Answers (Lesson 5-1 and Lesson 5-2)

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1 Answers (Lesson -1 and Lesson -) Lesson - - Stud Guide and Intervention Solving Inequalities b Multiplication and Division Solve Inequalities b Multiplication If each side of an inequalit is multiplied b the same positive number, the resulting inequalit is also true. However, if each side of an inequalit is multiplied b the same negative number, the direction of the inequalit must be reversed for the resulting inequalit to be true. For all numbers a, b, and c, with c 0, Multiplication Propert of Inequalities 1. if c is positive and a > b, then ac > bc; if c is positive and a < b, then ac < bc;. if c is negative and a > b, then ac < bc; if c is negative and a < b, then ac > bc. The propert is also true when > and < are replaced with and. Eample 1 Solve - Eample 8 1. Solve k < riginal equation (-8) (- 8) (-8)1 Multipl each side b -8; change to. k < 1 riginal equation ( ) k < ( ) 1 Multipl each side b. -96 Simplif. The solution is { -96}. k < 0 Simplif. The solution is {k k < 0}. Eercises Solve each inequalit. Check our solution n 0 >. h -. - p 6 < -6 { 1} {n n < -1100} {h h -} {p p > 6}. 1 n b < 1 7. m < - < h {n n 0} {b b > - 1 } {m m < - 1 } {h h.0} 9. g > - 9p 11. n a 7-6 {g g -10} {p p > 1} {n n } {a a -1} Define a variable, write an inequalit, and solve each problem. Check our solution Sample answer: Let n = the number Half of a number is at least 1. n 1; {n n 8} 1. The opposite of one-third a number is greater than ne fifth of a number is at most 0. n 0; {n n 10} n > 9; {n n < -7} Chapter 11 Glencoe Algebra 1-1 Enrichment Triangle Inequalities Recall that a line segment can be named b the letters of its endpoints. Line segment AB (written as AB ) has points A and B for endpoints. The length of AB is written without the bar as AB. AB > BC m A < m B The statement on the left above shows that AB is shorter than BC. The statement on the right above shows that the measure of angle A is less than that of angle B. These three inequalities are true for an triangle ABC, no matter how long the sides. a. AB + BC > AC b. If AB > AC, then m C > m B. c. If m C > m B, then AB > AC. B A C Use the three triangle inequalities for these problems. 1. List the sides of triangle DEF in order of increasing length. D DF, DE, EF 8. In the figure at the right, which line segment is the shortest? LM 60 F E K 6 60 L. Eplain wh the lengths cm, 10 cm, and 0 cm could not be used to make a triangle is not greater than 0. J M. Two sides of a triangle measure in. and 7 in. Between which two values must the third side be? in. and 10 in.. In triangle XYZ, XY = 1, YZ = 1, and XZ = 9. Which is the greatest angle? Which is the least? Z; Y 6. List the angles A, C, ABC, and ABD, in order of increasing size. C ABD, A, ABC, C 1 B D A Chapter 10 Glencoe Algebra 1 Chapter A Glencoe Algebra 1

2 Answers (Lesson -) Lesson - - Stud Guide and Intervention (continued) Solving Inequalities b Multiplication and Division Solve Inequalities b Division If each side of a true inequalit is divided b the same positive number, the resulting inequalit is also true. However, if each side of an inequalit is divided b the same negative number, the direction of the inequalit smbol must be reversed for the resulting inequalit to be true. Division Propert of Inequalities For all numbers a, b, and c with c 0, 1. if c is positive and a > b, then a b > b. if c is negative and a > b, then a c ; if c is positive and a < b, then a b < b c < b c ; if c is negative and a < b, then a c ; c > b c. The propert is also true when > and < are replaced with and. Eample Solve riginal inequalit Divide each side b -1 and change to Simplif. The solution is { -}. Eercises Solve each inequalit. Check our solution. 1. g c >. -8m < -6 {g g -} { - 1 } {c c < - } {m m > 8}. -6k < < -b 7. 0 < -n < 0.6w {k k > - 0} 1 {b b < -6} {n n < -10} {w w > -0.} 9. -m > -p 11. -n < 0.0h {m m -1 } 1 {p p > 6} {n n -.1} {h h > 700} > 10h 1. - n < p 16. > - {h h < -} {n n -9} {p p > -1} { > -8} Define a variable, write an inequalit, and solve each problem. Then check our solution Sample answer: Let n = the number. 17. Four times a number is no more than 108. n 108; {n n 7} 18. The opposite of three times a number is greater than 1. -n > 1; {n n < -} 19. Negative five times a number is at most n 100; {n n -0} Chapter 1 Glencoe Algebra 1 Answers - Skills Practice Solving Inequalities b Multiplication and Division Match each inequalit with its corresponding statement. 1. n < 9 d a. Three times a number is at most nine.. 1 n 9 f b. ne third of a number is no more than nine.. n 9 a c. Negative three times a number is more than nine.. -n > 9 c d. Three times a number is less than nine.. 1 n 9 b e. Negative three times a number is at least nine. 6. -n 9 e f. ne third of a number is greater than or equal to nine. Solve each inequalit. Check our solution. 7. 1g > w b r < 16 {g g > } {w w 7} {b b -6} {r r > -} p < 9 1. a p 7 > -9 {p p 6} { < 6} {a a -1} {p p < 6} 1. - t z < m > k k -17 {t t -7} {z z < -18} {m m < } {k k -8} < c h > d 1 { > -6} {c c 1} {h h -0} {d d < 1} Define a variable, write an inequalit, and solve each problem. Check our solution. 7. Sample answer: Let n = the number.. Four times a number is greater than -8. n > -8; {n n > -1}. ne eighth of a number is less than or equal to. 1 n ; {n n } 8. Negative twelve times a number is no more than 8. -1n 8; {n n -7} 6. Negative one sith of a number is less than n < -9; {n n > } 6 7. Eight times a number is at least 16. 8n 16; {n n } Chapter 1 Glencoe Algebra 1 Chapter A Glencoe Algebra 1

3 Answers (Lesson -) Lesson - - Practice Solving Inequalities b Multiplication and Division Match each inequalit with its corresponding statement. 1. -n d a. Negative four times a number is less than five.. n > f b. Four fifths of a number is no more than five.. n e c. Four times a number is fewer than five.. n b d. Negative four times a number is no less than five.. n < c e. Four times a number is at most five. 6. -n < a f. Four fifths of a number is more than five. Solve each inequalit. Check our solution a < h 9. b > 1p {a a > 70} {h h -} {b b -96} {p p < } 11. n > t < 1. - m k -10 {n n > -18} {t t > -} {m m 10} {k k -} 1. -b c > < j {b b -0.} {c c < 10} { -0} {j j > -9.} < 0..6v > -0.u. 7 8 f -1 { > - } 1 {v v -8} {u u > 0} {f f - 8 7} Define a variable, write an inequalit, and solve each problem. Check our solution.. Sample answer: Let n = the number.. Negative three times a number is at least 7. -n 7; {n n -19}. Two thirds of a number is no more than Negative three fifths of a number is less than n -10; {n n -1} n < -6; {n n > 10} 6. FLDING A river is rising at a rate of inches per hour. If the river rises more than feet, it will eceed flood stage. How long can the river rise at this rate without eceeding flood stage? no more than 8 h 7. SALES Pet Supplies makes a profit of $.0 per bag on its line of natural dog food. If the store wants to make a profit of no less than $ on natural dog food, how man bags of dog food does it need to sell? at least 90 bags Chapter 1 Glencoe Algebra 1 - Word Problem Practice Solving Inequalities b Multiplication and Division 1. PIZZA Tara and friends order a pizza. Tara eats of the 10 slices and pas $.0 for her share. Assuming that Tara has paid at least her fair share, what is the most the pizza cost? $1. EVENT PLANNING The Downtown Communit Center does not charge a rental fee as long as a rentee orders a minimum of $000 worth of food from the center. Antonio is planning a banquet for the Quarterback Club. If he is epecting people to attend, what is the minimum he will have to spend on food per person without paing a rental fee? $.. AIRLINES n average, at least,000 pieces of luggage are lost or misdirected each da b United States airlines. f these, 98% are located b the airlines within das. From a given da s lost luggage, at least how man pieces of luggage are still lost after das? at least 00 pieces. PHYSICS The densit of a substance determines whether it will float or sink in a liquid. The densit of water is 1 gram per milliliter. An object with a greater densit will sink and an object with a lesser densit will float. Densit is given b the formula d = m v, where m is mass and v is volume. Here is a table of common chemical solutions and their densities. Solution Densit (g/ml) concentrated calcium chloride % isopropl alcohol 0.9 Source: American Chemistr Council. SCHL Gil earned these scores on the first three tests in biolog this term: 86, 88, and 78. What is the lowest score that Gil can earn on the fourth and final test of the term if he wants to have an average of at least 8? 80 a. Plastics var in densit when the are manufactured; therefore, their volumes are variable for a given mass. A tablet of polstrene (a manufactured plastic) sinks in water and alcohol solution and floats in calcium chloride solution. The tablet has a mass of 0. grams. What is the most its volume can be? v < 0. b. What is the least its volume can be? v > 0.86 Chapter 1 Glencoe Algebra 1 Chapter A6 Glencoe Algebra 1

4 Answers (Lesson - and Lesson -) Lesson - - Enrichment Quadratic Inequalities Like linear inequalities, inequalities with higher degrees can also be solved. Quadratic Inequalities have a degree of. The following eample shows how to solve quadratic inequalities. Eample Solve ( + )( - ) > 0. Step 1 Determine what values of will make the left side 0. In other words, what values of will make either + = 0 or - = 0? = - or Step Plot these points on a number line. Above the number line, place a + if + is positive for that region or a - if + is negative for that region. Net, above the signs ou have just entered; do the same for. Step Below the chart, enter the product of the two signs. Your sign chart should look like the following: ( - )( + ) The final positive regions correspond to values for which the quadratic epression is greater than 0. So, the answer is Eercises < - or >. Solve each inequalit. 1. ( - 1)( + ) > 0. ( + )( + ) > 0 < - or > 1 < - or > -. ( - 1)( - ) < 0. ( + )( - ) 0 1 < < -. ( )( + ) 0 6. ( + )( - ) 0 - or - Chapter 16 Glencoe Algebra 1 Answers - Stud Guide and Intervention Solving Multi-Step Inequalities Solve Multi-Step Inequalities To solve linear inequalities involving more than one operation, undo the operations in reverse of the order of operations, just as ou would solve an equation with more than one operation. Eample 1 Solve Eample riginal inequalit Subtract from each side. - 1 Simplif Add to each side. 16 Simplif. 16 Simplif. The solution is { }. Divide each side b. Solve a - 1 > + a. a - 1 > + a riginal inequalit a a > + a - a Subtract a from each side. -a - 1 > Simplif. -a > + 1 Add 1 to each side. -a > 19 Simplif. -a - < 19 Divide each side b - - and change > to <. a < -9 1 The solution is {a a < } Simplif. Eercises Solve each inequalit. Check our solution n - 10 < 6 - n. q > - { -1 11} {n n < 1 {q q > -} }. 6n + 1 < 8 + 8n d > -1 + d 6. r - 6 > 8r - 18 {n n > } {d d < 0} {r r < } > m - < 18 - m { -6} { > 6} {m m > -} > n - n + > { < -1 } 1 {n n < } { - 1 } Define a variable, write an inequalit, and solve each problem. Check our solution Sample answer: Let n = the number. 1. Negative three times a number plus four is no more than the number minus eight. -n + n - 8; {n n } 1. ne fourth of a number decreased b three is at least two. 1 n - ; {n n 0} 1. The sum of twelve and a number is no greater than the sum of twice the number and n n + (-8); {n n 0} Chapter 17 Glencoe Algebra 1 Chapter A7 Glencoe Algebra 1

5 Answers (Lesson -) Lesson - - Stud Guide and Intervention (continued) Solving Multi-Step Inequalities Solve Inequalities Involving the Distributive Propert When solving inequalities that contain grouping smbols, first use the Distributive Propert to remove the grouping smbols. Then undo the operations in reverse of the order of operations, just as ou would solve an equation with more than one operation. Eample Solve a - (6a - ) > - (a + 6). a - (6a - ) > - (a + 6) riginal inequalit a - 1a + 8 > - a - 6 Distributive Propert -9a + 8 > - - a Combine like terms. -9a a > - - a + a Add a to each side. -a + 8 > - Combine like terms. -a > Subtract 8 from each side. -a > -10 Simplif. a < Divide each side b - and change > to <. The solution in set-builder notation is {a a < }. Eercises Solve each inequalit. Check our solution. 1. (t + ) 16. (d - ) - d > 16. h - 8 < (h - 1) {t t } {d d > } {h h < } > 8 - ( + 1)..6( -.) > ( + ) 1 { > - 7} { < -1.8} { - 7} 7. ( - ) - ( + 1) > (b + 1) < 1 - b 9. -(k - 1) > 8(1+ k) { > 6} {b b < 6} {k k < - } ( - ) > 0.(1 + ) 11. m (m - 1) { -10} {m m - } 1. n + 8 (n - ) - (1 - n) 1. ( - ) > - + {n n 18} 1. k (17 - k) 1. n - - ( + n) {k k is a real number} {n n - 1 } Define a variable, write an inequalit, and solve each problem. Check our solution Sample answer: Let n = the number. 16. Twice the sum of a number and is less than 1. (n + ) < 1; {n n < } 17. Three times the sum of a number and si is greater than four times the number decreased b two. (n + 6) > n - ; {n n < 0} 18. Twice the difference of a number and four is less than the sum of the number and five. (n - ) < n + ; {n n < 1} Chapter 18 Glencoe Algebra 1 - Skills Practice Solving Multi-Step Inequalities Justif each indicated step. 1. t - -1 t a.? ( t -1 t ) t -16 (-1) b.? a. Add to each side. b. Multipl each side b.. (k + 8) - 7 k a.? k + k b.? k -10 k -10 c.? k - a. Distributive Propert b. Subtract from each side. c. Divide each side b. Solve each inequalit. Check our solution.. -b + > d - 1 {b b < } { } {d d 8} 6. a - < 7. - t + 7 > - 8. j - 10 {a a < 1} {t t < } {j j 0} 9. - f + < p + p k + 1 > -k + {f f > 18} {p p 1} {k k > -} 1. (-m - ) (w + 1) < (w + ) 1. (q - ) {m m } {w w > -} {q q -} Define a variable, write an inequalit, and solve each problem. Check our solution Sample answer: Let n = the number. 1. Four more than the quotient of a number and three is at least nine. n + 9; {n n 1} 16. The sum of a number and fourteen is less than or equal to three times the number. n + 1 n; {n n 7} 17. Negative three times a number increased b seven is less than negative eleven. -n + 7 < -11; {n n > 6} 18. Five times a number decreased b eight is at most ten more than twice the number. n - 8 n + 10; {n n 6} 19. Seven more than five siths of a number is more than negative three. {n n > -1} 6 n + 7 > -; 0. Four times the sum of a number and two increased b three is at least twent-seven. (n + ) + 7; {n n } Chapter 19 Glencoe Algebra 1 Chapter A8 Glencoe Algebra 1

6 Answers (Lesson -) Lesson - - Practice Solving Multi-Step Inequalities Justif each indicated step > 8 > (8) > - 1 a.? 8 - > b.? > -1 > -1 c.? > - a. Multipl each side b 8. b. Subtract from each side. c. Divide each side b.. (h + ) < (h + ) - 1 h + < 6h a.? h + < 6h - h + - 6h < 6h - - 6h b.? -h + < - -h + - < - - c.? -h < -6 -h - > -6 - h > a. Distributive Propert b. Subtract 6h from each side. c. Subtract from each side. d. Divide each side b - and change < to >. d.? Solve each inequalit. Check our solution t 6-9. u - 6 6u > a - 1 {t t } {u u 7} {a a < 1} 6. w + < -8 {w w < -19} 7. f - 10 > 7 {f f > 1} 8. h 6h + {h h -} 9. (z + 1) + 11 < -(z + 1) {z z < -8} 10. r + (r + ) (6r + 1) {r r } 11. n - (n - 6) 0 {n n -9} Define a variable, write an inequalit, and solve each problem. Check our solution Sample answer: Let n = the number. 1. A number is less than one fourth the sum of three times the number and four. n, n + ; {n n < } 1. Two times the sum of a number and four is no more than three times the sum of the number and seven decreased b four. (n + ) (n + 7) - ; {n n -9} 1. GEMETRY The area of a triangular garden can be no more than 10 square feet. The base of the triangle is 16 feet. What is the height of the triangle? no more than 1 ft 1. MUSIC PRACTICE Nabuko practices the violin at least 1 hours per week. She practices for three fourths of an hour each session. If Nabuko has alread practiced hours in one week, how man sessions remain to meet or eceed her weekl practice goal? at least 1 sessions Chapter 0 Glencoe Algebra 1 Answers - Word Problem Practice Solving Multi-Step Inequalities 1. BEACHCMBING Ja has lost his mother s favorite necklace, so he will rent a metal detector to tr to find it. A rental compan charges a one-time rental fee of $1 plus $ per hour to rent a metal detector. Ja has onl $ to spend. What is the maimum amount of time he can rent the metal detector? no more than 10 hours. PLAYGRUND The perimeter of a rectangular plaground must be no greater than 10 meters, because that is the total length of the materials available for the border. The width of the plaground cannot eceed meters. What are the possible lengths of the plaground? less than or equal to 8 meters. MEDICINE Clark s Rule is a formula used to determine pediatric dosages of over-the-counter medicines. Weight of child ( lb) 10 adult dose = child dose. AGES Bobb, Bill, and Barr Smith are each one ear apart in age. The sum of their ages is greater than the age of their father, who is 60. What is the oungest age that the oldest brother can be? no ounger than a. If an adult dose of acetaminophen is 1000 milligrams and a child weighs no more than 90 pounds, what is the recommended child s dose? 600; no more than 600 mg. TAXI FARE Jamal works in a cit and sometimes takes a tai to work. The taicabs charge $1.0 for the first 1 mile and $0. for each additional 1 mile. Jamal has onl $.7 in his pocket. What is the maimum distance he can travel b cab if he does not intend to tip the cab driver? no more than mi b. This label appears on a child s cold medicine. What is the adult minimum dosage in milliliters? a 1.79 ml Weight (lb) Age (r) Dose under 8 under 6 call a doctor tsp or 10 ml c. What is the maimum adult dosage in milliliters? a 1. ml Chapter 1 Glencoe Algebra 1 Chapter A9 Glencoe Algebra 1

7 Answers (Lesson - and Lesson -) Lesson - - Enrichment Carlos Montezuma During his lifetime, Carlos Montezuma (186? 19) was one of the most influential Native Americans in the United States. He was recognized as a prominent phsician and was also a passionate advocate of the rights of Native American peoples. The eercises that follow will help ou learn some interesting facts about Dr. Montezuma s life. Solve each inequalit. The word or phrase net to the equivalent inequalit will complete the statement correctl. 1. -k > 10. r - 9 Montezuma was born in the state He was a Native American of the of?. Yavapais, who are a? people. a. k < - Arizona a. r - Navajo b. k > - Montana b. r - Mohawk c. k > 1 Utah c. r 1 Mohave-Apache q > 1 Montezuma received a medical As a phsician, Montezuma s field of degree from? in specialization was?. a. 9 Chicago Medical College a. q > - heart surger b. -9 Harvard Medical School b. q > 1 internal medicine c. 9 Johns Hopkins Universit c. q < -1 respirator diseases t < 7 + t For much of his career, he maintained In addition to maintaining his medical a medical practice in?. practice, he was also a(n)?. a. 9 New York Cit a. t > 7 director of a blood bank b. Chicago b. t > 0 instructor at a medical college c. -9 Boston c. t < -7 legal counsel to phsicians 7. a + 8 a n > 8n - 1 Montezuma founded, wrote, and Montezuma testified before a edited?, a monthl newsletter committee of the United States that addressed Native American Congress concerning his work in concerns. treating?. a. a - Yavapai a. n < 6 appendicitis b. a 18 Apache b. n > -6 asthma c. a 18 Wassaja c. n > -10 heart attacks Chapter Glencoe Algebra 1 - Stud Guide and Intervention Solving Compound Inequalities Inequalities Containing and A compound inequalit containing and is true onl if both inequalities are true. The graph of a compound inequalit containing and is the intersection of the graphs of the two inequalities. Ever solution of the compound inequalit must be a solution of both inequalities. Eample 1 Eample Graph the solution set of < and Graph <. Graph -1. Find the intersection. The solution set is { -1 < }. Solve -1 < + <. Then graph the solution set. -1 < + and + < -1 - < < - - < < Graph > -. Graph < Find the intersection. The solution set is { - < < 1}. Eercises Graph the solution set of each compound inequalit. 1. b > -1 and b. q -. > - and p <. - < d and d< p Solve each compound inequalit. Then graph the solution set. 7. < w p - < {w 1 < w } {p p < 7} < < and + 1 { -6 < -} { -1 < } n - > - and n + < 6 1. d - < 6d + 1 < d + {n -1 < n < } {d - < d < } Chapter Glencoe Algebra 1 Chapter A10 Glencoe Algebra 1

8 Answers (Lesson -) Lesson - - Stud Guide and Intervention (continued) Solving Compound Inequalities Inequalities Containing or A compound inequalit containing or is true if one or both of the inequalities are true. The graph of a compound inequalit containing or is the union of the graphs of the two inequalities. The union can be found b graphing both inequalities on the same number line. A solution of the compound inequalit is a solution of either inequalit, not necessaril both. Eample Solve a + 1 < 11 or a > a +. Then graph the solution set. a + 1 < 11 or a > a + a < 11-1 a - a > a - a + a < 10 -a > a 10 -a - < - < - a < a < -1 Graph a < Graph a < Find the union The solution set is {a a < }. Eercises Graph the solution set of each compound inequalit. 1. b > or b -. q or q 1. - or > p or p < 8. - < d or d < 6. - or Solve each compound inequalit. Then graph the solution set. 7. < w or w p or p < {w 1 < w} {p p or p < } or < 1 or - { } { < } n > - or n - < 6 + n 1. a + or 7 + a < a + 6 {n n is a real number} {a a < -1 or a 1} Chapter Glencoe Algebra 1 Answers - Skills Practice Solving Compound Inequalities Graph the solution set of each compound inequalit. 1. b > or b 0. z and z k > 1 and k >. < -1 or Write a compound inequalit for each graph.. - < < - or 1 < -1 or > Solve each compound inequalit. Then graph the solution set. 9. m + and m + < < - or - 1 {m m < } { < 1 or 6} < f + 6 and f + 6 < 1. w + 0 or w {f - < f < -1} {w w - or w } < b - < 1. p - - or p - > 1 {b - < b < 6} {p p 0 or p > } Define a variable, write an inequalit, and solve each problem. Check our solution Sample answer: Let n = the number. 1. A number plus one is greater than negative five and less than three. - < n + 1 < ; {n -6 < n < } 16. A number decreased b two is at most four or at least nine. n - or n - 9; {n n 6 or n 11} 17. The sum of a number and three is no more than eight or is more than twelve. n + 8 or n + > 1; {n n or n > 9} Chapter Glencoe Algebra 1 Chapter A11 Glencoe Algebra 1

9 Answers (Lesson -) Lesson - - Practice Solving Compound Inequalities Graph the solution set of each compound inequalit n 1. > 0 or < g < - or g. - p Write a compound inequalit for each graph or < or < - < < 0 Solve each compound inequalit. Then graph the solution set. 9. k - < -7 or k n < or n - > {k k < - or k } {n n > -} 11. < h c - > -6 and c + 1 < 1 {h 1 < h } {c -1 < c < } Define a variable, write an inequalit, and solve each problem. Check our solution Sample answer: Let n = the number. 1. Two times a number plus one is greater than five and less than seven. < n + 1 < 7; {n < n < } 1. A number minus one is at most nine, or two times the number is at least twent-four. n or n ; {n n 10 or n 1} 1. METERLGY Strong winds called the prevailing westerlies blow from west to east in a belt from 0 to 60 latitude in both the Northern and Southern Hemispheres. a. Write an inequalit to represent the latitude of the prevailing westerlies. {w 0 w 60} b. Write an inequalit to represent the latitudes where the prevailing westerlies are not located. {w w < 0 or w > 60} 16. NUTRITIN A cookie contains 9 grams of fat. If ou eat no fewer than and no more than 7 cookies, how man grams of fat will ou consume? between 6 g and 6 g inclusive Chapter 6 Glencoe Algebra 1 - Word Problem Practice Solving Compound Inequalities 1. WEATHER Ken saw this graph in the newspaper weather forecast. It shows the predicted temperature range for the following da. Write an inequalit to represent the number line graph. 0 F HEALTH The human heart circulates from 770,000 to 1,600,000 gallons of blood through a person s bod ever ear. How man gallons of blood does the heart circulate through the bod in one da? between 110 and 8 gal 68. PLS The ph of a person s ees is 7.. Therefore, the ideal ph for the water in a swimming pool is between 7.0 and 7.6. Write an inequalit to represent ph levels that could cause phsical discomfort to a person s ees. 7.0 or 7.6. STRE SIGNS In Rand s town, street-side signs themselves must be eactl 8 feet high. When mounted on poles, the signs must be shorter than 0 feet or taller than feet so that the do not interfere with the power and phone lines. Write a compound inequalit to represent the possible height of the poles.. HEALTH Bod mass inde (BMI) is a measure of weight status. The BMI of a person over 0 ears old is calculated using the following formula. weight in pounds BMI = 70 (height in inches) The table below shows the meaning of different BMI measures. BMI Weight Status less than 18. underweight normal 9.9 overweight more than 0 obese Source: Centers for Disease Control a. Write a compound inequalit to represent the normal BMI range. 18. < b. Write a compound inequalit to represent an adult weight that is within the health BMI range for a person 6 feet tall. 16. w 18.6 < 1 or > 7 Chapter 7 Glencoe Algebra 1 Chapter A1 Glencoe Algebra 1

10 Answers (Lesson -) Lesson - - Enrichment Some Properties of Inequalities The two epressions on either side of an inequalit smbol are sometimes called the first and second members of the inequalit. If the inequalit smbols of two inequalities point in the same direction, the inequalities have the same sense. For eample, a < b and c < d have the same sense; a < b and c > d have opposite senses. In the problems on this page, ou will eplore some properties of inequalities. Three of the four statements below are true for all numbers a and b (or a, b, c, and d). Write each statement in algebraic form. If the statement is true for all numbers, prove it. If it is not true, give an eample to show that it is false. 1. Given an inequalit, a new and equivalent inequalit can be created b interchanging the members and reversing the sense. If a > b, then b < a. a > b, a - b > 0, -b > -a, (-1)(-b) < (-1)(-a), b < a. Given an inequalit, a new and equivalent inequalit can be created b changing the signs of both terms and reversing the sense. If a > b, then -a < -b. a > b, a - b > 0, -b > -a, -a < -b. Given two inequalities with the same sense, the sum of the corresponding members are members of an equivalent inequalit with the same sense. If a > b and c > d, then a + c > b + d. a > b and c > d, so (a - b) and (c - d) are positive numbers, so the sum (a - b) + (c - d) is also positive. a - b + - d > 0, so a + c > b + d.. Given two inequalities with the same sense, the difference of the corresponding members are members of an equivalent inequalit with the same sense. If a > b and c > d, then a - c > b - d. The statement is false. > and >, but - -. Chapter 8 Glencoe Algebra 1 Answers - Stud Guide and Intervention Inequalities Involving Absolute Value Inequalities Involving Absolute Value (<) When solving inequalities that involve absolute value, there are two cases to consider for inequalities involving < (or ). Remember that inequalities with and are related to intersections. If < n, then > -n and < n. Eample Solve a + < 10. Then graph the solution set. Write a + < 10 as a + < 10 and a + > -10. a + < 10 and a + > -10 a + - < 10 - a + - > a < 6 a > -1 a < 6 a > -1 Now graph the solution set a < a > - The solution set is {a - < a < }. Eercises Solve each inequalit. Then graph the solution set. 1. <. - <. + { - < < } { 0 < < 8} { - -1} b +. w - 6. t + {b - b 1} {w - w 7} {t -6 t } p - 0. < 0. { - } { -1 1 } {p -0. < p < 0.7} Chapter 9 Glencoe Algebra 1 Chapter A1 Glencoe Algebra 1

11 Answers (Lesson -) Lesson - - Stud Guide and Intervention (continued) Inequalities Involving Absolute Value Solve Absolute Value Inequalities (>) When solving inequalities that involve absolute value, there are two cases to consider for inequalities involving > (or ). Remember that inequalities with or are related to unions. Eample Solve b + 9 >. Then graph the solution set. Write b + 9 > as b + 9 > or b + 9 < -. b + 9 > or b + 9 < - b > - 9 b < - -9 b > - b > -1 b - b -1 > > b > - b < -7 Now graph the solution set The solution set is {b b > -, or b < -7}. Eercises Solve each inequalit. Then graph the solution set. 1. c - > 6. - > 0. f + 10 {c c < - or c > 8} { > or < } { f f - or f - } { - or } { - or } { is a real number} 7. d ( - 1) 8 9. r + > - {d d -1 1 or d } 1 { - or 1} {r r is a real number} Chapter 0 Glencoe Algebra 1 - Skills Practice Inequalities Involving Absolute Value Match each open sentence with the graph of its solution set. 1. > a b < c. 6 Epress each statement using an inequalit involving absolute value.. The weatherman predicted that the temperature would be within of F.. Serena will make the B team if she scores within 8 points of the team average of The dance committee epects attendance to number within of last ear s 87 students. Solve each inequalit. Then graph the solution set < 0 8. c - < n t + 6 > w - < 1. k Chapter 1 Glencoe Algebra 1 Chapter A1 Glencoe Algebra 1

12 Answers (Lesson -) Lesson - - Practice Inequalities Involving Absolute Value Match each open sentence with the graph of its solution set a a < c b b c. - Epress each statement using an inequalit involving absolute value.. The height of the plant must be within inches of the standard 1-inch show size. h - 1. The majorit of grades in Sean s English class are within points of 8. g - 8 Solve each inequalit. Then graph the solution set. 6. z {z z } 7. - r > 7 {r r < - or r > } t + 6 < 9 {t - < t < 1} 9. g - 9 {g g - or g 7} Write an open sentence involving absolute value for each graph < + > RESTAURANTS The menu at Jeanne s favorite restaurant states that the roasted chicken with vegetables entree tpicall contains 80 Calories. Based on the size of the chicken, the actual number of Calories in the entree can var b as man as 0 Calories from this amount. a. Write an absolute value inequalit to represent the situation b. What is the range of the number of Calories in the chicken entree? 0 0 Chapter Glencoe Algebra 1 Answers - Word Problem Practice Solving pen Sentences Involving Absolute Value 1. SPEEDMETERS The government requires speedometers on cars sold in the United States to be accurate within ±.% of the actual speed of the car. If our speedometer reads 60 miles per hour while ou are driving on a highwa, what is the range of possible actual speeds our car could be traveling at? { }. CATS During a recent visit to the veterinarian s office, the Mrs. Van Allen was informed that a health weight for her cat is approimatel 10 pounds, plus or minus one pound. Write an absolute value equation that represents unhealth weights w for her cat. w BAKING Pete is making muffins for a bake sale. Before he starts baking, he goes online to research different muffin recipes. The recipes that he finds all specif baking temperatures between 0 F and 00 F. Write an absolute value epression to represent the possible temperatures t called for in the muffin recipes Pete is researching. t - 7. ARCHERY In an lmpic archer event, the center of the target is set eactl 10 centimeters off the ground. To get the highest score of ten points, an archer must shoot an arrow no further than.0 centimeters from the eact center of the target. a. Write an absolute value epression to represent the possible distances d from the ground an archer can hit the target and still score ten points. d STATISTICS The most familiar statistical measure is the arithmetic mean, or average. A second important statistical measure is the standard deviation, which is a measure of how far the individual scores deviate from the mean. For eample, in 007 the mean score on the mathematics section of the SAT test was 1 and the standard deviation was 11. This means that people within one deviation of the mean have SAT math scores that are 11 points higher or 11 points lower than the mean. a. Write an absolute value inequalit to find the range of 007 SAT mathematics test scores within one standard deviation of the mean b. What is the range of SAT mathematics test scores ± standard deviation from the mean? 87 to 7 b. Graph the solution set of the epression ou wrote in part a Chapter Glencoe Algebra 1 Chapter A1 Glencoe Algebra 1

13 Answers (Lesson -) Lesson - - Enrichment Precision of Measurement The precision of a measurement depends both on our accurac in measuring and the number of divisions on the ruler ou use. Suppose ou measured a length of wood to the nearest one-eighth of an inch and got a length of 6 8 in The drawing shows that the actual measurement lies somewhere between in. and 6 16 in. This measurement can be written using the smbol ±, which is read plus or minus. It can also be written as a compound inequalit. 6 8 ± 1 16 in in. m 6 16 in. In this eample, 1 16 in. is the absolute error. The absolute error is one-half the smallest unit used in a measurement. Write each measurement as a compound inequalit. Use the variable m ± 1 in...78 ± 0.00 kg ± 0.00 g 1 in. m in..77 kg m 7.10 g m 7.11 g.78 kg. 16 ± 1 d. ± 0. cm ± 1 in d m d 1. cm m. cm in. m 19 in. For each measurement, give the smallest unit used and the absolute error in. m 10. in in. m in. 1 in., 0. in. 1 in., 1 in cm. m 1 1 cm, 1 cm mm m 7.1 mm cm 0.1 mm, 0.0 mm Chapter Glencoe Algebra 1 - Graphing Calculator Activit Absolute Value Inequalities The TEST menu can be used to solve and graph absolute value inequalities b using the equivalent compound inequalities related to absolute value. Eample Graph and solve each inequalit. a. + 8 Enter the inequalit into Y1. Then enter the equivalent compound inequalit into Y and graph to view the results. Be sure to choose appropriate settings for the view window. Kestrokes: MATH ENTER + ) nd [TEST] 8 ENTER + nd [TEST] 6 ( ) 8 nd [TEST] + nd [TEST] 8 ENTER GRAPH. [-18, ] scl: b [-.1,.1] scl:1 Use TRACE to confirm the solution. When = 1 the statement is true, and when = 0 the statement is false. Thus, the solution is -1 or. b. + 7 Enter the inequalit into Y1 and the equivalent compound inequalities into Y. Then graph the solution set. Kestrokes: MATH ENTER ( + ) ) [-18, ] scl: b [-.1,.1] scl:1 nd [TEST] 6 7 ENTER ( + ) nd [TEST] ( ) 7 nd [TEST] ENTER ( + ) nd [TEST] 6 7 ENTER GRAPH. [-18, ] scl: b [-.1,.1] scl:1 The statement is true between -6 and.. Thus the solution is -6.. Eercise Graph and solve each inequalit. [ 18, ] scl: b [.1,.1] scl: >. + 8 < - Chapter Glencoe Algebra 1 Chapter A16 Glencoe Algebra 1

14 -6 Stud Guide and Intervention Graphing Inequalities in Two Variables Graph Linear Inequalities The solution set of an inequalit that involves two variables is graphed b graphing a related linear equation that forms a boundar of a half-plane. The graph of the ordered pairs that make up the solution set of the inequalit fill a region of the coordinate plane on one side of the half-plane. Eample Graph - -. Graph = - -. Since - - is the same as < - - and = - -, the boundar is included in the solution set and the graph should be drawn as a solid line. Select a point in each half plane and test it. Choose (0, 0) and (-, -) (0) - - -(-) is false is true. The half-plane that contains (-, -) contains the solution. Shade that half-plane. Eercises Graph each inequalit. 1. < > < < Chapter 6 Glencoe Algebra 1 Answers Answers (Lesson -6) Lesson -6-6 Stud Guide and Intervention (continued) Graphing Inequalities in Two Variables Solve Linear Inequalities We can use a coordinate plane to solve inequalities with one variable. Eample Use a graph to solve + > -1. Step 1 First graph the boundar, which is the related function. Replace the inequalit sign with an equals sign, and get 0 on a side b itself. + > -1 riginal inequalit + = -1 Change < to = = Add 1 to each side. + = 0 Simplif. Graph + = as a dashed line. Step Choose (0, 0) as a test point, substituting these values into the original inequalit give us > -. Step Because this statement is true, shade the half plane containing the point (0, 0). Notice that the -intercept of the graph is at Because the half-plane to the right of the -intercept is shaded, the solution is > Eercises Use a graph to solve each inequalit > > < - > > < < -17 < < Chapter 7 Glencoe Algebra 1 Chapter A17 Glencoe Algebra 1

15 Answers (Lesson -6) Lesson -6-6 Skills Practice Graphing Inequalities in Two Variables Match each inequalit to the graph of its solution < a. b > 1 c. d. Graph each inequalit.. < > > Use a graph to solve each inequalit > < Chapter 8 Glencoe Algebra 1-6 Practice Graphing Inequalities in Two Variables Determine which ordered pairs are part of the solution set for each inequalit , {(, ), (-, ), (-, -), (, -)} {(, ), (, -)}. +, {(6, ), (-, ), (, -), (, )} {(-, )}. - <, {(, -), (, ), (, ), (-, )} {(, ), (-, )} Graph each inequalit.. - < > - Use a graph to solve each inequalit > > > 1 1 > MVING A moving van has an interior height of 7 feet (8 inches). You have boes in 1 inch and 1 inch heights, and want to stack them as high as possible to fit. Write an inequalit that represents this situation BUDGETING Satchi found a used bookstore that sells pre-owned videos and CDs. Videos cost $9 each, and CDs cost $7 each. Satchi can spend no more than $. a. Write an inequalit that represents this situation b. Does Satchi have enough mone to bu videos and CDs? No, the purchases will be $9, which is greater than $. Chapter 9 Glencoe Algebra 1 Chapter A18 Glencoe Algebra 1

16 1. FAMILY Trone said that the ages of his siblings are all part of the solution set of >, where is the age of a sibling and is Trone s age. Which of the following ages is possible for Trone and a sibling? Trone is ; Maine is 1. no Trone is 18; Camille is 8. es Trone is 1; Francis is. es Trone is 11; Martin is 6. no Trone is 19; Paul is 9. es. FARMING The average value of U.S. farm cropland has steadil increased in recent ears. In 000, the average value was $190 per acre. Since then, the value has increased at least an average of $77 per acre per ear. Write an inequalit to show land values above the average for farmland. > SHIPPING An international shipping compan has established size limits for packages with all their services. The total of the length of the longest side and the girth (distance completel around the package at its widest point perpendicular to the length) must be less than or equal to 19 centimeters. Write and graph an inequalit that represents this situation.. FUNDRAISING Troop 00 sold cider and donuts to raise mone for charit. The sold small boes of donut holes for $1. and cider for $.0 a gallon. In order to cover their epenses, the needed to raise at least $100. Write and graph an inequalit that represents this situation. 1.d +.c c. INCME In 006 the median earl famil income was about $8,00 per ear. Suppose the average annual rate of change since then is $10 per ear. a. Write and graph an inequalit for the annual famil incomes that are less than the median for ears after 006. < , g b. Which of the following points is part of the solution set? (, 1,000) no (8, 69,00) no (, 0,000) es (10, 61,000) no Girth Length Chapter 0 Glencoe Algebra 1 Answers l Cider (gal) 1.d +.0c Donut holes d 8,000 6,000,000,000 0,000 8,000 6,000,000 Income ($1000) Lesson -6-6 Word Problem Practice Graphing Inequalities in Two Variables length girth l + g Years since Enrichment Linear Programming Linear programming can be used to maimize or minimize costs. It involves graphing a set of linear inequalities and using the region of intersection. You will use linear programming to solve the following problem. Eample Lane s Gift Shoppe sells at most 00 items per week. To meet her customers demands, she sells at least 100 stuffed animals and 7 greeting cards. If the profit for each stuffed animal is $.0 and the profit for each greeting card is $1.00, the equation P(a, g) =.0a g can be used to represent the profit. How man of each should she sell to maimize her profit? Write the inequalities: Graph the inequalities: a + g 00 a 100 g g a Find the vertices of the triangle formed: (100, 7), (100, 00), and (, 7) Substitute the values of the vertices into the equation found above:.0(100) + 1(7) =.0(100) + 1(00) = 60.0() + 1(7) = The maimum profit is $ Eercises The Spirit Club is selling shirts and banners. The sell at most 00 of the two items. To meet the demands of the students, the must sell at least 0 t-shirts and 100 banners. The profit on each shirt is $.00 and the profit on each banner is $1.0, the equation P(t, b) =.00t + 1.0b can be used to represent the profit. How man should the sell of each to maimize the profit? 1. Write the inequalities to represent this situation. t + b 00; t 0; b 100. Graph the inequalities from part 1. See graph at right.. Find the vertices of the figure formed. (0, 100), (0, 0), (00, 100) b What is the maimum profit the Spirit Club can make? $10 Chapter 1 Glencoe Algebra 1 t Answers (Lesson -6) Chapter A19 Glencoe Algebra 1

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