Solving Inequalities: Multiplying or Dividing by a Negative Number

Transcription

1 11 Slving Inequalities: Multiplying r Dividing by a Negative Number We slve inequalities the same way we slve equatins, with ne exceptin. When we divide r multiply bth sides f an inequality by a negative number, the directin f the inequality sign changes. We can see the prf f this in the fllwing examples. 5 is greater than 9, s we must reverse the sign t make the slutin true. 15 < ? > 9 2 < 8 2( 2)? 8( 2) 4 > 16 4 is greater than 16, s we must reverse the sign t make the slutin true. Therefre, when slving an inequality, if yu multiply r divide by a negative number, be sure t reverse the inequality sign. 2x + 13 < x < 14 2x > x > 7 Subtract 13 frm bth sides. Divide bth sides by ( 2) and reverse the inequality sign. Slutin. Remember: If yu must multiply r divide by a negative number, yu must als change the directin f the inequality sign. Slve the inequalities. Fr sme f them, yu will need t reverse the inequality sign. y z 1. a. 7x < 14 b. 8 > 12 c. + 5 >

2 Lessn 11 x 2. a. 9 < 0 b. 2y + 6 < 20 c. 3z + 5 > 4 2 We R e m e m b e r Cmplete the table shwing the number f different kinds f custmers served by Westside Equipment last year. Then cnstruct a rectangle graph f this infrmatin. 3. Kind f Custmer Number Sld New Retail 251 Repeat Retail 399 Repeat Whlesale 307 New Whlesale 43 Ttal Percent Reduce t simplest frm. p 2 q 4 4x 5 4. a. = b. = p 4 q 2 2x 4 5. The 94 miles frm Bill and Gail s hme t Bill s parents hme is 7 times farther than the distance t Gail s parents hme. Hw far d Bill and Gail live frm Gail s parents? Rund t the nearest mile. Simplify the expressins. 6. a. y -3 y -9 b. x 8 x 4 c. 2a 3 7b 2 d. 4z 4 6z -1 44

3 Lessn The middle number in an rdered list is the. 8. The frmula fr finding the vlume f a pyramid r cne is. 9. A number factr in a term used with a variable is knwn as a numerical. 10. a. 1 acre = feet 2 b. 1 mile 2 = acres 11. a. 1 galln liters b. 1 kilgram punds 3 8 M a s t e r y D r i l l 12. a. = % b. 169 = c. 2 4 = d. 5 3 = Write the number f significant digits in each bx. Underline the digit in the answer t shw which digit will be runded. Rund the answers accrding t the fewest number f significant digits. 9, a. 600 b. 5,716, ) 560 Simplify a. (9 + 6)(11 4) b. 40 (10 + 6) Slve. 16. a. 12b = 50 b. n = 14 c. 3 m = 8 d. p(p + 2) 2p = Bill arranged t have equipment hauled in frm an auctin 475 miles away. A hauler frm the auctin huse delivered the equipment fr $1.90 per mile. Hw much did it cst t have the equipment delivered? 45

4 Lessn x S k i l l B u i l d e r s 3 hr 40 min a. 9 b. 5 = c Find the mean, median, and mde. Rund t the nearest whle. 93, 84, 96, 89, 100, 84, 95, 84, 96, 98, a. mean b. median c. mde 20. The heavy-duty diesel il that Westside buys fr the equipment csts $15.99 fr 2 gallns r $34.99 fr 5 gallns. Which is the better buy? 22 Find the vlumes. Use 7 fr π. 3 m 7 ft 3 m 3 ft 3 m 21. a. b. 22. Westside cnsidered making quarterly payments instead f yearly payments fr liability insurance. The insurance cmpany wuld charge an additinal 4% fr the extra wrk that quarterly billings wuld make. Westside presently pays $6,265 per year. What wuld the yearly amunt be if they were t pay quarterly? 46

5 Lessns 11, When Bill and Gail visited his parents they traveled 94 miles in 1H hurs. What was their average speed? Use the d = rt frmula, and rund t the nearest whle. 24. One evening, the Weavers rde their bikes six miles n the rail trail and then back again. They averaged 9 miles per hur and rested 15 minutes by the stream where they turned arund. If they left at 6:15 p.m., what time did they return? Change t standard ntatin. 25. a = b = Cmplete this sentence. 26. Reverse the inequality sign if yu r bth sides by a negative. Slve. 27. a. 6a + 9 < 15 b. b + 12 < 11 3 c. 5c 7> Sine, Csine, and Tangent Applicatins Yu can use a trignmetry chart t find missing infrmatin abut right triangles. When slving stry prblems, sketch a triangle and fill in the given infrmatin t help yu decide which trignmetric rati t use. 47

6 Lessn 12 Example 1: LaRay wants t cut rafters fr a prch rf he is preparing t frame against the side f a huse. The prch is 8 feet wide, and he wants the rf t be 3 feet higher against the huse than at the edge f the prch. Hw many degrees shuld he cut the angle n the end f each rafter s it will fit snugly against the huse? What is the knwn infrmatin abut this triangle? The adjacent side is 3 ft, and the ppsite side is 8 ft. What is the desired infrmatin? Since he knws the lengths f the ppsite side and the adjacent side, LaRay needs t knw the tangent s he can figure ut what angle t cut the rafter ends. ppsite Frmula: tan = adjacent Slve fr the tangent: tan = a h 8 tan After finding the tangent in Example 1 abve, use the trignmetric chart n page 67 t find the clsest degree match fr the tangent. Since is between the tangent f and and is slightly clser t 2.605, the angle shuld be 69. Therefre, he will cut the rafters at 69. Example 2: Bill wants t cut dwn a large pine tree behind the barn at his parents farm. He needs t knw the height f the tree in rder t determine whether he can fell the tree withut hitting the rchard fence 65 feet dwnhill frm the tree. Carrying an astrlabe, Bill walks h dwn the hill until his eyes are level with the base f the tree. At this pint, 50 feet frm the tree, he sights the angle t the tp f the tree t be 55 degrees. Hw tall is the tree? Can astrlabe Bill cut dwn the tree withut taking ut the rchard 55 fence? a 50 Knwn infrmatin: The adjacent side is 50 ft, with a 55 angle. Desired infrmatin: What is the length f the ppsite side? Frmula: tan = ppsite adjacent Frm the chart n page 67, the tangent fr 55 is Substitute and slve ppsite 50 ppsite The tree is abut 71 feet tall ppsite Since the rchard fence is 65 feet away frm the tree and the tree is abut 71 feet tall, Bill will have t mve the fence befre he cuts the tree. 48

7 Lessn 12 Example 3: Delmar used a crane with a 102 bm t set equipment int the basement f a building under cnstructin. The gauge n the crane bm registered a 62 angle as he lwered a piece int place. What is the hrizntal distance between the pivt pint f the bm and the cable lwering the equipment? Knwn infrmatin: The hyptenuse is 102 ft with a 62 angle. 102 h Desired infrmatin: What is the length f the adjacent side? Frmula: cs = Csine fr 62 frm the chart: Substitute and slve. adjacent hyptenuse The cable is abut 48 feet frm the pivt pint. adjacent 102 adjacent adjacent 62 a Remember: If yu knw the ppsite and the adjacent, yu slve fr tangent. If yu knw the ppsite and the hyptenuse, yu slve fr sine. If yu knw the adjacent and the hyptenuse, yu slve fr csine. Use the chart n page 67 t find the angle measurement t the nearest whle degree. 1. a. Knwn infrmatin: The lengths f the and the sides b. Desired infrmatin: The fr the marked angle. c. Set up the frmula. d. Substitute and slve. e. Hw many degrees is the marked angle? h 20 a 18 Use sine, csine, and tangent t find the angle measurement t the nearest whle degree. 2. Bill wants t put slar panels n the huse rf fr heating water. Bill needs t knw hw many degrees are in the lwer angle f the rf in rder t make his panel munting brackets. The rf rises 5 inches fr every 12 hrizntal inches. 49

8 Lessn 12 a. Knwn infrmatin: The lengths f the and the sides. b. Desired infrmatin: The fr the marked angle. c. Set up the frmula. d. Substitute and slve. e. Hw many degrees are in the lwer angle f the rf? 5 a h 12 Use the chart n page 67 t find the missing length t the nearest tenth. 3. a. Knwn infrmatin: The length f the and the number f degrees fr the marked angle. 20 h 48? a b. Desired infrmatin: What is the length f the side? c. The csine fr 48 frm the chart is d. Set up the frmula. e. Substitute and slve. f. What is the length f the adjacent side? We Remember Cmplete the sentence. 4. Reverse the inequality sign if yu r bth sides by a negative. Slve. 5. a. w x >6 b. 6 > c. y 1< 2 d. 4z 5> 7 50

9 Lessn 12 Find the interest cmpunded yearly n $600 savings at 6% interest fr 4 years. 6. Year 1 principal a. interest b. 7. Year 2 principal a. interest b. 8. Year 3 principal a. interest b. 9. Year 4 principal a. interest b. 10. Ending balance a. Ttal interest b. M a s t e r y D r i l l 11. The frmula fr finding the area f a trapezid is. 12. Parts f an expressin separated by psitive r negative signs are called. 13. a. The repeating decimal fr is. b. The decimal fr is. ppsite side hyptenuse 1 6 ppsite side adjacent side 14. a. is the. b. is the. 15. a. 1 galln = fluid unces b. 1 pint = fluid unces 7 8 Simplify and slve. 16. a. 24y + 10( 2) = 3y ( 6 + 8) b. 8(x 6) = 3( 3x 2) c. 2x = 99 Cmplete the table shwing the number f different ways Westside Equipment resld used equipment they had bught. Use this infrmatin t cnstruct a rectangle graph. 17. Hw Used Equipment Was Sld As peratinal equipment Pieces Sld 12 As a unit fr parts 7 As parts nly 16 Ttal Percent 51

10 Lessns 12, 13 Find the vlume f the sphere. Use 3.14 fr pi. Rund t the nearest whle yd 3. V = Jπr 3 6 yd Vlume f an Irregular Slid Yu have learned t find the vlume f rectangular and triangular prisms and cylinders using the frmula V=Bh. Yu have als learned t find the vlume f a pyramid r cne using the frmula V=2Bh, and the vlume f a sphere using V=Jπr 3. Many familiar bjects cmbine tw r mre f these regular shapes int ne irregular slid. Grain bins may cmbine a cylinder and a cne r a rectangular prism and a pyramid. Buildings ften cmbine a rectangular prism fr the lwer part f the building with a triangular prism fr the rf. Other bjects may be a little mre cmplex. A prpane tank may be a cylinder with a hemisphere (half a sphere) n each end. A huse rf with n gable cmbines a triangular prism with half a pyramid at each end. Example 1: The grain bin has a ttal height f 12 feet frm tp t pint. The cylinder part is 5 feet high. The radius f bth the cne and the cylinder is 3 ft.hw many bushels will this grain bin hld? a. Find the vlume f the cne. b. Find the vlume f the cylinder. c. Find the ttal vlume. d. Find the number f bushels. a. V = 2Bh V = 2(πr 2 )h V = 2( )7 V = 2( ) V = V = b. V = Bh V = (πr 2 )h V = ( )5 V = V = c. Upper part + lwer part = ft 3 12 feet d. 1 ft bushels ft = bushels radius 3 feet The grain bin hlds abut (r almst 166) bushels. 5 feet 7 feet