13.1.7: Chapter 7
( \newcommand{\kernel}{\mathrm{null}\,}\)
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.
There is no solution.
x=3x=3
y=7y=7
There is no solution.
There is no solution.
There is no solution.
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
1.
ⓐ z=0z=0 ⓑ p=56p=56
ⓒ n=−4,n=2n=−4,n=2
3.
ⓐ y=0y=0, ⓑ x=−12x=−12, ⓒ u=−4,u=7u=−4,u=7
5.
−45−45
7.
2m23n2m23n
9.
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.
44
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.
11
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.
ⓐ The domain is all real numbers except x≠2x≠2 and x≠4.x≠4. ⓑ x=1,x=6x=1,x=6
ⓒ (1,1),(6,1)(1,1),(6,1)
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.
