13.1 Simplifying Rational Expressions
1. x = 0
2. x = –4
3. a = –2
4. b = 5
5. x = –8, 3
6. y = –9, 2
7. c = 4, 6
8. z = 0, –1, 4
9.
5 x
10. 3 2
11. a + 7
12. 2 13. x – 1
14. x + 1
15. 2a + 1
16. 1
1 2− c
17. 5
1
x x
+
− ; x ≠ 1, 9
18. 6
1
x x
−
Section 13.1
20. –1; x ≠ ±y
21. 3x
x − y ; x ≠ ±y
22. 2 2( 2)
3
x y x x
−
− ; x ≠ 0, ±3
23. x 5
x
− ;
x ≠ 0, –2
24. ( 5)
3 y x−
; x ≠ 0, –4
25.
4
π ≈ 78.5%
26. 1
4 = 25%
27. 2
π ≈ 63.7%
28. 3
9
π ≈ 60.5%
29. 5 x
x
−
30. 5
x
31.
5 x
32. –6x – 5; x ≠ 1
8
33.
2( 1) z
34. ( 2)( 1)
( 2)
x x
x x
− +
+ ; x ≠ 0, –2 35. The simplified expression
has only one excluded value (x ≠ 0), but the original
expression has two excluded values
4 – , 0
3 x
⎛ ≠ ⎞
⎜ ⎟
⎝ ⎠ .
36.
6
π ≈ 52.4%
37. 1
4l
38. Press the string down at 1
8l.
39. m = r
p
40. s =
2 4 2
m m nq
n
− ± −
41. 8
15
42. 30
9
x
43. 11, 0 0, 11
3 5
⎛ ⎞ ⎛ ⎞
−
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
44. y = 3
5x – 11
5 ; m = 3 5
Section 13.1
46. direct
47. direct; y = 3x
48. inverse; y = 24
13.2 Multiplying and Dividing Rational Expressions
1. 19
70
2. 6 3. a
4.
2
5 54
xy
5. 5
6. 4
4
x x
+ −
7. –3
2
8.
3
2
32
xy a b
9.
2
2 2
a b a −b
10. 2 1
2 3
x − x −
11. 7
2
12. 3
2
13.
2
7 3
Section 13.2 14. 2 2 4 b y a x
15. x + y
16. 4
15
17. 2
2 x + 18. 2 ( 6) 1 x x x + −
19. (x + 8)(x + 3) 20. 1
1
x −
21. 2
3 4 x y x y + − 22.
( 3)( 4)
y
x − x +
23. 2
( )
x y
x y
+
−
24. 2
3( 3)( 3)
x x x + + − 25. 1 x x +
26. ( 9)( 2)
( 3)( 4)
x x
x x
− −
− +
27. (3 )( 2 )
( )
x y x y
x y
+ +
−
28. 9
2
x x
+
29. a 2b
a +
30. x 3y
x −
31. 1 32. 332
( )
m n
m n m n
+
−
33. (x + 2)(x – 2)
34. 3
( )
x y
x y
−
+
35. (4 7 )( 2 )
( 3 )( )
x y x y
x y x y
− +
+ +
36. 4.5 km/hr
37. 1.875 gal/min 38. 15 mi/hr
39a. ≈ 0.28 chapters/hr
39b. ≈ 3.6 hr/chapter
40. x – 3
41. 4 ( )( 2 )
( 7 )
b a b a b
a a b
− −
+
42.
3
2
3( 6) ( 4)
x x
− −
43. 2 1
1
x x
− −
44. 1
Section 13.2
45. 6 2 3 26−
46.
47.
48. x = –335 2
49. x = 1 241 6
− ±
50. 60
51. m = –7 5
52. m = 1 8
53. 65
y
x
13.3 Adding and Subtracting Expressions
with Common Denominators
1. 14
x
2. 12
2
x +
3.
5
x
−
4. 2x2
y −
5. 1
c −
6. x
8. 4 9. 29x
x + y
10. – 12 2
x +
11. 1 12. 4
13. 3(3x y)
yz +
Section 13.3
15. b, d 16. b, c 17. – 3
4
m−
18. 14
2
x −
19. 1 20. a b
b a
+
−
21. 3(2 a)
a b
−
+
22. 3
a + b
23. – 2 2
3 2
a + a −
24. 2 6
3 9
x − x −
25. 22 7
4 7
x
x x
−
− −
26. 22( 1)
3 9
a
a a
+
− +
27. 3m n
m n
+
−
28. 7 3
5
x x
−
+
29. 7
9
x x
+
30. a(a + 1) 31.
2
2
3 11 15
8 15 x x x x − − − + 32. 2
2( 2 7)
( 6)( 2)
x x
x x
− +
− +
33. 1
a −b
34. 1
x y − + 35. 2 2 x x x + − 36. 2 6 x x x − − 37. 3 2 2
2 4 8
x x x
x
+ − −
38. x – 2 39. 3 1
4
x x
− +
− or
1 3 4 x x − −
40. 4 8
2 6 x x + + = 2(2 4) 2( 3) x x + + ; distributive property;
2 (2 4) 2
x +
(x + 3) =
2 4 3 x x + + ; Multiplicative Identity Property; 4 7 2 6 x x + + = 4 7 2( 3) x x + + ;
Section 13.3
41. 3
4l
42. 2
3l
43a.
43b. Since the interval of a 5th is one step closer to an octave (an interval of an 8th), the 5th must be at 2
3l and the
interval of the 4th, which is closer to the original tone, is at 3
4l, the arithmetic mean.
44. 6 : 12 = 1
2, an octave;
8 : 12 =2
3, a 5
th
; 9 : 12 = 3
4, a 4
th
45. 20 7 4 14
23 +
46. 2 53
47. 5x – 4 48. 3 7
2 ±
49. 3x
50. 13
15
51. 1 52. 11
8
x
53. 5
3
x y
− −
or – 5 3
x + y
54. 23 19
15
13.4 Adding and Subtracting Expressions
with Different Denominators
1. 24x3
2. 12ab2c
3. 6(r + 6)
4. (3x – 1)(x – 3) 5. (a + b)(a – b)2
6. (t – 1)(t + 1)(t – 2) 7. 12 9
16
x +
8. 29
63
x
−
9. 2 3
15
x + y
10. 11 57
36
y +
11.
2
7 6 45
10
a + a −
12.
2
3
4 1
3
x x
−
13.
(
)
3 1c c c
− + +
14. 7 312
4
m m
15. 2 3 2 20 5 2 x y x y + 16. 2 2
2x xy 4
x
− −
17.
(
)(
)
2
2 11 15
4 5
b b
b b
− −
− +
18.
(
)
(
)
22 3 2 g g − − − 19.
(
) (
)
2 22 23 36
7 4
y y
y y
− −
− +
20.
(
)
(
)(
)(
)
8 2 3
2 3 5
x
x x x
− + + − 21.
(
)(
)
2 6 2 4 x x x − − + −22. 3 7
5
b b
− + −
23.
(
)
(
)(
)
286 4 4
d d d d + + − 24.
(
)
2 3 6 4 x x − − +25.
(
)
(
)(
)
7 1
12 3 4
x
x x
+
− +
26.
(
)
Section 13.4
27.
(
)(
)
5x 5y 3x y x y
− +
+ −
28.
(
)
(
)(
)(
)
6 3 22
9 6 3
x
x x x
−
− − +
29.
(
)
(
)(
)
2
2
2 15
2 5
y y y
y y
+ − −
+ +
30. 1
m− n
31. 3
(
2)
5
x x
− +
32.
(
)
(
)(
)
2
8 17
5 3
x x x
x x + + + + 33.
(
)(
)
8 25 5 5 a a a − − − + 34.(
)(
)(
)
22 5 28
8 4 2
x x
x x x
+ −
+ − −
35. 1
2
x −
36. 3
1
x +
37. Factoring out –1 from both 2 – x and from 5 – x results in a factor of (–1)2 = 1, so the sign of the second term does not change.
38.
(
)(
)
2 9
3 4
x
x x
−
− −
39. 5
6l; 4 5l
40.
41. harmonic; arithmetic 42. 258
4
y x
43.
2
2
16
a b
−
44. 1
3
x +
45. a + b
46. 4(x – 5)(x + 5) 47. 3a(a + 3)(a + 4) 48. y = 2, 8
49. x = –2, 0, 3
50. 44°, 54°, and 82°
51. 2 109 ≈ 20.9 ft
1 2
Octave 5th 4th
major 3rd
minor 3rd
2 3
3 4
13.5 Complex and Mixed Expressions
1. 2x 1
x
+
2. 2 1
2
y +
3.
2
2 2 1
1
y y
y
+ + +
4.
2
3 9 1
3
x x
x
− − −
5. 2ab a
b
−
or a
(
2b 1)
b
−
6.
2 2
3b 3ba a b a
− +
−
7. 2
5
8. 77
30
9. ad
bc
10. x
yz
11. xz
y
12. s
13.
3
2
ac d
14. xy4
15. 2 y
c + cy
16. 5x 5y
x y − + 17. 2 2 1 x x x + +
or
(
)
2 1 x x + 18. 2 3 3 2 y y y − + − 19. 2
2 5 2
1
y y
y
+ +
+ or
(
2 1)(
2)
1 y y y + + + 20. 2 8 9 3 x x x − −
− or
(
9)(
1)
3
x x
x
− +
−
21. Ac b
c +
22.
( )
( )
( )
( )
p x d x n x
Section 13.5
25. x
x − y
26. a b
a b
−
+
27.
(
)
2 1x − y
28. (x – y)(x – y) 29. 1
4
30. (m + n)2 31. 2 2
(x + 2)
32. 2(x + 3)2
33.
(
)(
)
65 3
x
x x
− +
+ +
34. sf
s − v
35. 429 Hz 36. 374 Hz 37. 4 33
3
x x
−
38. x
y
39. 48 hi/hr 40. 1 2
1 2
41. There is a negative exponent in the denominator; 3x2. 42. The numerator and
denominator have a common factor; x – 1. 43. The radicand contains a
perfect square factor. 25 2⋅
= 5 2
44. There is a fraction in the radicand; 10
5 .
45. x = 8
9
46. y = 2 47. a = 4 48. d = – 7
11
49. 24x3y2z5
13.6 Solving Rational Equations
1. m = 27 2. x = 11 3. x = 1
4
4. y = 1
2
5. x = 2
3
6. a = 45
7
7. no solution 8. x = 3
9. a = 9
2
10. no solution 11. m = 22
12. a = 69
7
13. a = –241
8
14. x = 47
7
15. x = –1
16. x = 13
4
17. x = –103
2
18. a = 5 161
2
±
19. no solution 20. no solution 21. x = –3
22. x = 0 23. m = 51
10
24. x = –6, –2
25. 36 in. × 18 in.
26. $112 27. 13 28. 23
29. 4 and 10 30. 36 and 48
31. a. 25 made free throws b. 8 missed free throws 32. x = 3 3 17
2
±
33. x = 3 137
4
Section 13.6
34. x = 11
3 −
35. x = 11
5 −
36. a = 1
2l, b = l, A = 3 4l, H = 2
3l 2 1 3 2 3 4 l l l l = 1 3
2l l ⋅ = 4 2 2 l ⋅ 3 l 2 2 1 1
2l 2l =
37.
2
2
ab
a a b
a b b
+ = +
a b
ab = +
2
2
⋅ ab
a + b
ab = ab
38. x = 5
3
−
39. 3
(
3)
5
x x
+ +
40. 2 4
42. r = d
t
43. y =
6
nq t p
+
44. (3, –1) 45. y = 3
4x – 3
46.
47. 3 cm, 4 cm, and 5 cm
y
13.7 Applying Rational Equations
1.
r t d
Going x + 3
200 3
x+ 200
Returning x
200
x 200
2.
r t d
against
wind x – 5
50 5
x− 50
with
wind x + 5
50 5
x+ 50
3.
Time (hours)
Rate of Work
Pete 3
1 3
Adam 6
1 6
Together x
4.
Time (hours)
Rate of Work
Sheila 1 1
Keisha 1 1
Together x
1 x
5. 2.5 mi 6. 41
2 hr
7. 29.8 hr 8. 2.4 hr
9. 21.3 hr 10. 4.5 hr
11. slower plane: 750 mi; faster plane: 1500 mi 12. 165 mi
13. walking: 3 mi/hr; riding: 10 mi/hr 14. Gabriel: 4 mi/hr;
train: 40 mi/hr 15. 15 mi/hr
Section 13.7
19. Liam: 75 min or 11
4 hr;
Lucas: 150 min or 21
2 hr
20. 0.9 hr
21. Fran: 6.8 hr; Rachel: 9.8 hr 22. 13.3 hr
23. 5.8 mi/hr
24. going: 6 mi/hr;
returning: 10 mi/hr
25. 1.2 hr
26. chores: 2 hr 14 min ≈ 2.2 hr;
golf: 1 hr 32 min ≈ 1.5 hr
27. going: 40 mi/hr;
returning: 50 mi/hr; 2.5 hr 28. 9.4 hr
29. Carla: 7.1 hr; Pamela: 9.1 hr 30. going: 5.5 hr, 50 mi/hr;
31.
r t d
Going 20 mi/hr 5
2 hr 50 mi
Returning 30 mi/hr 5
3 hr 50 mi
32. 25 mi/hr 33. 24 mi/hr 34. 24 mi/hr
35.
36.
y
x y
Section 13.7
37.
38. (12, 6) 39. (–8, –10)
40. f(x) = (x + 2)2 – 1
41. x + 2 + 6
1
x −
42. 879 and 71
43. u48% – au ≤ 3%;
45% ≤ a ≤ 51%
44. 500 mL of the 28% solution 300 mL of the 60% solution
y
13.8 Graphing Rational Functions
1. b
2. e
3. a
4. d
5. c
6. f
7. b
8. f
9. e
10. b
12. x = –3
13. center: (0, 0)
vertical asymptote: x = 0 horizontal asymptote: y = 0
14. center: (4, 0)
vertical asymptote: x = 4 horizontal asymptote: y = 0
15. center: (1, 7)
vertical asymptote: x = 1 horizontal asymptote: y = 7
16. center: (5, –1)
vertical asymptote: x = 5
Section 13.8
17. center: 1, 1 2
⎛ ⎞
⎜ ⎟
⎝ ⎠
vertical asymptote: x = 1
2
horizontal asymptote: y = 1
18. center: (0, –9)
vertical asymptote: x = 0
horizontal asymptote: y = –9
19. (0, 0); x = 0 and y = 0
20. (–1, –2); x = –1 and y = –2
21.
22.
y
x
y
23.
24.
25.
26.
y
x
y
x
y
x
y
Section 13.8
27. 10 units right
28. 14 units left
29. 7 units up
30. 3 units down
31. 4 units left and 9 units down
32. 4 units right and 2 units up
33.
34. The current decreases.
35. The current increases.
36. inverse variation
37. f(x) = 1
1
x − + 2;
y
x
0 10 20 30 40 50 60 70 80 90 100 110 120 0
38. f(x) = 4
2
x
−
+ + 3;
y
x
39. The equation is not defined for those values of x.
40. The solutions approach but never reach the horizontal asymptote.
41. Answers will vary. It is similar to a hyperbola with the branch in the third
quadrant being reflected
across the x-axis. The “bend” in the curve is also “sharper.”
y
x
Section 13.8
43a. ab
44b. 2ab
a + b
44a. H = 4
1 1 1 1
24 + 18 + 16 + 20
44b. 720
44c. 19.1 mi/hr
45. x ≤ 2
5 ~ x > 4 5
46. ∅
47. x = ± 4
48. (7x2 + 4) (x2 + 1)
49. (x – 3)(x + 3)(3x2 – 4)
50. (x4 + y4)(x2 + y2)(x + y) (x – y)
51. m = –2
3
52. 13
53. 20 lb
Chapter 13 Review
1. x = 3
2. x = –2, 6
3. x = 0, ±1
4. x + 4
5. 3
2
6. 1
2x −5
7. 5
3
x x
+ −
8. 1
−
9. 56x4y
10. 4(x – 5)(x + 5)
11. ac
b
12.
2
2 5
x
x −
13. 3 10
3 5
x x
+ +
14.
2
2c d
c
+
Chapter 13 Review
16. 3
4
x x
− +
17. 3(x + 11)
18. 2 2 1 x y x y − − −
19. 1
x − y
20.
(
)
(
)(
)
2
4 4 2
2 3
x x
x x
+ +
+ +
21. – x
y
22.
(
)
(
)
2
6 1
3 3 5
x x x
+ −
23. 4
1
x −
24. y2
25. 0
26. (x 2)(x 1)
x
+ −
27.
(
)(
)
(
)
2 1 5 3 x x x x − − +28. 2 3
3 5
x x
+ −
29. x 2
x
−
30. 9 2
27
x xy y y
31.
(
3 2)(
2)
x x + x +
32.
(
)(
)
2 2 8
x
x+ x −
33. 1
xy
34. 3
x − y
35. x = –3
2, 8
36. x = 5
37. x = 26
38. x = 28
39. x = 11
6
40. x = –1, –2
41. x = 7
42. no solution
43. 7(x – 5); 4(x + 5); 7(x – 5) = 4(x + 5)
44. 40
3
x + ;
50
x ;
40 3
x + +
50
Chapter 13 Review
45. 1
5; 1 2;
1
x ;
1 5 +
1 2 =
1
x
46. 1
x ;
1 2x;
1 5
x + ;
1 8; 1
x +
1 2x +
1 5
x+ =
1 8
47. 17 8 hr
48. 18 hr
49. Micah: 24 mi/hr; Alexis: 60 mi/hr
50. before: 50 mi/hr;
after: 35 mi/hr; 45 mi/hr
51. 3
6
π ≈
90.6%
52. 5
53. (–3, 0); x = –3; y = 0
54. (3, 4); x = 3; y = 4
55. y
56.
57. b
58. e
y
Sequences—Harmonic Sequences
1. yes 2. no 3. no 4. yes 5. An =
1 2n − 1
6. An =
1 4n
7. An = 1
5n − 7
8. An =
1 15 − 6n
