13.1.7: Chapter 7

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Be Prepared

6y6y

425425

154154

37303730

27x−323627x−3236

8x+268x+26

2323

154154

x=4x=4

x=−1x=−1

n=9,n=−4n=9,n=−4

y=10−5x2y=10−5x2

n=−2n=-2

The speed of the bus is 28 mph.

x=107x=107

ⓐ 1; ⓑ −8; ⓒ 0

x>−2x>−2

[−3,5)[−3,5)

Try It

ⓐ x=0x=0 ⓑ n=−13n=−13
ⓒ a=−1,a=−3a=−1,a=−3

ⓐ q=0q=0 ⓑ y=−23y=−23
ⓒ m=2,m=−3m=2,m=−3

x+1x−1,x+1x−1, x≠2,x≠2, x≠1x≠1

x−5x−1,x−5x−1, x≠−2,x≠−2, x≠1x≠1

2(x−3y)3(x+3y)2(x−3y)3(x+3y)

5(x−y)2(x+5y)5(x−y)2(x+5y)

−x+1x+5−x+1x+5

−x+2x+1−x+2x+1

x−22(x+3)x−22(x+3)

3(x−6)x+53(x−6)x+5

x−4x−5x−4x−5

−(b+2)(b−1)(1+b)(b+4)−(b+2)(b−1)(1+b)(b+4)

2(x2+2x+4)(x+2)(x2−2x+4)2(x2+2x+4)(x+2)(x2−2x+4)

2zz−12zz−1

x+24x+24

2y+52y+5

2(m+1)(m+2)3(m+4)(m−3)2(m+1)(m+2)3(m+4)(m−3)

(n+5)(n+9)2(n+6)(2n+3)(n+5)(n+9)2(n+6)(2n+3)

The domain of R(x)R(x) is all real numbers where x≠5x≠5 and x≠−1.x≠−1.

The domain of R(x)R(x) is all real numbers where x≠4x≠4 and x≠−2.x≠−2.

R(x)=2R(x)=2

R(x)=13R(x)=13

R(x)=x−24(x−8)R(x)=x−24(x−8)

R(x)=x(x−2)x−1R(x)=x(x−2)x−1

x+2x+2

x+3x+3

x−11x−2x−11x−2

x−3x+9x−3x+9

y+3y+2y+3y+2

3n−2n−13n−2n−1

ⓐ (x−4)(x+3)(x+4)(x−4)(x+3)(x+4)
ⓑ 2x+8(x−4)(x+3)(x+4)2x+8(x−4)(x+3)(x+4),
x+3(x−4)(x+3)(x+4)x+3(x−4)(x+3)(x+4)

ⓐ (x+2)(x−5)(x+1)(x+2)(x−5)(x+1)
ⓑ 3x2+3x(x+2)(x−5)(x+1)3x2+3x(x+2)(x−5)(x+1),
5x−25(x+2)(x−5)(x+1)5x−25(x+2)(x−5)(x+1)

7x−4(x−2)(x+3)7x−4(x−2)(x+3)

7m+25(m+3)(m+4)7m+25(m+3)(m+4)

5m2−9m+2(m+1)(m−2)(m+2)5m2−9m+2(m+1)(m−2)(m+2)

2n2+12n−30(n+2)(n−5)(n+3)2n2+12n−30(n+2)(n−5)(n+3)

1x−21x−2

−3z−3−3z−3

5x+1(x−6)(x+1)5x+1(x−6)(x+1)

y+3y+4y+3y+4

1(b+1)(b−1)1(b+1)(b−1)

1(x+2)(x+1)1(x+2)(x+1)

v+3v+1v+3v+1

3ww+73ww+7

x−7x−4x−7x−4

x2−3x+18(x+3)(x−3)x2−3x+18(x+3)(x−3)

23(x−1)23(x−1)

12(x−3)12(x−3)

14111411

10231023

y+xy−xy+xy−x

abb−aabb−a

b(b+2)(b−5)3b−5b(b+2)(b−5)3b−5

3c+33c+3

7373

103103

b+aa2+b2b+aa2+b2

y−xxyy−xxy

3(x−2)5x+73(x−2)5x+7

x+216x−43x+216x−43

35x+2235x+22

2(2y2+13y+5)3y2(2y2+13y+5)3y

xx+4xx+4

x(x+1)3(x−1)x(x+1)3(x−1)

y=−157y=−157

x=1513x=1513

x=−3,x=5x=−3,x=5

y=−2,y=6y=−2,y=6

x=23x=23

y=2y=2

There is no solution.

x=3x=3

y=7y=7

There is no solution.

ⓐ The domain is all real numbers except x≠3x≠3 and x≠4.x≠4. ⓑ x=2,x=143x=2,x=143
ⓒ (2,3),(143,3)(2,3),(143,3)

ⓐ The domain is all real numbers except x≠1x≠1 and x≠5.x≠5. ⓑ x=214x=214 ⓒ (214,4)(214,4)

y=mx−4m+5y=mx−4m+5

y=mx+5m+1y=mx+5m+1

a=bcb−1a=bcb−1

y=3xx+6y=3xx+6

y=33y=33

z=14z=14

The pediatrician will prescribe 12 ml of acetaminophen to Emilia.

The pediatrician will prescribe 180 mg of fever reducer to Isabella.

The distance is 150 miles.

The distance is 350 miles.

The telephone pole is 40 feet tall.

The pine tree is 60 feet tall.

Link’s biking speed is 15 mph.

The speed of Danica’s boat is 17 mph.

Dennis’s uphill speed was 5 mph and his downhill speed was 10 mph.

Joon’s rate on the country roads was 50 mph.

Kayla’s biking speed was 15 mph.

Victoria jogged 6 mph on the flat trail.

When the two gardeners work together it takes 2 hours and 24 minutes.

When Daria and her mother work together it takes 2 hours and 6 minutes.

Kristina can paint the room in 12 hours.

It will take Jordan 6 hours.

ⓐ c=4.8tc=4.8t ⓑ He would burn 432 calories.

ⓐ d=50td=50t ⓑ It would travel 250 miles.

ⓐ h=130th=130t ⓑ 123123 hours

ⓐ x=3500px=3500p ⓑ 500 units

(−∞,−4)∪[2,∞)(−∞,−4)∪[2,∞)

(−∞,−2]∪(4,∞)(−∞,−2]∪(4,∞)

(−32,3)(−32,3)

(−8,4)(−8,4)

(−∞,−4)∪(2,∞)(−∞,−4)∪(2,∞)

(−∞,−4)∪(3,∞)(−∞,−4)∪(3,∞)

(2,4)(2,4)

(3,6)(3,6)

(−4,2](−4,2]

[−1,4)[−1,4)

ⓐ c(x)=20x+6000xc(x)=20x+6000x
ⓑ More than 150 items must be produced to keep the average cost below $60 per item.

ⓐ c(x)=5x+900xc(x)=5x+900x ⓑ More than 60 items must be produced to keep the average cost below $20 per item.

Section 7.1 Exercises

ⓐ z=0z=0 ⓑ p=56p=56
ⓒ n=−4,n=2n=−4,n=2

ⓐ y=0y=0, ⓑ x=−12x=−12, ⓒ u=−4,u=7u=−4,u=7

−45−45

2m23n2m23n

83(n≠2)83(n≠2)

11.

x+5x−1x+5x−1

13.

a+2a+8a+2a+8

15.

p2+4p−2p2+4p−2

17.

4b(b−4)(b+5)(b−8)4b(b−4)(b+5)(b−8)

19.

3(m+5n)4(m−5n)3(m+5n)4(m−5n)

21.

−1−1

23.

−5y+4−5y+4

25.

w2−6w+36w−6w2−6w+36w−6

27.

−z−54+z−z−54+z

29.

310310

31.

x38yx38y

33.

p(p−4)2(p−9)p(p−4)2(p−9)

35.

y−53(y+5)y−53(y+5)

37.

−4(b+9)3(b+7)−4(b+9)3(b+7)

39.

(3c−1)(c+5)(3c+1)(c−5)(3c−1)(c+5)(3c+1)(c−5)

41.

−(m−2)(m−3)(3+m)(m+4)−(m−2)(m−3)(3+m)(m+4)

43.

−1v+5−1v+5

45.

3ss+43ss+4

47.

4(p2−pq+q2)(p−q)(p2+pq+q2)4(p2−pq+q2)(p−q)(p2+pq+q2)

49.

x−28x(x+5)x−28x(x+5)

51.

2a−752a−75

53.

3(3c−5)3(3c−5)

55.

4(m+8)(m+7)3(m−4)(m+2)4(m+8)(m+7)3(m−4)(m+2)

57.

(4p+1)(p−4)3p(p+9)(p−1)(4p+1)(p−4)3p(p+9)(p−1)

59.

x≠5x≠5 and x≠−5x≠−5

61.

x≠2x≠2 and x≠−3x≠−3

63.

R(x)=2R(x)=2

65.

R(x)=x+52x(x+2)R(x)=x+52x(x+2)

67.

R(x)=3x(x+7)x−7R(x)=3x(x+7)x−7

69.

R(x)=x(x−5)x−6R(x)=x(x−5)x−6

71.

Answers will vary.

73.

Answers will vary.

Section 7.2 Exercises

75.

3535

77.

3c+54c−53c+54c−5

79.

r+8r+8

81.

2ww−42ww−4

83.

3a+73a+7

85.

m−22m−22

87.

p+3p+5p+3p+5

89.

r+9r+7r+9r+7

91.

93.

x+2x+2

95.

z+4z−5z+4z−5

97.

4b−3b−74b−3b−7

99.

ⓐ (x+2)(x−4)(x+3)(x+2)(x−4)(x+3)
ⓑ 5x+15(x+2)(x−4)(x+3)5x+15(x+2)(x−4)(x+3),
2x2+4x(x+2)(x−4)(x+3)2x2+4x(x+2)(x−4)(x+3)

101.

ⓐ (z−2)(z+4)(z+2)(z−2)(z+4)(z+2)
ⓑ 9z+18(z−2)(z+4)(z+2)9z+18(z−2)(z+4)(z+2),
4z2+16z(z−2)(z+4)(z+2)4z2+16z(z−2)(z+4)(z+2)

103.

ⓐ (b+3)(b+3)(b−5)(b+3)(b+3)(b−5)
ⓑ 4b−20(b+3)(b+3)(b−5)4b−20(b+3)(b+3)(b−5),
2b2+6b(b+3)(b+3)(b−5)2b2+6b(b+3)(b+3)(b−5)

105.

ⓐ (d+5)(3d−1)(d−6)(d+5)(3d−1)(d−6)
ⓑ 2d−12(d+5)(3d−1)(d−6)2d−12(d+5)(3d−1)(d−6),
5d2+25d(d+5)(3d−1)(d−6)5d2+25d(d+5)(3d−1)(d−6)

107.

21y+8x30x2y221y+8x30x2y2

109.

5r−7(r+4)(r−5)5r−7(r+4)(r−5)

111.

11w+1(3w−2)(w+1)11w+1(3w−2)(w+1)

113.

2y2+y+9(y+3)(y−1)2y2+y+9(y+3)(y−1)

115.

b(5b+10+2a2)a2(b−2)(b+2)b(5b+10+2a2)a2(b−2)(b+2)

117.

−mm+4−mm+4

119.

3(r2+6r+18)(r+1)(r+6)(r+3)3(r2+6r+18)(r+1)(r+6)(r+3)

121.

2(7t−6)(t−6)(t+6)2(7t−6)(t−6)(t+6)

123.

4a2+25a−6(a+3)(a+6)4a2+25a−6(a+3)(a+6)

125.

−6m−6−6m−6

127.

p+2p+3p+2p+3

129.

3r−23r−2

131.

4(8x+1)10x−14(8x+1)10x−1

133.

x−5(x−4)(x+1)(x−1)x−5(x−4)(x+1)(x−1)

135.

1(x−1)(x+1)1(x−1)(x+1)

137.

5a2+7a−36a(a−2)5a2+7a−36a(a−2)

139.

c−5c+2c−5c+2

141.

3(d+1)d+23(d+1)d+2

143.

ⓐ R(x)=−(x+8)(x+1)(x−2)(x+3)R(x)=−(x+8)(x+1)(x−2)(x+3) ⓑ R(x)=x+1x+3R(x)=x+1x+3

145.

ⓐ 3(3x+8)(x−8)(x+8)3(3x+8)(x−8)(x+8)
ⓑ R(x)=3x+8R(x)=3x+8

147.

Answers will vary.

149.

ⓐ Answers will vary.
ⓑ Answers will vary.
ⓒ Answers will vary.
ⓓ x+yxyx+yxy

Section 7.3 Exercises

151.

a−42aa−42a

153.

12(c−2)12(c−2)

155.

12131213

157.

20572057

159.

n2+mm−n2n2+mm−n2

161.

rtt−rrtt−r

163.

(x+1)(x−3)2(x+1)(x−3)2

165.

4a+14a+1

167.

118118

169.

1919

171.

c2+cc−d2c2+cc−d2

173.

pqq−ppqq−p

175.

2x−103x+162x−103x+16

177.

3z−193z+83z−193z+8

179.

43a−743a−7

181.

2c+295c2c+295c

183.

2p−552p−55

185.

m(m−5)(4m−19)(m+5)m(m−5)(4m−19)(m+5)

187.

13241324

189.

2(a−4)2(a−4)

191.

3mnn−m3mnn−m

193.

(x−1)(x−2)6(x−1)(x−2)6

195.

Answers will vary.

Section 7.4 Exercises

197.

a=10a=10

199.

v=4021v=4021

201.

m=−2,m=4m=−2,m=4

203.

p=−5,p=−4p=−5,p=−4

205.

v=14v=14

207.

x=−45x=−45

209.

z=−145z=−145

211.

q=−18,q=−1q=−18,q=−1

213.

no solutionno solution

215.

no solutionno solution

217.

b=−8b=−8

219.

d=2d=2

221.

n=1n=1

223.

no solutionno solution

225.

s=54s=54

227.

x=−43x=−43

229.

no solution

231.

ⓐ The domain is all real numbers except x≠−2x≠−2 and x≠−4.x≠−4. ⓑ x=−3,x=−145x=−3,x=−145 ⓒ (−3,5),(−145,5)(−3,5),(−145,5)

233.

ⓐ The domain is all real numbers except x≠2x≠2 and x≠5.x≠5. ⓑ x=92,x=92, ⓒ (92,2)(92,2)

235.

r=C2πr=C2π

237.

w=2v+7w=2v+7

239.

c=b+3+2aac=b+3+2aa

241.

p=q4q−2p=q4q−2

243.

w=15v10+vw=15v10+v

245.

n=5m+234n=5m+234

247.

c=Em2c=Em2

249.

y=20x12−xy=20x12−x

251.

Answers will vary.

Section 7.5 Exercises

253.

x=49x=49

255.

p=−11p=−11

257.

a=16a=16

259.

m=60m=60

261.

p=30p=30

263.

ⓐ 162 beats per minute ⓑ yes

265.

99 ml

267.

159159 calories

269.

325325 Canadian dollars

271.

33 cups

273.

4 bags

275.

ⓐ 6 ⓑ 1212

277.

950 miles

279.

680 miles

281.

2323 foot (88 in.)

283.

247.3247.3 feet

285.

160160 mph

287.

2929 mph

289.

3030 mph

291.

2020 mph

293.

44 mph

295.

6060 mph

297.

650650 mph

299.

5050 mph

301.

5050 mph

303.

3 mph

305.

22 hours

307.

22 hours and 4444 minutes

309.

77 hours and 3030 minutes

311.

1010 min

313.

y=143xy=143x

315.

p=3.2qp=3.2q

317.

ⓐ P=2.5gP=2.5g ⓑ $82.50$82.50

319.

ⓐ m=8vm=8v ⓑ 1616 liters

321.

ⓐ L=3d2L=3d2 ⓑ 300300 pounds

323.

y=20xy=20x

325.

v=3wv=3w

327.

ⓐ g=92,400wg=92,400w ⓑ 16.8 mpg

329.

ⓐ t=1000rt=1000r ⓑ 2.52.5 hours

331.

ⓐ c=2tc=2t ⓑ 11 cavity

333.

ⓐ c=2.5mc=2.5m ⓑ $55

335.

Answers will vary.

337.

Answers will vary.

Section 7.6 Exercises

339.

(−∞,−4)∪[3,∞)(−∞,−4)∪[3,∞)

341.

[−1,3)[−1,3)

343.

(−∞,1)∪(7,∞)(−∞,1)∪(7,∞)

345.

(−5,6)(−5,6)

347.

(−52,5)(−52,5)

349.

(−∞,−3)∪(6,∞)(−∞,−3)∪(6,∞)

351.

[−9,6)[−9,6)

353.

(−∞,−6]∪(4,∞)(−∞,−6]∪(4,∞)

355.

(−∞,−4)∪(−3,∞)(−∞,−4)∪(−3,∞)

357.

(1,4)(1,4)

359.

(−∞,−3)∪(52,∞)(−∞,−3)∪(52,∞)

361.

(−∞,23)∪(32,∞)(−∞,23)∪(32,∞)

363.

(−∞,0)∪(0,4)∪(6,∞)(−∞,0)∪(0,4)∪(6,∞)

365.

[−2,0)∪(0,4][−2,0)∪(0,4]

367.

(−4,4)(−4,4)

369.

[−10,−1)∪(2,∞)[−10,−1)∪(2,∞)

371.

(2,5](2,5]

373.

(−2,6](−2,6]

375.

Answers will vary.

Review Exercises

377.

a≠23a≠23

379.

y≠0y≠0

381.

3434

383.

x+3x+4x+3x+4

385.

1616

387.

−3x2−3x2

389.

3x(x+6)(x+6)3x(x+6)(x+6)

391.

−111−w−111−w

393.

5c+45c+4

395.

R(x)=3R(x)=3

397.

399.

y+5y+5

401.

x+4x+4

403.

q2−2q−3(q+5)(q+1)q2-2q-3(q+5)(q+1)

405.

15w+26w−115w+26w−1

407.

3b2+19b−16b2−493b2+19b−16b2−49

409.

(a+2)(a−5)(a+4)(a+2)(a−5)(a+4)

411.

(3p−1)(p+6)(p+8)(3p−1)(p+6)(p+8)

413.

11c−12(c−2)(c+3)11c−12(c−2)(c+3)

415.

5x2+26x(x+4)(x+4)(x+6)5x2+26x(x+4)(x+4)(x+6)

417.

2(y2+10y−2)(y+2)(y+8)2(y2+10y−2)(y+2)(y+8)

419.

2m−7m+22m−7m+2

421.

4a−84a−8

423.

R(x)=x+8x+5R(x)=x+8x+5

425.

R(x)=2x+11R(x)=2x+11

427.

x−22xx−22x

429.

(x+2)(x−5)2(x+2)(x−5)2

431.

118118

433.

z−521z+21z−521z+21

435.

x=67x=67

437.

b=32b=32

439.

no solution

441.

443.

l=Vhwl=Vhw

445.

z=y+5+7xxz=y+5+7xx

447.

x=125x=125

449.

s=15s=15

451.

11261126 calories

453.

b=9;x=213b=9;x=213

455.

23 feet

457.

4545 mph

459.

1616 mph

461.

4848 minutes

463.

1212 days

465.

x=7x=7

467.

301301 mph

469.

288288 feet

471.

99 tickets

473.

(−4,3](−4,3]

475.

[−6,4)[−6,4)

477.

(−∞,−2]∪[4,∞)(−∞,−2]∪[4,∞)

479.

(−∞,2)∪[5,∞)(−∞,2)∪[5,∞)

481.

ⓐ c(x)=150x+100000xc(x)=150x+100000x
ⓑ More than 10,000 items must be produced to keep the average cost below $160$160 per item.

Practice Test

483.

a3ba3b

485.

x+33xx+33x

487.

x−3x+9x−3x+9

489.

3n−2n−13n−2n−1

491.

n−mm+nn−mm+n

493.

z=12z=12

495.

[−3,6)[−3,6)

497.

(−∞,0)∪(0,4]∪[6,∞)(−∞,0)∪(0,4]∪[6,∞)

499.

R(x)=1(x+2)(x+2)R(x)=1(x+2)(x+2)

501.

(−3,52)(−3,52)

503.

y=8116y=8116

505.

Oliver’s dad would take 445445 hours to split the logs himself.

507.

The distance between Dayton and Columbus is 64 miles.

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