),, or. ( ) ( )( + ),, or. ( )( + + ), or + + ± ()() ± ± i ± i + i or i Thus the solutions are, + i, i. + ( ) ( + ) + +

15 Section ( ) ( + ) + ( + ) ( + ) + + ( + ) + No solution because each side is undefined when.. + r r r ( ) ( ) r r + r r r ( r ) + r r + r r+ r r r r r No solution because each side is undefined when r.. ( )( ) ( )( ) ( ) ( ) ( ) ( + ) ( + )( ) ( + )( ) ( )( + ) + +. Check The solution is.. + ( ) ( ) + + ( )( ) or Check + + The solution is. + + (No)

16 Chapter : Equations and Inequalities. + ( ) (+ + ) ( ) ( + ) ( )( ). + ( + ) (+ ) () ( ), or Check () + Check () + () checks as the solution. () + The solution is. (No). + + ( + ) ( ) + + ( ) (). + ( ) ( + ) + (+ )( ), or Check checks as a solution. Check + + (No) The solution is. + +

17 Section ( )( + ), or Check () + () +. + Let u. u u+ ( u )( u ) u or u ± ± The solutions are,,,. ( ) + ( ) ( ) + The solutions are and.. + Let u. u u + (u )( u ) u ± ± or u ±. + Let u. u + u ( u )( u+ ) u or u The solutions are and. The solutions are,,,.. / / + / Let u. u u + ( u )( u ) u / or / The solutions areand. u. / / / Let u. u u (u + )( u ) u / or The solutions are u / and.

18 Chapter : Equations and Inequalities. Let u. u u + (u )(u ) u ± ± The solutions are and.. / / / / / ( ) ± () ± The solutions areand.. + Let u. u u + (u )( u ) u or u The solutions are and.. Let number of games the golfer needs to play. () + + ( + ) games. Let the number of hours the assistant would take to build the fence working alone. The worker does of the job per hour; the assistant does of the job per hour. worker assistant ( ) ( ) () + () ( ) + + ( ) +. SMOG w+ w+ hours w w words with three or more syllables.

19 Section... L π r r + h π π r r + r r + r ( r + ) r + r Let u r. u + u ( u+ )( u ) u or u (No) r r The radius is in. d mm, d mm Vc s Vs π r s π () + π () π ( + ). s. mm The side is approimately. mm.. d. h. h () h h h The height is approimately ft.

20 Chapter : Equations and Inequalities... Connecting Concepts. a. s (a + b + c) ( + + ) abc r s( s a)( s b)( s c) ()()() r ( )( )( ) ()() r ()()() ())(). The radius is approimately. in.. T s + s T + T. seconds b. a side, a b c, r a s (a+ b+ c ) (a+ a+ a) abc a a a r s( s a)( s b)( s c) s( s a)( s a)( s a) aaa a a a a a a a a a a a a a a a a a Each side is in.... Prepare for Section.. { > }. ( ) ( ) +. +., It is undefined for.. + (+ )( ). + ( )( )

21 Section. Section.. + < < < < { < }. + > + > < { < }. ( + ) + +. ( ) > ( ) + > > < { < }. + > and + > and > and > { } <.. + < or + < or { < } { } { < or }. + > or + < > or < < or < { < } { < } { < }. > < or > < or > < or >,,.,. (Note: The absolute value of any real number is greater than or equal to.) (, ). + + or + (, ] [, ). or,, ( ]. (Note: No absolute value is less than.) {}

22 Chapter : Equations and Inequalities. + > ( + ) > The product + ( + ) is a critical value. is positive. is a critical value.. ( )( + ) The product ( )( + ) is negative or zero. is a critical value. + is a critical value. + ( ) ( )( + ) (, ) (, ) [, ]. + + < ( + )( + ) < The product ( + )( + ) is negative. + is a critical value. + ( + )( + ) is a critical value.. ( )( + ) The product ( )( + ) is positive or zero. is a critical value. + is a critical value. (, ) ( )( + ) (, ] [, ). + < + The quotient is negative. + The critical values are and. + (, ). + + ( + ) The quotient is positive or zero. + + The critical values are and. + The denominator cannot equal zero.,

23 Section (+ ) The quotient is positive or zero. + + The critical values are and.. ( + )( ) < ( + )( ) The quotient is negative. + The critical values are,, and. ( + )( ) (, ) (, ) + The denominator cannot equal zero., ( ) The quotient is negative or zero. + The critical values are and. + The denominator cannot equal zero.. > ( + )( ) > ( + )( ) The quotient is positive. + The critical values are,, and. ( + )( ),, (, ) [, )

24 Chapter : Equations and Inequalities. + ( )( ) ( )( ) The quotient is negative or zero. The critical values are and. ( )( ) The denominator cannot equal zero. (, ) {}. Plan A: +. Plan B: < +. <.. < Plan A is less epensive if you use more than checks.. Let h the height of the package. length + girth length + (width) + (height) + () + h + + h + h h h The height must be more than but less than or equal to inches.. F C + C () C () C. Solve h (. f +.). for h where f.. h (. f +.). h [.(.) +.]. h (. +.). h.. h.. or h.. h. h. The height, to the nearest. cm, is from. cm to. cm.. Plan A: + Plan B: +. + > +.. > >. Plan A pays better if at least sales are made.. < + ( + ) + ( + ) < < + < < < < < must be even, thus or. Therefore, the numbers are {,, } or {,, }.. R > ( ) > The product is positive. Critical values are and. ( ) (, )

25 Section... +, <.+, <. <, >. At least books must be published.... Connecting Concepts.. C +.. π r... r π π. r. The radius of the cylinder must be between. inches and. inches.. ( ) > ( ) The quotient is positive. Critical values are and. ( ) ( ) (, ) (, ) (, ). ( ) ( + ) The quotient is positive or zero. + Critical values are and. ( ) ( + ) Denominator not. (, ). < < if if < < < < < > > (, ) (, ). < if if < < < > (, ] [, ). < α < δ, δ > if a < a < δ α < < δ + α if a< < ( a) < δ > a > δ a > > a δ ( a δ, a) ( a, a + δ )

26 Chapter : Equations and Inequalities. s t + vt + s, s >, v, s t + t > t + t > ( t t + ) > ( t )( t ) > The product is positive. The critical values are and. ( t )( t ) second < t < seconds The ball is higher than ft between and seconds.... Prepare for Section.. k() k. k. k. k (). k. (.)... The area becomes times as large.. No. The volume becomes times as large. Section.. d kt. k y. m knp. V klwh. A ks. kmm F d. y k k k k. r kt k k k k

27 Section.. T krs k k k. k. V klwh k k k. V kt. k.. k k.. Thus V T (.). liters. j k d k () k p () p. fl oz. s k q k k p p semester hours. T k l. k. k. k a.. T... seconds b. T k l T l k l. l.. ft l

28 Chapter : Equations and Inequalities. k r t k k r r revolutions per minute. k l d k k k l l decibels. a.. b. V kr h V k( r) h ( kr h) V Thus the new volume is times the original volume. V kr ( h) ( kr h) V Thus the new volume is times the original volume. c. V k( r) ( h) kr h ( kr h) V Thus the new volume is times the original volume.. V knt P k( n) T V p knt p V Thus the new volume is times larger than the original volume.. For Randy Johnson, ERA kr i k(). (). k For Tom Glavine, () ERA (.).. F kws r k k k. k (.) Thus F pounds... Connecting Concepts. / T kd k k / / / Thus d / / / d / / d / d million miles d

29 Chapter Review... Chapter True/False Eercises. False, ( ), therefore is also a solution.. False, has solution. has solutions and.. True. True. False, > but >/.. False, the discriminant is b ac.. False, if a, b, and c, then a + b + + c, but a + b + c.. True. False, also has roots of and.. True... Chapter Review. ( ) ( + ) + + [.]. ( ) ( ) [.]. ( ) ( ) + + [.]. ( ) ( ) () + + [.]. + + ( ) + + ()( + ) ( + ) + + [.]. y y+ y y yy ( + ) ( ) yy+ y+ y yy ( ) yy ( + ) ( y+ ) y y y y y+ y y [.]. + [.] ( )( ) or. + [.] ( )(+ ) or +. [.] ( ) ± ± + ( ) ()( ) or ()

30 Chapter : Equations and Inequalities. + [.] ( ) ± ± ± i + i ( ) ()() () or i. [.] ( ) or. [.] ( ) ( )( + ),, or. + [.] Let u. u u + (u )(u ) u ± ± ± or u ± ± ±. + [.] Let u. u + u (u )( u + ) u Thus,. or u No solution.. [.] + ( + )( ) [ ] or. ( )( + ) or [.] Check ( ) ( ) () The solutions are and. Check () () ( ) ( ) The solutions are and.

31 Chapter Review ( ) ( ) + + ( )( ) or ( )( ) ( )( ) ( )( ) ( + ) ( + )( + ) [ + ] ( )( ) ( )( + ) or Check () () (No) The solution is. [.] Check () + + ( ) + + i i i i i (No) The solution is. [.]. + + ( )( ) ( )( ) [ ] ( )( ) + ( + ) + + ( )( + ) or [.] Check + + (No) ( ) ( ) The solution is.

32 Chapter : Equations and Inequalities [.] (+ )(+ ) (+ )(+ ) ( )( ) + + [ + ] (+ )(+ ) ( + ) or Check () + () + + ( ) + ) ( ) + + (No) The solution is.. [.] ( y+ ) ( y + ) y + y + y + y + ( y + )( y + ) y or y. (s ) [.] (s s + ) s s + s s + (s )(s ) s or s. [.] or. + [.] + or +. + [.] + or +. [.] or. / / ( + ) + ( + ) [.] / ( + ) / ( + ) [ + ( + ) ] [ + + ] / ( + ) ( + + ) / ( + ) ( + ) / ( + ) or ( + ) + +. / / ( ) + ( ) [.] / ( ) ( + ) / ( ) or + ( + )( ) + or. + [.] (, ]. + + [.], )

33 Chapter Review. + [.] ( + )( ) The product is negative or zero. + Critical values are and. ( + )( ) [, ]. C + [.] C C,. > [.] ( + )( ) > The product is positive. + Critical values are and. ( + )( ) (, ) (, ). < ( F ) < [.] < F < < F < (, ). + [.]. + > [.] ( + ) ( )( ). The product is negative or zero. The critical values are,, and. ( + ) > ( + )( ) >. The product is positive. + The critical values are,, and. ( )( ) ( + )( ) (, ] [, ] (, ) (, ). + > [.] The quotient is positive. + The critical values are and. + (, ) (, ). ( ) [.] + The quotient is negative or zero. + The critical values are, and. ( ) + Denominator. (, ) [, ]

34 Chapter : Equations and Inequalities. [.] ( ) + The quotient is negative or zero. The critical values are and. Denominator (, (, ).. [.] ( ) + The quotient is positive or zero. The critical values are and. Denominator, ).. < [.] < < < < < < (, ). < < [.] If, then < <. If <, then < < (, ) (, ) > >.. [.] or (,] [, ). < a < b [.] If a, then a< < a+ b. If a<, then < a< b [ a b, a) ( a, a+ b] a > > a b.. V π r h [.] V h π r. P A + rt P( + rt) A P+ Pr t A Pr t A P t A P Pr [.]. A h ( b+ b) A hb ( + b) A hb+ hb A hb hb A hb b h [.]. P ( l+ w) P l+ w P l w P l w [.]. e mc e m c [.]. mm F G s Fs Gmm Fs m Gm [.]

35 Chapter Review. Let the number [.] P l + w (w ) + w w + w w w w () width ft, length ft [.]. d rt d t ( t) t t t t t d ( t) d () nautical miles [.]. Let cost last year Cost last year + raise Let the number The cost last year was$.. [.]. % %. +.( ). +.. $ in the % account $ in the % account [.]. Let price of battery + price of calculator Price of calculator is $.. Price of battery is $.. [.]. Let monthly maintenance cost per owner ( ) The total monthly maintenance cost is $. [.]. P A l + w l + w l w lw ( w) w w w w w + ( w )( w ) w or w l or l l l Length in. and width in., or length in. and width in. [.]

36 Chapter : Equations and Inequalities. Time Mason Apprentic e Part completed In hour + ( ) + ( ) ( ) ( )( ) or (Note : mason's time hours. Thus.) Apprentice takes hours to build the wall. [.]. Let number of adult tickets number of student tickets + ( ), +,, adult tickets, student tickets [.]. h d V d d r r π V r h d π d π d () d π d ft [.]. R, R > > > + + < + < ( )( ) < The product is negative. Critical values ( )( ) (, ) are and. The revenue is greater than $ when the price is between $ and $. [.]

37 Chapter Review. a. B > [.] b. B > B > or B > (, ) (, ) B < B < [.]. Let the fee for the second apartment. < < or > < > The second apartment costs more than $ but less than $. [.]. Let h the height of the package. length + girth length + (width) + (height) + () + h + + h + h h h. The height must be more than but less than or equal to. inches. [.]. Let C the circumference, r the radius, and d the diameter. C π r π d. C.. π d... d π π. d. The diameter of the basketball is from. to. inches. [.]. Let the score on the fifth test The student needs to earn a score in the interval [, ] to receive a C grade for the course. [.] ( ) ( ) ()() ± () ±. or. More than. mi but less than. mi from the city center. [.]. k(. ). (,,).(,, ) k(. ).(,, ). A km r k k. A. r m (. )(. ) A (,, ) A. meters/sec [.]. L kd h k(.) ( ) k(.) ( ) k (.) k. L.d h.() L L. tons [.]

38 Chapter : Equations and Inequalities... Chapter Test. ( ) + ( ) [.]. [.] or. ( )(+ ) [.] or ( ) ± ± ± ± ± ± ± [.]. a, b, c [.] ( ) ± ( ) ()( ) () ± + ±. + + [.] a, b, c b ac () ()() The discriminant,, is a positive number. Therefore, there are two real solutions.. a c c( d) a c c cd a c c cd a ( c) c cd c cd, a c a c [.]. ( ) ( ) + ( ) ( ) + + ( )( ) Check (No) The solution is. [.]

39 Chapter Test. / / + [.] / Let u u + u (u )( u+ ) u or u / / / / ( ) ( ) ( ) ( ). [.] + + ( ) ( ) () ( + ) (). a. or + > b. < and + > < < < { } { < } { } [.] { < } { } [, ) [.]. + + ( + )( ) + The quotient is positive or zero. + + Critical values are,, and. ( + )( ) +. The range is from in. to in. [.] Denominator. [, ) [, ) [.]. Let the rate of the current. Rate with current +. Rate against current. d rt ( + ) t ( ) t t t + + ( ) ( + ) + The current is mph. [.].. Remove amount of %. Add amount of % (.) (.) + (.) (.) liters [.]

40 Chapter : Equations and Inequalities. Let number of hours the assistant needs to cover the parking lot. + () + () + The assistant takes hours to cover the parking lot. [.]. +. > +.. > > If you drive more than miles, then company A is less epensive. [.] (.) (.) (.)(.) ± (.).±...±... or. More than. ft but less than. ft from a side line. [.]. + ( + ) + + ( )( ). More than. mi but less than mi from the city center. [.]. k v d k k v v. miles/second [.]... Cumulative Review. + ( ) [P.]... [P.]. ( ) ( + )( ) ( + ) ( ) [P.]. + ( )( + ) [P.]. + + [P.]. / / / + / / a a a a [P.]. ( + i)( i) i + [P.]. ( ) + ( ) [.]

41 Cumulative Review. [.] ( ) ± ( ) ()( ) ± ± (). [.] or. ( ) ( ) + + ( ) or [.] Check : No Check : ( ) ( + )( ) The solutions are,,. [.]. > or + > > The solution is { or > }. [.]. + Let u. u u+ (u )( u ) u or u u ± ± u ± The solutions are,,, [.]. The solution is (, ] [, ). [.] or. ( ) + + Solve + and to find the critical values. + The critical values are and. Test, in the interval (, ) : The intervals are (, ), (, ) ( ) and, which is false. (), Test., in the interval, :..., which is true. (.). Test, in the interval ( ) The denominator cannot equal zero The solution is { } [.], :, which is false. () <..

42 Chapter : Equations and Inequalities. w w + Perimeter (Length) + (Width) ( w+ ) + w w+ + w w w w w + The width is feet; the length is feet. [.]. Let time for assistant to do the job alone. The worker does of the job per hour. The assistant does of the job per hour. ( ) ( ) + + ( ) + + ( ) The assistant takes hours working alone. [.]. Let the score on the fourth test < and + < + < < [, ) [, ] [, ] The fourth test score must be from to. [.]. p p p ( p) p, p p, p. p and p p ( p) p, p p, p. They can epect to ticket from.% to.% of the speeders. [.]