Selected Answers to Math 390 Review Exam 4
1. Write in standard form.
a. - (-7/2) = 7/2 b. (x - 3)/(-2) = ( -(x - 3) )/2 c. - ( (2-y)/3 ) = ( -(2 - y) )/3
2. Reduce.
a. 16/28 = 4/7 b. 32a³ b/36a² b³ = (8a)/(9b² )
c. (4x² + 20x + 25)/(4x² - 25) = (2x + 5)² /( (2x + 5)(2x - 5) ) = (2x + 5)/(2x - 5)
3. Express as an equivalent fraction with indicated denominator.
a. (x - 1)/(x + 2) = ?/(x² - 4)
(x - 1)/(x + 2) = ?/( (x + 2)(x - 2) ) Building Factor: x - 2
(x - 1)/(x + 2) = ( (x - 1)(x - 2) )/( (x + 2)(x - 2) )
b. (x + 7)/(x - 2) = ?/(x² + x - 6)
(x + 7)/(x - 2) = ?/( (x + 3)(x - 2) ) Building Factor: x + 3
(x + 7)/(x - 2) = ( (x + 7)(x + 3) )/( (x - 2)(x + 3) )
4. Perform the indicated operations. Be sure to reduce final answer to lowest terms.
a. (2x)/( 5(x + 3) ) + 6/( 5(x + 3) ) = ( 2x + 6 )/( 5(x + 3) ) = ( 2(x + 3) )/ ( 5(x + 3) ) = 2/5
b. x/(x² - 9) - 3/(x² - 9) = (x - 3)/(x² - 9) = (x - 3)/( (x + 3)(x - 3) ) = 1/(x + 3)
c. (4x)/(3y² ) + x/(6y) = (4x)/(3y² ) + x/(2(3)y) = ( (4x)(2) )/( 2(3)y² ) + (xy) /(2(3)y² )
= ( 8x + xy )/(6y² ) = ( x(8 + y) )/(6y² )
d. (5x)/(x² + 8x + 16) - (3x)/(x² + 6x + 8) = (5x)/( (x + 4)² - (3x)/( (x + 4)(x + 2)
= ( 5x(x + 2) )/( (x + 4)² (x + 2) ) - ( 3x(x + 4) )/( (x + 4)² (x + 2) )
= ( 5x² + 10x )/( (x + 4)² (x + 2) ) - ( 3x² + 12x )/( (x + 4)² (x + 2) )
= ( ( 5x² + 10x ) - (3x² + 12x) )/( (x + 4)² (x + 2) )
= ( 5x² + 10x - 3x² - 12x )/( (x + 4)² (x + 2) )
= ( 2x² - 2x )/( (x + 4)² (x + 2) )
= ( 2x(x - 1) )/( (x + 4)² (x + 2) )
e. (2x - 1)/(x² + 7x + 10) - (x + 1)/(x² + 8x + 15) = (2x - 1)/( (x + 5)(x + 2) ) - (x + 1)/( (x + 5)(x + 3) )
= ( (2x - 1)(x + 3) )/( (x + 2)(x + 3)(x + 5) ) - ( (x + 1)(x + 2) )/( (x + 2)(x + 3)(x + 5) )
= ( (2x² + 5x - 3) - (x² + 3x + 2) )/( (x + 2)(x + 3)(x + 5) )
= ( 2x² + 5x - 3 - x² - 3x - 2)/( (x + 2)(x + 3)(x + 5) )
= ( x² + 2x - 5)/( (x + 2)(X + 3)(x + 5) )
f. (4a² )/(9b² c)(6bc² )/(5a) = (8ac)/(15b)
g. ( (15a³ b)/(7c) ) / ( (5a) /(14c² ) ) = ( (15a³ b)/(7c) )((14c² )/(5a) ) = 6a² bc
h. ( (2x + 14)/(x² + 2x - 15) )/( (x² + 8x + 7)/(x² - 2x - 3) )
= ( 2(x + 7)/( (x + 5)(x - 3) )( (x - 3)(x + 1)/( (x + 7)(x + 1) ) )
= 2/(x + 5)
i. ( (2x - 10)/(x² - 1) )( ( x² + 6x + 5)/(4x² ) )( (4)/(x² - 25) )
= [ ( 2(x - 5) )/( (x + 1)(x - 1) ) ][ ( (x + 5)(x + 1) )/ (4x² ) ][ 4/( (x + 5)(X - 5) ) ]
= 2/(x² (x - 1))
5. Write as a simple fraction in lowest terms. The main fraction bar is the one in bold face type.
a. (2/3)/(7/5) = (2/3)(5/7) = 10/21
b. ((4x)/3)/((2x)/5) = ( (4x)/3 )( 5/(2x) ) = 10/3
c. (x - 1/x)/(x + 1/x) = ( x² /x - 1/x )/( x² /x + 1/x )
= ( (x² - 1)/x )/( (x² + 1)/x )
= ( (x² - 1)/x)( x/(x² + 1) )
= (x² - 1)/(x² + 1)
6. Use long division technique to divide each of the following.
a. (2x³ + x² - 5)/(x + 3) = (2x² - 5x + 15) + (-50)/(x + 3)
b. (2x² - 3x + 4)/(x - 2) = (2x + 1) + 6/(x - 2)
7. Solve for x.
a. (5x² )/8 - (3x)/4 = 1
8( (5x² )/8 ) - 8( (3x)/4 ) = 8(1)
5x² - 6x = 8
5x² - 6x - 8 = 0
(5x + 4)(x - 2) = 0
Either 5x + 4 = 0 or x - 2 = 0
5x = -4 or x = 2
x = -4/5 or x = 2
b. 4/(x - 4) + 3/(x + 1) = 20/(x² - 3x - 4)
[(x - 4)(x + 1) ][ 4/(x - 4) ] + [(x - 4)(x + 1) ][ 3/(x + 1) ] = [(x - 4)(x + 1)][ 20/( (x - 4)(x + 1) )]
(x + 1)4 + (x - 4)3 = 20
4x + 4 + 3x - 12 = 20
7x - 8 = 20
7x = 28
x = 4
8. Solve each of the pollowing stated problems.
a. Antoine traveled 36 miles downstream by motorboat to visit his parents and then returned
home. Antoine spent 5 hours traveling, and the speed of the current is 3 miles per hour.
How fast does the motorboat travel in still water?
Let x denote speed of boat in still water.
rate time distance
downstream x + 3 36/(x + 3) 36
upstream x - 3 36/(x - 3) 36
36/(x + 3) + 36/(x - 3) = 5
[ (x + 3)(x - 3) ][ 36/(x + 3) ] + [ (x + 3)(x - 3) ][ 36/(x - 3) ] = [ (x + 3)(x - 3) ](5)
(x - 3)(36) + (x + 3)(36) = (x² - 9)5
36x - 108 + 36x + 108 = 5x² - 45
72x = 5x² - 45
0 = 5x² - 72x - 45
0 = (5x + 3)(x - 15)
Either 5x - 3 = 0 or x - 15 = 0
Either 5x = 3 or x = 15
Either x = -3/5 or x = 15
Since speed in still water can't be negative, the speed is 15 miles per hour.
b. It takes Professor Gilbert 3 hours to grade a set of homework papers, and it takes her
teaching assistant 5 hours to grade a set. If they work together, how long will it take
them to grade a set of homework papers?
Let x denote the time in hours it takes them to do job if they work together.
rate of doing work time work fractional part of job done
Gilbert 1/3 job per hour x hours x/3
Assistant 1/5 job per hour x hours x/5
x/3 + x/5 = 1
15(x/3) + 15(x/5) = 15(1)
5x + 3x = 15
8x = 15
x = 15/8
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