Chapter 1 Equations and Inequalities

Transcription

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

2 Chapter : Equations and Inequalities. + Conditional equation. or. or. or. + + or +. or () (). + + Contradiction. There is no solution.. a b, b> a b or a b a+ b a b. d t a+ b a b t or t t t t t.. hours hours minutes Ruben will be eactly miles from Barstow after hours and after hours and minutes or Revenue The revenue first eceeded $ billion in.. + Cost + Rounded to the nearest yard, sq yards can be carpeted for $.., 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 + The case + < has no solution since there is no real number such that < and. { }. + ( + ) if + + ( + ) + { } if + < < + + an identity. + + if if + < < + ( + ) an identity + The case + < has no solution since there is no real number such that < and.... Prepare for Section. PS. PS. l + w (l + w) Distributive property PS. bh PS. b h bh Associative property of multiplication PS. + + PS. ab + b + a b + a b+ a a+ b a b a b b a ab ab ab Section.. π V r h V π r h V π r h. I Pr t I t Pr. Gmm F Fd Fd Gm d Gm m m

4 Chapter : Equations and Inequalities. an a + ( n ) d a a ( n ) d n an a n d. a S r S( r) a S Sr a S a Sr S a r S. qb rating [.(. ) +.(. ) +.(.) + (..(.))].. SMOG +.. GFI.(. +.).. Let the number W W. P L + W (W ) + W W + W W W ft L W () ft. 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. + + () The shortest side is cm. The longer sides are each cm. 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 d t t hours

5 Section.. 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 rate time distance Up To Jon s y Level house y Down z z Up Back home Level y y Down z z y y + + z z y z y z () + y+ z+ + y+ z + y+ z ( + y+ z) + y+ z The distance to Jon s house is miles.. Let the number of sunglasses. Profit Revenue Cost,..,. The manufacturer must sell sunglasses to make a profit of $,. A score of will produce an average of.. Let cost last year.... The cost of a computer last year was $... Let amount invested at %. (, ) amount invested at.%.. +.(, ). +.., $ was invested at %. $ was invested at.%...% % % +.() +..( + ) $ additional investment.

6 Chapter : Equations and Inequalities ( ).(). +.. g pure silver + ( ), +,,, tickets at $ each tickets at $ each.. or or + + () ( + ) + () ( + ) + () ( + ) ().( ).... liters of water. $ X $ $ + ( ) () + lb of $ coffee lb of $ coffee. 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

7 Section.. Let t the time it takes both painters working together to paint the kitchen. The painter can paint of the kitchen every hour. The apprentice can paint of the kitchen every hour. t+ t t+ t t+ t t t hours. Let t the time it takes the older machine to finish the job. The new machine does of the job every hour. The old machine does of the job every hour. The new machine works for hours: ( ). The old machine completes the job. + t t t hours. 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. PS. ( + )( ) PS. (+ )( ) PS. + + i PS. ( ) ( ) ( )() ( ) PS. ( ) + ( ) ()() + () PS. ( 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 ± (.) (.)(.) (.).±.+..±..±., or. + ± ()( ) () ± + ± + or. + ( ) ± ( ) ()() () ± ± ± i + i, or i. ( ) ± ( ) ()( ) () ± + ± ±, or. b ac ( ) ()( ) + Two real solutions

11 Section.. + b ac ( ) ()() No real solutions. + b ac ( ) ()() One real solution. + b ac () ()( ) + Two real solutions. + + b ac () ()() No real solutions. 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.

12 Chapter : Equations and Inequalities.,. +,. +, a., b, c, ± (.)(,) (.) ± ±.. + or.... or, racquets must be sold.. Solve D + + for with D a, b, c ± ( )( ) ( ) ± ±. +. or.... mile. miles. Solve h t + t for t where h. t + t t + t a, b, c ± ( )( ) t ( ) ± ±. t +. or t.... seconds. seconds. Let w width of region Then w length. Area length(width) w w w w w w+ ( w w+ ) w w+ ( w )( w ) w or w w w w () () w The region is either feet wide and feet long, or feet wide and feet long.. Solve h t +.t+ for t where h. t +.t+ t +.t+. ± (.) ( )() t ( ). ±. t. or t. He was in the air for. s.. 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.

13 Section.. Solve h n ( n )for n where h. nn ( ) nn ( ) n n n n ( n )( n+) n n people or n + n (no). Solve R. t+. for t where R.. t+.. t. a., b, c. ± ( ) (.)(.) t (.) ±, ±. t +. or t... (not in the future) Revenue for cellular phone software will first reach $ billion. 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.

14 Chapter : Equations and Inequalities... Connecting Concepts. A line etending from A to MN is equal in length to b. Draw a line from A to Z (a point on MP) that is parallel to line OP and perpendicular to MP. Draw a line from A to N. Since AN is also a radius of the circle, then its length is a. AZ is equal in length to OP, so OP AZ b. We have a right triangle AZN, where angle AZN is the angle. Also, ZN ZP - NP, where Theorem, ( AZ) + ( NZ) ( AN) a ( NP) a ( ) b + ( ZP NP) b + a b + a anp + ( NP) a or ( NP) anp + b a+ b, where NP Thus NP is a solution to ab+ b. ZP a. Using the Pythagorean Draw line AM. Since AM is also a radius of the circle, its length is a. We have a right triangle AZM, where angle AZM is the angle. Also, MZ MP - ZP, where Theorem, ( AZ) + ( MZ) ( AM) ( MP a) a ( ) b + ( MP ZP) b + a b + ( MP) amp + a a or ( MP) amp + b a+ b, where MP. Thus MP is a solution to ab+ b. ZP a. Using the Pythagorean

15 Section.. + ; b a ( ) c ( ) ( ) ; a Yes, and are roots of.. b ( ) ( + i ) + ( i) ; a c () ( + i )( i) i + ; a Yes, + i and i are roots of Prepare for Section. PS. PS. ( ) ( ) ( + )( ) ( + )( ) PS. / ( ) PS. / ( ) PS. ( + ) + + ( ) PS. ( + ) () + + ( + ) ( ) ( )( + ),, or Section.. + ( ) ( ) ( )( ) ( )( )( + ),, or. ( ) ( )( + ),, or. ( ) ( + )( ),, or. ( ) ( )( + ),, or. ( )( + + ), or + + ± ()() ± ± i ± i + i or i Thus the solutions are, + i, i. + ( ) ( + ) + +

16 Chapter : Equations and Inequalities ( ) ( + ) + ( + ) ( + ) + + ( + ) + 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)

17 Section.. + ( ) (+ + ) ( ) ( + ) ( )( ). + ( + ) (+ ) () ( ), or Check () + Check () + () checks as the solution. () + The solution is. (No). + + ( + ) ( ) + + ( ) (). + ( ) ( + ) + (+ )( ), or Check checks as a solution. Check + + (No) The solution is. + +

18 Chapter : Equations and Inequalities ( )( + ), or. / ( /) ( ) Check () + () + ( ) + ( ) ( ) + The solutions are and.. / ( /) ( ) / /,. / ( /) ( ) / /. / / / ( /) ( ) / /,. / / / ( /) ( ) / /. / / / ( /) ( ) / /. + Let u. u u+ ( u )( u ) u or u ± ± The solutions are,,,.. + Let u. u u+ (u )( u ) u or u ± ± ± The solutions are,,,.. + Let u. u + u ( u )( u+ ) u or u The solutions are and.

19 Section.. / / + / Let u. u u + ( u )( u ) u / or / The solutions areand.. Let u. u u + (u )(u ) u ± ± u The solutions are and.. + Let u. u u + (u )( u ) u or u The solutions are and.. / / / Let u. u u (u + )( u ) u / or The solutions are. / / / / / ( ) ± () ± u / and. The solutions areand.. 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 ( ) ( ) () + () + ( ) + ( ) + hours

20 Chapter : Equations and Inequalities. Let number of rounds the golfer needs to play. () + + ( + ) rounds.. SMOG w + w + w w words with three or more syllables.. 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.

21 Section.... Connecting Concepts. a. s (a + b + c) ( + + ) r abc ss ( a)( s b)( s c) ()()() r ( )( )( ) ()() ())() r. ()()() The radius is approimately. in. 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) a a a a a a a a a a a a a a a a a aaa Each side is in.. T s + s T s + s Let s u. u + u T ± () ()( ) T u u ±,, +, T +,, +, T s +,, +, T s + + T s... Prepare for Section. PS. { > } PS. ( ) ( ) + PS. + PS., It is undefined for. PS. + (+ )( ) PS. + ( )( )

22 Chapter : Equations and Inequalities 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.) {}

23 Section.. + > ( + ) > 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.,

24 Chapter : Equations and Inequalities. + + (+ ) The quotient is positive or zero. + + The critical values are and. + The denominator cannot equal zero,.. ( + )( ) < ( + )( ) The quotient is negative. + The critical values are,, and. ( + )( ) (, ) (, ) ( ) 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. ( + )( ),, (, ) [, )

25 Section.. + ( )( ) ( )( ) 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 >. >. >. months after September is in January.. µ. < <... <. µ <.. < µ <.. > µ >.. R. < µ <. lb > ( ) > The product is positive. Critical values are and. ( ) ($, $)

26 Chapter : Equations and Inequalities.. +, <.+, <. <, >. 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 )

27 Section.. 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. PS. k() k PS. k. k PS. k () PS. k. (.).. PS. The area becomes times as large. PS. No. The volume becomes times as large.. d kt. k y Section.. m knp. V klwh. A ks. kmm F d. y k k k k. r kt k k k k. T krs k k k. k. V klwh k k k. V kt. k.. k k Thus V. T. (.). liters

28 Chapter : Equations and Inequalities. s k q k k p p semester hours. j k d k () k () p p. fl oz. T k l. k. k. k a. T.... seconds b. T k l T l k l. l.. ft l. r k t k k r r revolutions per minute. l k 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

29 Chapter Review... Connecting Concepts. / T kd / / k k / Thus d / / / / d d d million miles d / /... Assessing Concepts. No. The solution set of is {} but the solution set of is {, }.. a <. a< b. Equal to. False. ( ) c. b. a. g. f... Chapter Review. ( ) ( + ) + + [.]. ( ) ( ) [.]. ( ) ( ) + + [.]

30 Chapter : Equations and Inequalities. [.]. + + ( + ) + ()( + ) + ( ) ( ) () + + ( + ) + + [.] + +. y y+ y y y( y ) y( y ) + + y+ y y( y ) y( y+ ) ( y+ ) y y y y y+ y y [.]. + [.] ( )( ) or. + [.] ( ) ± ( ) ()() () ± ± i + i or i. + [.] ( )(+ ) or +. [.] ( ) or. [.] ( ) ± ( ) ()( ) () ± + or. [.] ( ) ( )( + ),, or. + [.] Let u. u u+ (u )(u ) a ± or ± ± u ± ± ±. + [.] Let u. u + u (u )( u+ ) u Thus,. or u No solution.

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

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

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

34 Chapter : Equations and Inequalities. + > [.] The quotient is positive. + The critical values are and. + (, ) (, ). ( ) [.] + The quotient is negative or zero. + The critical values are, and. ( ) + Denominator. (, ) [, ]. [.] ( ) + 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.

35 Chapter Review. V π r h V h π r [.]. P A + rt P( + rt) A P+ Prt A Prt A P t A P Pr [.]. A h ( b + b) A hb ( + b) A hb + hb A hb hb A hb h b [.]. P ( l + w) P l + w P l w P l w [.]. e mc e m c [.]. mm F G s Fs Fs Gm Gm m m [.]. 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 $.. [.]

36 Chapter : Equations and Inequalities. 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. Time Mason Apprentic e Part completed In hour + ( ) + ( ) ( ) ( )( ) or (Note : mason's time hours. Thus.) Apprentice takes hours to build the wall. [.] Length in. and width in., or length in. and width in. [.]. Let number of adult tickets number of student tickets + ( ), +,, adult tickets, student tickets [.]

37 Chapter Review. h d V d r r d π V r h π d 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 $. [.].. < <.. <. <.. < <. < <, where is an integer. µ. < <... < µ <.. < µ <.. > µ >.. < µ <. lb. 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. [.].. µ. < <... <. µ <.. < µ <.. > µ >.. < µ <. lb. 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. [.]

38 Chapter : Equations and Inequalities. 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 [.]... Quantitative Reasoning QR. + ± ()( ) ± + ± () QR. Answers will vary. QR. Answers will vary.... Chapter Test. ( ) + ( ) [.]. [.] or. ( )(+ ) [.] or ( ). ± ± ± ± ± ± ± [.]

39 Chapter Test. [.] 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. [.]. / / + [.] Let u / u + u (u )( u+ ) u or u / / ( /) ( ) ( /) ( ). a. or + > > < { } { < } { } [.]. + + ( + ) ( + ) + + () ( + ) () b. < < and + < { < } { } [, ) [.] [.]

40 Chapter : Equations and Inequalities. + + ( + )( ) + 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. [.]. Let number of hours the assistant needs to cover the parking lot. + () + () + The assistant takes hours to cover the parking lot. [.].. Remove amount of %. Add amount of % (.) (.) + (.) (.) > +.. >... liters [.] > 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. [.]

41 Cumulative Review. v k d k k v v. miles/second [.]... Cumulative Review. + ( ) [P.]... [P.]. ( ) ( + )( ) ( + ) ( ) [P.]. + ( )( + ) [P.]. + + [P.]. / / / / / a a a a [P.] +. ( + i)( i) i + [P.]. ( ) + ( ) [.]. [.] ( ) ± ( ) ()( ) ± ± (). [.] or. ( ) ( ) + + ( ) or [.] Check : No Check : The solution is.. ( ) ( + )( ) The solutions are,,. [.]. + Let u. u u+ (u )( u ) u or u u ± ± u ± The solutions are,,,. [.]

42 Chapter : Equations and Inequalities. > > > or + The solution is { or > }. [.]. or The solution is (, ] [, ). [.]. ( ) + + Solve + and to find the critical values. + The critical values are and. The intervals are,,, and, Test, in the interval (, ):, which is false. () Test., in the interval, :..., which is true. (.). Test, in the interval, :, which is false. () The denominator cannot equal zero. The solution is <. [.]. w w + Perimeter (Length) + (Width) ( w+ ) + w w+ + w w w w w + The width is feet; the length is feet. [.]. P R C. (+,). +, Profits must be greater than or equal to,.. +,,. +, ± () (.)(,) (.) ±. or, to, printers should be manufactured. [.]. 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.% to.% of the speeders. [.]