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( x 7 ) 4 ÷ ( x 7 ) 3 x + 1

( x 7 ) ( x + 1 )

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( 4 x + 9 ) 6 ÷ ( 4 x + 9 ) 2 ( 3 x + 1 ) 4

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5 x + 2 x 2 + 1 x 4

7 x 2 20 x + 1 ( x 4 )

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y 2 + 4 y + 4 y 2 + 10 y + 21 ÷ ( y + 2 )

( y + 2 ) ( y + 3 ) ( y + 7 )

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2 x 3 + 4 x 2 + x 1 x 1

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3 x + 1 x 2 + 3 x + 2 + 5 x + 6 x 2 + 6 x + 5 3 x 7 x 2 2 x 35

5 x 3 26 x 2 192 x 105 ( x 2 2 x 35 ) ( x + 1 ) ( x + 2 )

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5 a + 3 b 8 a 2 + 2 a b b 2 3 a b 4 a 2 9 a b + 2 b 2 a + 5 b 4 a 2 + 3 a b b 2

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3 x 2 + 6 x + 10 10 x 2 + 11 x 6 + 2 x 2 4 x + 15 2 x 2 11 x 21

13 x 3 39 x 2 + 51 x 100 ( 2 x + 3 ) ( x 7 ) ( 5 x 2 )

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Rational equations ( [link] )

For the following problems, solve the rational equations.

4 x 5 + 3 x 1 15 = 29 25

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6 a 7 + 2 a 3 21 = 77 21

a = 4

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5 x 1 6 + 3 x + 4 9 = 8 9

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4 y 5 4 + 8 y + 1 6 = 69 12

y = 2

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4 x 1 + 7 x + 2 = 43 x 2 + x 2

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5 a + 3 + 6 a 4 = 9 a 2 a 12

a = 1

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5 y 3 + 2 y 3 = 3 y 3

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2 m + 5 m 8 + 9 m 8 = 30 m 8

No solution; m = 8 is excluded.

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r + 6 r 1 3 r + 2 r 1 = 6 r 1

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8 b + 1 b 7 b + 5 b 7 = 45 b 7

No solution; b = 7 is excluded.

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Solve z = x x ¯ x  for  s .

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Solve A = P ( 1 + r t )  for  t .

t = A P P r

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Solve 1 R = 1 E + 1 F  for  E .

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Solve Q = 2 m n s + t  for  t .

t = 2 m n Q s Q

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Solve I = E R + r  for  r .

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Applications ( [link] )

For the following problems, find the solution.

When the same number is subtracted from both terms of the fraction 7 12 , the result is 1 2 . What is the number?

2

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When the same number is added to both terms of the fraction 13 15 , the result is 8 9 . What is the number?

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When three fourths of a number is added to the reciprocal of the number, the result is 173 16 . What is the number?

No rational solution.

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When one third of a number is added to the reciprocal of the number, the result is 127 90 . What is the number?

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Person A working alone can complete a job in 9 hours. Person B working alone can complete the same job in 7 hours. How long will it take both people to complete the job working together?

3 15 16  hrs

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Debbie can complete an algebra assignment in 3 4 of an hour. Sandi, who plays her radio while working, can complete the same assignment in 1 1 4 hours. If Debbie and Sandi work together, how long will it take them to complete the assignment?

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An inlet pipe can fill a tank in 6 hours and an outlet pipe can drain the tank in 8 hours. If both pipes are open, how long will it take to fill the tank?

24 hrs

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Two pipes can fill a tank in 4 and 5 hours, respectively. How long will it take both pipes to fill the tank?

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The pressure due to surface tension in a spherical bubble is given by P = 4 T r , where T is the surface tension of the liquid, and r is the radius of the bubble.
(a) Determine the pressure due to surface tension within a soap bubble of radius 1 2 inch and surface tension 22.
(b) Determine the radius of a bubble if the pressure due to surface tension is 57.6 and the surface tension is 18.

(a) 176 units of pressure; (b) 5 4 units of length

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The equation 1 p + 1 q = 1 f relates an objects distance p from a lens and the image distance q from the lens to the focal length f of the lens.
(a) Determine the focal length of a lens in which an object 8 feet away produces an image 6 feet away.
(b) Determine how far an object is from a lens if the focal length of the lens is 10 inches and the image distance is 10 inches.
(c) Determine how far an object will be from a lens that has a focal length of 1 7 8 cm and the object distance is 3 cm away from the lens.

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Dividing polynomials ( [link] )

For the following problems, divide the polynomials.

a 2 + 9 a + 18 by a + 3

a + 6

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c 2 + 3 c 88 by c 8

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x 3 + 9 x 2 + 18 x + 28 by x + 7

x 2 + 2 x + 4

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y 3 2 y 2 49 y 6 by y + 6

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m 4 + 2 m 3 8 m 2 m + 2 by m 2

m 3 + 4 m 2 1

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3 r 2 17 r 27 by r 7

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a 3 3 a 2 56 a + 10 by a 9

a 2 + 6 a 2 8 a 9

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y 3 + y 2 y by y + 4

y 2 3 y + 11 44 y + 4

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5 x 6 + 5 x 5 2 x 4 + 5 x 3 7 x 2 8 x + 6 by x 2 + x 1

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y 10 y 7 + 3 y 4 3 y by y 4 y

y 6 + 3

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4 b 7 3 b 6 22 b 5 19 b 4 + 12 b 3 6 b 2 + b + 4 by b 2 + 6

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x 3 + 1 by x + 1

x 2 x + 1

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a 4 + 6 a 3 + 4 a 2 + 12 a + 8 by a 2 + 3 a + 2

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y 10 + 6 y 5 + 9 by y 5 + 3

y 5 + 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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