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Proficiency Exam

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary:

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.

A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.

The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.

This module contains the proficiency exam for the chapter "Rational Expressions".

Proficiency Exam

Exercise 1

((Reference)) Find the domain of 5a+1 a 2 5a24 . 5a+1 a 2 5a24 .

Solution

a3,8 a3,8

Exercise 2

For the following problems, fill in the missing term.
((Reference)) 3 x+4 = x+4 3 x+4 = x+4

Solution

3 3

Exercise 3

((Reference)) 2x+5 x+1 = x1 2x+5 x+1 = x1

Solution

2x5 2x5

Exercise 4

For the following problems, reduce to lowest terms.
((Reference)) 30 x 6 y 3 ( x3 ) 2 ( x+5 ) 2 6x y 3 ( x+5 ) 30 x 6 y 3 ( x3 ) 2 ( x+5 ) 2 6x y 3 ( x+5 )

Solution

5 x 5 ( x3 ) 2 ( x+5 ) 5 x 5 ( x3 ) 2 ( x+5 )

Exercise 5

((Reference)) x 2 +10x+24 x 2 +x30 x 2 +10x+24 x 2 +x30

Solution

x+4 x5 x+4 x5

Exercise 6

((Reference)) 8 x 2 +2x3 4 x 2 +12x7 8 x 2 +2x3 4 x 2 +12x7

Solution

4x+3 2x+7 4x+3 2x+7

Exercise 7

((Reference)) Replace N N with the proper quantity.
x+2 x1 = N x 2 4x+3 x+2 x1 = N x 2 4x+3

Solution

( x3 )( x+2 ) ( x3 )( x+2 )

Exercise 8

((Reference)) Assume that a 2 +a6, a 2 +a6, a 2 a12, a 2 a12, and a 2 2a8 a 2 2a8 are denominators of rational expressions. Find the LCD.

Solution

( a+2 )( a2 )( a+3 )( a4 ) ( a+2 )( a2 )( a+3 )( a4 )

Exercise 9

For the following problems, perform the operations.
((Reference)) 3a+4 a+6 2a1 a+6 3a+4 a+6 2a1 a+6

Solution

a+5 a+6 a+5 a+6

Exercise 10

((Reference)) 18 x 3 y 5 a 2 · 15 a 3 b 6 x 2 y 18 x 3 y 5 a 2 · 15 a 3 b 6 x 2 y

Solution

9abx 9abx

Exercise 11

((Reference)) y 2 y12 y 2 +3y+2 · y 2 +10y+16 y 2 7y+12 y 2 y12 y 2 +3y+2 · y 2 +10y+16 y 2 7y+12

Solution

( y+3 )( y+8 ) ( y+1 )( y3 ) ( y+3 )( y+8 ) ( y+1 )( y3 )

Exercise 12

((Reference)) y2 y 2 11y+24 + y+4 y 2 +3y18 y2 y 2 11y+24 + y+4 y 2 +3y18

Solution

2( y 2 22 ) ( y8 )( y3 )( y+6 ) 2( y 2 22 ) ( y8 )( y3 )( y+6 )

Exercise 13

((Reference)) 9 2x+7 + 4 6x1 9 2x+7 + 4 6x1

Solution

62x+19 ( 2x+7 )( 6x1 ) 62x+19 ( 2x+7 )( 6x1 )

Exercise 14

((Reference)) 16 x 5 ( x 2 1 ) 9x9 ÷ 2 x 2 2x 3 16 x 5 ( x 2 1 ) 9x9 ÷ 2 x 2 2x 3

Solution

8 x 4 ( x+1 ) 3( x1 ) 8 x 4 ( x+1 ) 3( x1 )

Exercise 15

((Reference)) ( m+3 )÷ 2m+6 5m+1 ( m+3 )÷ 2m+6 5m+1

Solution

5m+1 2 5m+1 2

Exercise 16

((Reference)) 3y+10 8 y 2 +10y3 5y1 4 y 2 +23y6 3y+10 8 y 2 +10y3 5y1 4 y 2 +23y6

Solution

7 y 2 +15y+63 ( 4y1 )( 2y+3 )( y+6 ) 7 y 2 +15y+63 ( 4y1 )( 2y+3 )( y+6 )

Exercise 17

((Reference)) Solve 1 x+3 + 3 x3 = x x 2 9 . 1 x+3 + 3 x3 = x x 2 9 .

Solution

x=2 x=2

Exercise 18

((Reference)) Solve 12 m4 +5= 3m m4 . 12 m4 +5= 3m m4 .

Solution

No solution; m=4 m=4 is excluded.

Exercise 19

((Reference)) When the same number is added to both the numerator and denominator of the fraction 5 3 , 5 3 , the result is 6 5 . 6 5 . What is the number that is added?

Solution

7

Exercise 20

((Reference)) Person A, working alone, can complete a job in 20 hours. Person B, working alone, can complete the same job in 30 hours. How long will it take both people, working together, to complete the job?

Solution

12 hours

Exercise 21

((Reference)) The width of a rectangle is 1 foot longer than one half the length. Find the dimensions (lengh and width) of the rectangle if the perimeter is 44 feet.

Solution

8 ft by 14 ft

Exercise 22

((Reference)) Simplify the complex fraction 4 3 x 4+ 3 x . 4 3 x 4+ 3 x .

Solution

4x3 4x+3 4x3 4x+3

Exercise 23

((Reference)) Simplify the complex fraction 1 5 x 6 x 2 1+ 6 x + 5 x 2 . 1 5 x 6 x 2 1+ 6 x + 5 x 2 .

Solution

x6 x+5 x6 x+5

Exercise 24

((Reference)) Perform the division: x 3 +10 x 2 +21x18 x+6 . x 3 +10 x 2 +21x18 x+6 .

Solution

x 2 +4x3 x 2 +4x3

Exercise 25

((Reference)) Perform the division: 2 x 3 +5x1 x2 . 2 x 3 +5x1 x2 .

Solution

2 x 2 +4x+13+ 25 x2 2 x 2 +4x+13+ 25 x2

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Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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