2 9 } 2 1 } 2 5 } 2 7 } 2 2 } Prerequisite Skills for the chapter Solving and Graphing Linear. x 0 1 } y 3 5 } x } y 1 }

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1 Chapter Solving and Graphing Linear Inequalities Prerequisite Skills for the chapter Solving and Graphing Linear Inequalities. Sample answer: (, ) 8 8() () No; because the solution of 7 is, the two equations do not have the same solution.. The absolute value of a number a is the distance between a and on a number line.. m 8 m Make a table of values Make a table of values. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. r r r 7. t 7t t t t ?.7 >..8?..8 <...?.. >...7?..7 < For ever value of, the value of is 8. The graph of the equation 8 is a vertical line 8 units to the left of the -ais. 8 Algebra Worked-Out Solution Ke

2 . For ever value of, the value of is. The graph of the equation is a horizontal line units above the -ais.. < < m m m Lesson. Solve Inequalities Using Addition and Subtraction Guided Practice for the lesson Solve Inequalities Using Addition and Subtraction. Let temperature in Antarctica 8 7. < p. < p <. 8. m m. 7.. >. >. 8.. w w. You can add no more than. pounds to the bag. Eercises for the lesson Solve Inequalities Using Addition and Subtraction Skill Practice. To graph < 8, ou draw an open circle at 8, and ou draw an arrow to the left of 8.. B solving the inequalit 7 8 using subtraction, ou see that, So, 7 8 is not equivalent to.... > < 8. >.. n 7 n. 8. v > 7. v >.8. w. >.7. r < w > 7. r < 7. s 8 s s p p p 7. q > q >.. c c c 8.8. d. > 8.7 d > In order to solve the inequalit, 8 needed to be subtracted from both sides. It was mistakenl added to the right side. 8 < 8 8 < 8 <. The solution is or, so the graph should be shaded to the right, not the left. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

3 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. m >. n 7 m > n. c < c < 7... <..7 <. < It is not possible to check all the numbers that are solutions of an inequalit because there are an infinite number of real number solutions. Checking one solution does not guarantee ou have solved an inequalit correctl because that solution could satisf man inequalities, so ou cannot be sure that our solution is correct based on one correct solution Not solutions: < 7.7 Problem Solving 7. Let points before redemption > > 8 8 You must have more than 8 points before making a redemption.. Goals so far Additional goals Gretzk s record The plaer must make at least more goals in order to match or break Wane Gretzk s records.. a. Your first score Your second score > Competitor s first score Competitor s second score. > > 7. > 7. You must score more than 7.. b. If ou earn 8. points, ou will win since 8. > 7.. If ou earn.78 points, ou will not win since.78 ò 7.. If ou earn 7. points, ou will not win, rather ou will be tied with our competitor.. C; p. Answers will var.. ales: w,, w,8 ales: w,, w,8 ales: w,, w,8 ales: w, 8, w,8 A vehicle with ales weighing, pounds cannot carr, pounds of contents because the total weight of, pounds would be greater than the maimum weight allowed,, pounds. 7. a. Original Price ($) Final Price ($),,,8,8, 7,,7,7,, b. 7,, Your friend will consider buing cars originall priced up to $,. 8. a. last ear s time: () The last athlete must run meters in no more than. seconds. b. The team could fail to meet their goal. The last athlete has to run at least as fast as his average speed of. seconds. If his average speed is. seconds, that means his slowest speed is slower than. seconds. So running faster than his slowest speed could still be slower than the speed necessar for the team to win.. () () 7,,8 7,,8 The station needs to raise at least another $,8 to meet their goal. Algebra Worked-Out Solution Ke 7

4 Investigating Algebra Activit for the lesson Solve Inequalities Using Multiplication and Division. Divide both sides b Write original inequalit. 8 Add to each side Subtract 8 from each side. 8 Divide each side b. Rewrite inequalit with on the left side.. It does not give the solution found in Eercise. You must also change the direction of the inequalit smbol.. You do not need to change the direction of the inequalit smbol when ou divide b a positive number, but ou do need to change it when ou divide b a negative number > 8. 7 < < < Lesson. Solve Inequalities Using Multiplication and Division Guided Practice for the lesson Solve Inequalities Using Multiplication and Division. > 8. m 8 p > 8 p 8 p m 8 8 p () > m >. p..() p < () < 8 8 m. <.. v 7 7 p m 7 > 7(.) v. 8 m >. v n < n > n > 8. r r r 7 The student can afford to pa no more than $7 per hour. Eercises for the lesson Solve Inequalities Using Multiplication and Division Skill Practice. You would use the division propert of inequalit to solve.. < p > () > The are not equivalent inequalities because < simplifies to >, not <. When ou multipl each side of an inequalit b a negative number, ou must reverse the direction of the inequalit smbol.. p. p p 7 < p > p >. < > >. > w 7. () > p w > w w < q < 7 8 p q < p r q < 8 7 r 8 r r Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 8 Algebra Worked-Out Solution Ke

5 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. g > p g > () g >. m m m. t t. t. t.. n < 7 p n < p () n < 7. s. s s s v 8 8 p v p 8 v. 8. f >.. 8. f 8. <. 8. f <... d p d (8.) d < c. > c. > c c < k 7 7(.7) 7 p k 7. k k z.. z. z z.. 7. p (7.) 7... < j... > j. > j j < p..() r <.8 r p > (.8) r >. 8.. p.8.. p.8 p p.8 m. >. p m. >.() m > t >.. t <. t < It is similar because ou have to divide both sides b a in both inequalities. It is different because, when a <, ou also have to reverse the direction of the inequalit smbol. Algebra Worked-Out Solution Ke

6 8. Because ou have to divide each side b a negative number, the direction of the inequalit sign needed to be reversed (division propert of inequalit). > < <. The error is that the inequalit sign did not need to be reversed because both sides of the inequalit were multiplied b a positive, not a negative, number. 7 p p (7). 8 > 8 8 > 8 > v < 8 p v > p (8) v > w p w w. Sample answer: <. a. a <, b > a > b < b a Because is less than a negative number, the solution consists of onl negative numbers. b. a >, b > a > b > b a Because is greater than a positive number, the solution consists of onl positive numbers. c. a >, b < a > b > b a Because is greater than a negative number, the solution consists of both positive and negative numbers, as well as zero. d. a <, b < a > b < b a Because is less than a positive number, the solution consists of both positive and negative numbers, as well as zero. Problem Solving. Cost of a CD p Number of CDs Total amount You can bu no more than CDs. Total words 7. words per minute Time (min) p () You must tpe at least words in minutes. 8. a. books b.. You can stack a maimum of books on the shelf. c., d. Answers will var.. 8 w. w The width must be at least. inches.. minutes.7 hours d < 8(.7) d <. A beginner water skier can travel a distance of less than. miles. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

7 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. a. length p width Area reserved per horse number of horses 8 p 8 h h p h h h. h h. The corral can hold no more than horses. b. Increasing the length and width b feet each will increase the total area b more than square feet, so the corral will be able to hold more than one more horse. c. New area (original length )(original width ) A (8 )(8 ) (7) square feet After finding the new area, use the verbal model from part (a) to create a new inequalit to represent the new corral size. h h p h h h. h h. The corral can hold up to horses.. Let sale price of the laptop The laptop must be decreased in price at least $.. Quiz for the lessons Solve Inequalities Using Addition and Subtraction and Solve Inequalities Using Multiplication and Division. 8. < < v v v 8. w > w > 8. < r. < s > r < s 8 > r < s r < 8 s > 8 7. m 8. m. n > 7 p n < (7) m n < 8 8 c p c (8) c 8. Sandwiches per minute p Time (minutes) Total number of sandwiches 8 t t It will take ou at least minutes to make the number of sandwiches ou need. Lesson. Solve Multi-Step Inequalities Guided Practice for the lesson Solve Multi-Step Inequalities (p ) > p > p > p <.. (m ) < m 7 8 m < m 7 < 7; False. 8s (s ) 8s 8s ; True Algebra Worked-Out Solution Ke

8 7. $.. $..g.g g. You can bu up to. gallons of gasoline. 8. Weeks saving p Amount saved per week Amount alread saved 8.8 You must save, at least, $.8 per week. Total cost Skill Practice. The inequalities <, <, and < are called equivalent inequalities.. When ou solve the inequalit and all variables are eliminated, if ou end up with a true statement, the solution is all real numbers. If, however, ou end up with a false statement, the inequalit has no solution.. > 7 > >.. 8v 8v 8 v. (w ) < w < w < w < 7. 7(r ) 7r 7r 8 r 7 8. (s ) s 8 s 8 s. m > 7 m m > n > 7n > n > n n < 7 7. p > p 8 p > 8 p <. q q 8 8 q q q 8. d < d 8 d < 8 d < d >. 8 f > f 8 f > f > f > 8 8. The error is that the inequalit smbol was not reversed when dividing each side b. 7. The distributive propert was not used properl. In step, () should have given a positive. ( ) < 8 8 < 8 8 < > 7. p > p p 7 p > p 7 > 7 The statement is true, so the solutions are all real numbers. 8. d 8d d d d d d d. (s ) (s ) s s s s Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

9 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. (t ) > t 8 t > t 8 > 8 The statement is true so, the solutions are all real numbers.. (b ) b b b The statement is true, so the solutions are all real numbers.. (c 7) 8(c ) 8c 8c The statement is true, so the solutions are all real numbers.. ( ) < 8 8 < 8 8 < 8 The statement is false, so the inequalit has no solution.. ( ) The statement is false, so the inequalit has no solution...h. (.h.).h..h... The statement is false, so the inequalit has no solution... j. j ( j ) j j The statement is true, so the solutions are all real numbers. 7. (m ) < m m < m < The statement is false, so the inequalit has no solution. 8. (8n ) < ( n) n < n < The statement is false, so the inequalit has no solution.. < < < 8. ( 8). > 7 > > <. ( ) ( 8) If a >, b > a b > a > b > b a If a <, b < a b > a > b < b a If a >, b < a b > a > b > b a C; a >, b <. ( ) > 8 8 > 8 > > 7 is greater than 7 feet.. p 8( ) ( ) is less than or equal to.. ( ) a a a a a Algebra Worked-Out Solution Ke

10 a a a All solutions are less than or equal to when a. Problem Solving 7. Minutes per song p Number of songs Minutes alread burned > Total minutes n 7 n n. You cannot burn more than additional songs. 8. Price per ornament p Number of ornaments Cost of supplies > Zero 8.n > 8.n > n >. The woodworker must sell more than ornaments to make a profit.. a. Area for first swans Area for each additional swan p Number of swans Area of habitat n () n n n B solving the inequalit for n, we find that the zookeeper can have up to additional swans after the first, so the habitat can hold a total of swans. b. n (7) n 8 n n 8. 8 The habitat can now hold up to more swans than the original habitat.. C; ( ) <. a. b. Average Pitches per Inning (p) Average pitches per inning Total Pitches (t) p innings Pitches in first innings Maimum pitches allowed p p p p 7. The pitcher can make no more than an average of 7 pitches per inning in the net innings.. a.. ( 7).. The ta rate is %. This result was obtained b first finding the difference of the price and 7. This is the taable amount. We know that whatever the ta rate is, the amount paid on ta is $.. From here an equation can be set up and solved to find the ta rate. b. p ( p 7)(.) p.p 8.7.p 8.7 p 8. The price of the coat must be no more than $8.. c. p.p < p ( p 7)(.). p < p. p 8.7. p <. p <.p 87 < p The % ta rate would be cheaper for items more than $87. Test: p $.()? < ( 7)(.) <. Scores from each game. League Average Number of games You must score no less than on our net game. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

11 Problem Solving Workshop for the lesson Solve Multi-Step Inequalities. Number of batches Number of cookies You must bake at least batches of cookies.. Total spent (dollars) Number of games You can bu up to games. Games Amount of mone left ($) 7 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. Total number of cookies Number of cookies 7 Cookies per batter 8 Number of batches p Number of batters You must bake at least batches of cookies. Games Amount of mone left ($) 8 You can bu up to games. Total cost Cost per game p Number of games Cost of console. 8 7 You can bu up to games. Games Amount of mone left ($) You can withdraw mone for up to 7 weeks. Total left in account 7 Original amount in account Amount withdrawn per week p Number of weeks Algebra Worked-Out Solution Ke

12 Account value (dollars) Number of $ withdrawals You can withdraw mone for up to 7 weeks.. h min 8 min Total time Time for Time for first miles remaining 7 miles...7 < 7..7 <..7 So, <.7 Check:.().7 <? 7. Cost of a PPV movie ($) p. < 7 Number of PPV movies Graph on a graphing calculator. 8 Cable TV package cost ($) (.7, )..7 Amount budgeted ($) Total time (minutes) Average time for remaining 7 miles (minutes per mile) Your average running time for the remaining miles must be slightl less than 8 minutes per mile in order to have a total time of less than hour and minutes. Chapter Etension for the etension Solve Linear Inequalities b Graphing. > > So, >. Check: () >?. 7 7 So,. > Check: ()? (, ) (, ) 7 X=.8 Y= Trace to find the -intercept. -intercept ø. You can afford up to pa-per-view movies per month. Investigating Algebra Activit for the lesson Solve Compound Inequalities. Answers will var. 7. True 8. False Lesson. Solve Compound Inequalities Guided Practice for the lesson Solve Compound Inequalities. < or. <. c < or c >.. 7 < <. 7 < and < < and < < < 8. 8 and and and Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 8 Algebra Worked-Out Solution Ke

13 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. 7 < z < 7 < z and z < < z and z < > z and z > < z < < 8 < 7 < < 7 8. t t t t 8. h < or h > 7 8 h < or h > h < or h > 8. c or c > 7 c or c > c or c >. C 7 (F ) 7. F F Possible temperatures include 8F, 8F, 8F (eamples will var). Eercises for the lesson Solve Compound Inequalities Skill Practice. A compound inequalit is an inequalit that consists of two inequalities joined b and or or.. The graph of is shaded between and. The graph of or is shaded to the left of and to the right of.. < <. 8 or > 8.. < or < or > < <. 7 > 8 > <. m m m. < n < < n < < n < 7. (p ) < p < p < p <. 7 > (q 8) 7 > q > q > q q <. r < 7 or r r < or r 7 r < or r Algebra Worked-Out Solution Ke 7

14 . < s or s < s or s > s or s 7. v < 8 or 8v < 8 v < or v > 8 8. > w or w > w 7 7 > w or w > w < 7 or w > 7. g > g or > g 8 g > 7 or > g g < 7 or < g 7. h 7 > h or h > 8 h < or or (h 8) h h 7 h < or h 8 8. In Step, should have been subtracted from all three epressions of the inequalit. < < < < > >. The graph should be shaded to the left of and to the right of 7. > or 8 < > 7 or < 7. < 8 or > < or > 8 7. < < < <. 8 ( ) or < 8 7. C; or > or < 8 a > 8 or a > Tr : > 8 or > 8 > or 7 > 8 > or 7 < Tr : > 8 or > > or > > or < Tr : > 8 or > < < > or > > or < 8. The statement is false. A countereample is because it is not greater than or equal to.. The statement is true.. The converse is true. If a is a solution of < and, then a is also a solution of <.. The converse is false. For eample, is a solution of > or, but it is not a solution of >.. a. < 7 < 7 7 < < < < b. < < c., 8, (Eamples will var.). 8 < and < < and < This inequalit has no solution because no numbers are less than and greater than.. 7 and > and > 7 and > 7 7 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 8 Algebra Worked-Out Solution Ke

15 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. m or m > m or m > m or m >. n or n > n or n > n or n < Problem Solving C (F ) F 7 F. B; euro was worth:. v.7 v <. or v >.7. p 7,8.. p. f... f. A pound deer can eat. to. pounds of food per da...h < or.h > h <. or h > 8.7 The shop does not provide skis for skiers who are under. centimeters or over 8.7 centimeters.. a. C < or C > (F ) < or (F ) > F < or F > 8 F < or F > Water is not a liquid when it is less than 8F or more than 8F. b. Temperature (8F) Temperature (8C) 8..a..a. a Possible air temperatures range from 8F to 8F. t. a. 8 t Total amplification needs to be from watts to watts. b. p t 8. p t 7. The amplifier will be strong enough for people because the amplification per person would be 8. watts, which falls within the required range of 8 to watts. The amplifier will not be strong enough for people because the amplification of 7. watts per person does not fall in the required range. t c. 8 8 t The least amount of amplification that our amplifier should provide is 8 watts. This will provide the minimum amplification of 8 watts per person for a maimum crowd of.. Let t total cost of meals t.t.t. t.7 You can spend $. to $.7 on meals before the tip is added. Quiz for the lessons Solve Multi-Step Inequalities and Solve Compound Inequalities. ( ) >. 8 > 8 > < 8 8. r 7 r r 7 r r 8 Water is not a liquid at 8F or 8F. Algebra Worked-Out Solution Ke

16 . (s ) s 8 s s 8 8 The statement is true, so the solution is all real numbers.. a or a < a or a < 8 8. > c > 8 > c > < c < 7. 7 m 8 m n > 8 or (n ) > n > or n > n > or n > n > or n < 8 Graphing Calculator Activit for the leson Solve Compound Inequalities. The graphs look the same on a graphing calculator.. You have to know if the problem used less than/greater than or less than or equal to/greater than or equal to inequalit smbols since the graphs on the calculator look identical... < < 8.. > 8 or > 7. 8 or 8. 8 or Mied Review of Problem Solving for the lessons Solve Inequalities Using Addition and Subtraction, Solve Inequalities Using Multiplication and Division, Solve Multi-Step Inequalities, and Solve Compound Inequalities. a. C (F ) F F The thermometer can measure temperatures from 8F to 8F. b. 8 c. The thermometer cannot measure a temperature of 8F because that is out of the range it can measure a You must score at least 78 on our sith test. b. 8 It is not possible for ou to have an average of after our sith test because ou would have to score 8 on that test, which is not possible.. Let number of cartons.8 You must bu at least cartons.. a. Let amount spent on socks (8) 7 You can spend up to $ on socks. b. Let pairs of socks You can bu up to 7 pairs of socks.. Answers will var.. a. Let average weight of each adult () 8 8. The average weight of each adult cannot eceed. pounds. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

17 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. b. Yes, the raft can accommodate all of the people and baggage because none of the adults are heavier than the average weight of each of them. 7. a..().(,) 8 b. Their incomes were greater than $, because the paid more taes than those with incomes from $ to $, (from part (a))..(,). 7,,,,,, Their incomes were from $, to $,. c. Let total income. <.(,).(,). <.. < >, Based on the inequalit shown above, the % ta rate would result in less taes for those with incomes greater than $,. Lesson. Solve Absolute Value Equations Guided Practice for the lesson Solve Absolute Value Equations. a. or b. or. r 7 r 7 or r 7 r or r The solutions are and.. s. 8. s.8 s 7. s 7. or s 7. The solutions are 7. and 7... t t t t or t t or t The solutions are and.. m m m There are no solutions.. n 7 n n n or n n or n The solutions are and or or. The solutions are.8 and.. Eercises for the lesson Solve Absolute Value Equations Skill Practice. The equation 7. is an eample of an absolute value equation. The absolute deviation of from is... or or. v.7. w. v.7 or v.7 w. or w. 7. r r or r. m 7 m 7 or m 7 m or m. n 8 n 8 or n 8 n or n n or n. p 7 p 7 or p 7 p or p p or p. q 8 q 8 or q 8 q or q. d 7 d 7 or d 7 d or d 8 d or d. f 8 f 8 or f 8 f or f 8. s 7 s 7 or s 7 Algebra Worked-Out Solution Ke

18 . t t t or t t 8 or t 8 t or t. b 7 7 b b b or b b 7 or b 7. c 7 c c c or c c 7 or c 7 c j 7 j j 8 or c j 8 or j 8 j or j j or j. k k 8 k k or k k or k k or k. g g g g or g g or g g or g. The absolute value equation has to be rewritten as two equations. or or 7. The absolute value of a number is never negative, so this equation has no solution.. This equation has no solution.. 8 This equation has no solution.. m. m. m. or m. m. or m.. 8 n 8 n. The equation has no solution. 7. v v v or v v or v v or v 8. w w w or w w or w 8 w or w. r 7 r r r r r. s 8 s or r or r or r s The equation has no solution.. p 8 p 7 p p or p p or p p or p Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

19 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. D; There is no solution or 8 or. or or. (.)... or.. 7. or.7. (.).7..7 or..7. or or 7 or 8. 8 or 7 or. () or or or. The are equivalent when a is positive or zero. The are not equivalent when a is negative. Eample: Let a, let a 8 a p 8 Let a, let a 8 a p 8 Let a, let a a p. a b c d if a >, c d a b b b b The equation has one solution. If a <, c > d a b c d a b d c (negative number) b (positive number) b positive or b negative The equation has two solutions. Problem Solving. or or The minimum height is inches, and the maimum height is inches.. () or or The least amount of time the program can last is seconds, and the most the program can last is seconds.. C;. inches.... or... or.. a or or. The least score was., and the greatest score was.888. b or or This ear s greatest score is. points more than last ear s...(). or.. or. The minimum weight is. carats, and the maimum weight is. carats. 7. a. s p b. s s or s s or s You can have a score of or a score of. Algebra Worked-Out Solution Ke

20 8. a. p. t.8. t t.7 t t.7 or t.7 t.7 or t... In, foreign-born residents accounted for about % of all residents. b..7.7 In around, foreign-born residents will again account for % of all residents. c.. t.8.8. t.8 t Foreign-born residents did not account for % of all residents during this time because the given equation has no solution when p.. a. p. m m m 7.8 m 7 m 7.8 or m 7.8 m.8 or m. The average price was $. in Jul. the average price will again be $. in around December. b. p p.7 The stock s lowest price during this period was $.7. Difference of times. t average of two times min.s.s min 7.8s 7.8s Difference t.7. t.7.7 t.7.7 or t.7.7 t 7.8 or t. Etension for the etension Graph Absolute Value Functions. g () f() u u g() u u. g () g () 7 7. g () 7 g() u u f() u u g () 7. g () f() u u g() u u 7 g () g() u u f() u u The graph is units to the left of the graph of f (). The graph is units above the graph of f (). The graph is 7 units below the graph of f (). The graph is narrower than the graph of f (). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. g () Algebra Worked-Out Solution Ke

21 . g (). g ().... f() u u. g () g().u u g () f() u u The graph is wider than the graph of f (). The graph is narrower than the graph of f (), and it opens down instead of up.. > 8 < 8 or > 8 < or > 8 8. w < < w < < w < < w <. m 8 m m 7 7 m 7 m. m computer prices meet this condition. 8.. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 7. (, ) g() u u The domain of the function is all real numbers, and the range is. The verte is (, ). The maimum value occurs at the verte, so the maimum value is. Lesson. Solve Absolute Value Inequalities Guided Practice for the lesson Solve Absolute Value Inequalities. 8. u < < u < v > v < or v >.. Eercises for the lesson Solve Absolute Value Inequalities Skill Practice. The inequalities > 8 and > 8 or < 8 are equivalent inequalities.. is equivalent to, but is equivalent to or.. < < <. or. h >. h <. or h >.. p <.. < p < t t 8. j j or j. d d or d d 7 or d 7 8 Algebra Worked-Out Solution Ke

22 . b < < b < < b <. m > m < or m > m < or m > 8. s 7 < m > or m < 8 < s 7 < < s < 8 < s <. 7 8 c 7 or c 7 c or c c or c. n n n. n n.. r > r > r < or r > r < or r > r < 8 or r >. s 7 8 > s 7 > s 7 < or s 7 > s < or s > 8 s < or s > u 8 u u or u. u 8 or u u 8 8. w 8 < w 8 < 8 w 8 < or u 8 < w 8 < < w < w <. v > v > 7 v > v < v < or or v > v > 8 v < or v >. 7 f 7 f 8 f f or f f 8 or f f or f 8. B; > Tr 8 > 8 > 8 < or 8 > < or > Tr > > < or > < or > > or < Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

23 . If the absolute value is less than (or less than or equal to) a number, then it is equivalent to a compound inequalit with and. If the absolute value is greater than (or greater than or equal to) a number, then it is equivalent to a compound inequalit with or.. Because the absolute value is greater than, it is equivalent to a compound inequalit with or, not and. > < or > < 7 or >. Because is an absolute value, it is equivalent to a compound inequalit. < < < < <.. The statement is false. Countereample: a is a solution of is not a solution of The statement is true.. Solve each inequalit < and > 8 < < and < 8 or > 8 < < 7 and < or > Graph each inequalit Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. (7) 7 or 7 or 8 or > 8 7 > 7 7 < 7 or 7 > 7 < or > > or <. 8. < 8 < < < 7 < <. The statement is true.. The statement is false. Countereample: a is a solution of > 8. > 8 > 8 > 8 is not a solution of > 8. > 8 ò 8 < Look for where the two graphs overlap. The onl values that fulfill both inequalities are between and 7. So the solution is < < 7.. a b < c, c < This inequalit has no solution because an absolute value cannot be negative, and therefore cannot be less than a negative number. a b > c, c < The solution to this inequalit is all real numbers. The absolute value of an number is nonnegative, so if c is a negative number, a b > c. Problem Solving. < < < 7 < < The essa can have between 7 and words.. >. <. or >. <. or >. The water is highl corrosive when the saturation inde is less than. or greater than You should continue to preheat the oven because the oven temperature is at most 88F, so it still needs to heat to 8F. Algebra Worked-Out Solution Ke 7

24 8. a. Measured compression (pounds) Absolute deviation (pounds) b. p p p The organization will allow compressions of to pounds. The values of p that are solutions of the inequalit are and 7.. a The mean is. m/sec. b.. d.. d.88 d. a. 8,(.) The absolute deviation is. b. 8, 8,,, There are from, to, antelope in Nevada. c. The actual population would not necessaril be greater. The possible numbers of antelope would cover a greater range. So there could be more, less, or the same amount.. < > and < or > and < or > and 8 8 < or < The skater receives a. penalt point if he/she finishes from seconds, up to but not including, seconds or from, but not including, seconds to seconds. Investigating Algebra for the lesson Graph Linear Inequalities in Two Variables Activit for the lesson Graph Linear Inequalities in Two Variables Answers will var. Lesson.7 Graph Linear Inequalities in Two Variables Guided Practice for the lesson Graph Linear Inequalities in Two Variables. < 8; (, ). < 8; (, ) () < 8 () < 8 < 8 8 < 8. < 8; (, ) () < 8 < 8 7 < 8 (, ) is a solution.. Test (, ). (). >. Test (, ). Test (, ). > 7. < 8. Test (, ). < Basketball pa rate > p Basketball time < Swimming pa rate Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. p Swimming Time Total earnings 8 Algebra Worked-Out Solution Ke

25 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. Test (, ). ().() Swimming (hours) Basketball (hours) Basketball time (hours) Swimming time (hours) Total earnings (dollars) 77. Eercises for the lesson Graph Linear Inequalities in Two Variables Skill Practice. The ordered pair (, ) is a solution of > 7.. When ou graph a linear inequalit in two variables, ou have to graph the line and then shade the half plane that represents the man solutions to the inequalit.you do not have to shade when ou graph linear equations.. < ; (, ). ; (8, ) < 8 < (, ) is not a solution. (8, ) is a solution.. > ; (, ). ; (, ) () > () () > (, ) is not a solution. (, ) is a solution > 8; (, ) 8. < ; (, ) () 7() > 8 () () < > 8 8 < (, ) is not a solution. (, ) is a solution...8. ; (, )... >.; (, ).8().().().() >. >. (, ) is a solution. (, ) is not a solution.. ; (, ). 8; (, 7) 7 8 (, ) is not a solution. (, 7) is a solution.. < ; (8, ) (8) () < < < (8, ) is a solution.. ; (, ) () (, ) is a solution.. C; (, ) < Tr (, ). () < Tr (, ). < () < Tr (, ). < () < <. A; 7. > 8. >. <. 8 <. < 8. < 8 8 Algebra Worked-Out Solution Ke

26 . 8 >. 8 >. ( ) > 8 > 8 > < ( ) > 8. ( ). ( ) > 7 > > < 7 7 ( ) > 7 8 ( ). < 7. <. ( ) < 8 < 8 < 7 < ( ) < 8 8. () <. <. <. >. > (). ( ) 8 8 ( ) 7. The graph should have been shaded above the line, not below. 8. The vertical line at should have been a solid line because the inequalit smbol is less than or equal to not less than.. You cannot use (, ) as a test point because the line representing passes through the origin, and since the inequalit smbol is greater than, that line will be a dashed line indicating that points on the line are not solutions to the inequalit.. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 7 Algebra Worked-Out Solution Ke

27 .. > 7 > > 7. (7, ), (, 8) m 8 () (7) 8 m b 8 () b b < <. < Problem Solving Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. < <. m () m b () b b >. m () () 7 m b 7 () b 7 b 7 7. m m b () b b > 7. Answers will var.. (, ), (, ) () m m b () b b <. 8 8 A possible solution is the weight of the bobsled being pounds and the weight of the athletes being pounds. (Eamples will var.) Weight of athletes (pounds) Weight of bobsled (pounds). If the elevator s maimum capacit is pounds, it can hold passengers. (Eamples will var.) Number of passengers Maimum weight (pounds). a. Let hours tutoring Spanish. Let hours tutoring French. b. French (hours) 8 7 Possible combinations include: hours tutoring Spanish and hours tutoring French, hours tutoring Spanish and 7 hours tutoring French, or hours tutoring Spanish and hours tutoring French. (Eamples will var.) 7 8 Spanish (hours) Algebra Worked-Out Solution Ke 7

28 c. Hours Tutoring Spanish. B; 7. a. m l Number of loaves of bread l Hours Tutoring French Total Earned (dollars) 7 l m l m m l 7 8 m Number of muffins b. If ou make loaves of bread, ou can make up to muffins. 8. a.. Fat calories 7 Total calories. b. ( p ) You can consume up to more fat calories toda.. a. Bedroll weight (pounds) Duffel weight (pounds) Possible combinations: pound duffel and a pound bedroll, pound duffel and a pound bedroll, or a pound duffel and a pound bedroll. (Eamples will var.) b. (, ) and (, ) are solutions to the inequalit, but the do not make sense in this situation because neither the duffel nor the bedroll could weigh pounds.. a. > Percent invested in stocks 8 > 8 Age (ears) b. A ear-old agressive investor should invest over 8% in stocks. c. Prior to age, none of the solutions make sense because the sa to invest over %.. a. Sinks: m v >, m > v Floats: m v <, m < v Mass (grams) m 7 7 m > V V Volume (cm ) Mass (grams) m 7 7 m < V V Volume (cm ) b. Volume pr h p ( )() 78. cm m.g Since its mass is greater than its volume, it will sink. Quiz for the lessons Solve Absolute Value Equations, Solve Absolute Value Inequalities, and Graph Linear Inequalities in Two Variables. or. c 8 c 8 or c 8 c or c. r r r or r r 8 or r. > < or >. t < < t < < t < 8 < t < Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 7 Algebra Worked-Out Solution Ke

29 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. s 7 7 s 7 s 7 s 7 or s 7 s 7 or s s < <. 8 8 or s < 7 8 Mied Review of Problem Solving for the lessons Solve Absolute Value Equations, Solve Absolute Value Inequalities, and Graph Linear Inequalities in Two Variables. a. Let number of apple pies. Let pints of applesauce. 7 b. Number of pints of applesauce Number of pies c. Possible combinations include: pie and pints of applesauce, pies and pints of applesauce, or pies and pints of applesauce. (Eamples will var.). a Each scoop must weigh from. to. ounces. b. 8 of the scoops must weigh in the correct range in order for 8% to meet the weight requirement. 8 of our scoops do meet the weight requirement, so ou can start working at the shop The maimum possible duration of our presentation is. minutes.. Answers will var.. a. Let air temperature. Let water temperature. < Water temperature (8F) 8 < 8 Air temperature (8F) b. If the water temperature is 8, a protective suit is recommended for air temperatures below 88F. c. The -intercept would be, and the -intercept would be, instead of. This would represent the sum of the air and water temperatures being less than 8F.. a The mean price is $.. b You will consider phones priced from $7. to $7.. c. You will consider of the phones on the website. Chapter Review for the chapter Solving and Graphing Linear Inequalities. 8. (, ), (, ), (, ) Answers will var.. The boundar line is a solid line if the inequalit sign is less than or equal to or greater than or equal to. The boundar line is a dashed line if the inequalit sign is less than or greater than. You know which half-plane Algebra Worked-Out Solution Ke 7

30 to shade b choosing a point in one of the half-planes and testing to see if it is a solution of the inequalit. If it is a solution, shade the half-plane where that point lies. If it is not a solution, shade the other half-plane.. >. > > 8. m m 7. s.7 < 8. p s < p 7. 8(t ) > 8 8t 8t 8 > 8 8t 8 > 8 The statement is false, so the inequalit has no solution. 8. (n ) 8n n 8n n n. Let number of tickets You can bu up to tickets.. t t. 7. n. < 8 n >. > 7 < g < g <. r r g < 7 7 r 7... (q ) < q 7 q < q 7 < The solution is all real numbers. 8 t. < 8 < < < > > < <. s < or s > s < 8 or s > s < or s >. w or w >. r 7 w or w > w or w > r 7 or r 7. a a or a a or a 8. c c or c c or c c 8 c Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 7 Algebra Worked-Out Solution Ke

31 7. or or 8. q q q q or q q or q q. p p q j > j > j < or j > j < or j > j < or j > 8 p Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. p p 7 p 7.. or p or p or p. or.. or.87 The minimum height is.87 inches, and the maimum height is. inches.. m 8 m 8 or m 8 8. k 8 k or k k or k k. g < < g < < g < 7 or k < g <. d 8 7 > 8 d 8 > d 8 > d 8 < or d 8 > d < or d > ; (, 8) () (8) (, 8) is a solution. 8. ; (, ) () () (, ) is not a solution.. ; (, ) () () (, ) is a solution.. ; (, ) () () 7 (, ) is not a solution.. >. > Algebra Worked-Out Solution Ke 7

32 . <. < > < Chapter Test for the chapter Solving and Graphing Linear Inequalities. <.. < 7. > 8 or <.. > 7 > < z z 8. m z m 7 7. n <. n < 8 r r 8. s < s s <. t 7 s >.... t. 8 > v > v < v v >. (w ) < w w < w w <. p > (p ) p > p > 7 w < 7 8 The inequalit has no solution.. q (q ) q q The solution is all real numbers. 7. a a a a 8. 7 < c < < c < < c < 8. h or h 7 7 h or h 7 h or h h 7 or h. d 8 > 7 8 d 8 < or d 8 > d < or d > d <.. 8. f 7 < f 7 < f 7 < or d > < f 7 < < f < < f < Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. t 7 Algebra Worked-Out Solution Ke

33 . j 7. j 7 j 7 j 8 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. The equation has no solution because an absolute value cannot be equal to a negative number.. or or or. z z z The equation has no solution because an absolute value cannot be equal to a negative number.. < ; (, ) () () < < (, ) is not a solution. 7., (, ) () (, ) is not a solution. 8. ; (, 7) 7 (, 7) is a solution.. <. <. Let r revenues ($). r r Your friend must earn at least $ The widths of the biccle chains can be.7 inches to. inches.. a..() Weight of supplies (pounds) Weight (pounds) b. A possible solution is that ou weigh pounds and our supplies weigh pounds. (Eamples will var.) Etra Practice for the chatper Solving and Graphing Linear Inequalities. > > > Algebra Worked-Out Solution Ke 77

34 . m < m < m <.. >... >... >.. n n n. n < 8. n < 8 8 n < > z 7 8 > z 7 8 > z k k k. 8. t 8. t. t. r <.7 r <.7 r < m..8.8 m..8. p 7 m. m.. p 7 p. t >.. 8 t < t < p p 8 < p > p > 7. m m m 8. n p () p n n 8 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 78 Algebra Worked-Out Solution Ke

35 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved...z..z... z 8 8. >.s. <.s. < s..8z..8z.8..8 z..d <.8..d. <.8. d <. >. p. >.() 8 >...8t <.8t.8 >.8 t >.. 8. z < z < 8 z < 7. 8(t ) > 8 8t > 8 8t > t > n n 7 7 n 7 n n. 8(m ) < ( m) 8m < 8m < All real numbers are solutions because < is true.. d d d d 8 d > 8 > 8 > 8. ( p) (p ) 8p 8p All real numbers are solutions because is true.. (z ) > (z ) z > z > There are no solutions because > is false.. < 7 < 7 < 8. 7 < < 7 7 < < 7 < < Algebra Worked-Out Solution Ke 7

36 . < z < < z < < z < < z < < z < 7. < (8a ) p < p (8a ) p 7 < 8a 7 < 8a < 8a 8 < 8a 8 8 < a. 8. v > or v < v > or v < v > or v < v > or v < 8. r 7 < or 7r r 7 < or 7r r 7 7 < 7 or 7r r < or 7r r < or 7 7r 7 r < or r. m < 8 or m < m < 8 or m < m > 8 or m < m > or m < m < m <. t t or < t t t or < t t t t t or p > p t t or 8 > t t 8 t. n > or n 8 > n 8 n > or n 8 > n 8 n > or n n 8 > n n 8. 8 n > n 8 > 8 n < n 8 8 > 8 8 n < n > The distance between and is 8. So 8 and 8. The solutions are 8 and 8.. The absolute value of a number is never negative. So, there is no solution.. m m or m m or m The solutions are and.. z z or z z or z z or z The solutions are and. 7. t 7 t 7 or t 7 t 8 or t The solutions are and z z z or z z or z The solutions are and.. s s The absolute value of a number is never negative. So, there is no solution. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 8 Algebra Worked-Out Solution Ke

37 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. r r r or r r or r The solutions are and.. r r r or r r or r The solutions are and.. s s 7 s 7 or s 7 s or s s or s The solutions are and.. 7v 7v 8 The absolute value of a number is never negative. So, there is no solution.. w 8 w w w or w The solutions are and.. w or w w or w The distance between and is less than or equal to. So,.. The distance between and is greater or equal to. So, or. 7. s >. The distance between s and is greater than.. So, s <. or s > q < The distance between q and is less than. So, < q <.. > > or < 8 > or < m < < 8 m < < m < > m > < m <. n n 7 or n 7. p n 8 or n n or n p p p Algebra Worked-Out Solution Ke 8

38 . q 8 q q or q. a q or q q or q a 8 a a a 7. Test (, ) in. ()? ñ Shade the half-plane that does not contain (, ). 7. ( 8) Test (, ) in ( 8) ( 8)? () 8 Shade the half-plane that contains the point (, ). a. c < c < < c < ` 8 < c < 7 7 < c < 7. Test (, ) in ()? Shade the half-plane that contains the point (, ). 7. > 8 Test (, ) in > 8 ò 8 Shade the half-plane that does not contain the point (, ). > 8 7. Test (, ) in.? () ò The side that does not contain (, ) should be shaded. 8. < Test (, ) in <. <? ñ < Shade the half-plane that does not contain (, ).. > Test (, ) in >. () () >? ò Shade the half-plane that does not contain (, ). > 7. ( ) Test (, ) in ( ) ( )? () à Shade the half-plane that does not contain the point (, ) Test (, ) in 8 8? 8 Shade the half-plane that contains the point (, ). 7. Test (, ) in ()? () Shade the half-plane that contains the point (, ). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 8 Algebra Worked-Out Solution Ke

39 77. ( 8) > Test (, ) in ( 8) > ( 8)? ò Shade the half-plane that does not contain the point (, ). > 78. ( ) Test (, ) in ( ) (() )? () à Shade the half-plane that does not contain the point (, ). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke 8