Solutions with detailed explanations to compass math test practice questions in sample 7.

For all x not equal to 1, which of the following is equivalent to the rational expression below?
x^{2} + 5x  6

 
x  1 
A) x  6
B) x  1
C) x + 6
D)  x  6
E) 6  x
Solution
Factor the numerator
x^{2} + 5x  6

 
x  1 

(x  1)(x + 6) 
=  
x  1 

For x not equal to 1, the given rational expression could be simplified to
= x + 6

Which of the following is a simplified expression equal to
for all x not equal to 5?
A) 1/2
B) 1 / (x  5)
C) 2
D)  1 / (x  5)
E) 1/2
Solution
Factor denominator 2x  10 in the given expression as follows

5  x 
=  
2(5  x) 

For x not equal to 5, the above could be simplified to
=  1 / 2

For all x not equal to  4, which of the given expressions is equivalent to the expression below?
16  x^{2}

 
x + 4 
A) x  4
B) 16  1
C) x + x
D)  x  4
E) 4  x
Solution
Factor the numerator in the given expression as follows
16  x^{2}

 
x + 4 

(4  x)(4 + x) 
=  
x + 4 

For x not equal to 4, the above could be simplified to
= 4  x

Simplify the following rational expression.
x + 2 
 = 
x^{2} + 2x

Solution
Factor the denominator in the given expression as follows
x + 2 
 = 
x^{2} + 2x


x + 2 
 
x(x + 2) 

For x not equal to 2, the above could be simplified to
= 1/x

For all x not equal to 3, which of the given expressions is equivalent to the expression below?
3  x 
 
x^{2} x  6

A)  1 / (x + 2)
B) 1 / (x  2)
C) 1 / (x  2)
D) 1 / (x  3)
E) 1 / (x  3)
Solution
Factor the numerator and denominator in the given expression as follows
3  x 
 
x^{2} x  6


(x  3) 
=  
(x + 2)(x  3) 

For x not equal to 3, the above could be simplified to
=  1 / (x + 2)

x^{3}  x

 = 
x^{2}  1

Solution
Factor the numerator and denominator in the given expression as follows
x^{3}  x

 = 
x^{2}  1


x(x^{2}  1)

 
x^{2}  1


Since x^{2}  1 = 0 for x = 1 or x = 1, for all x not equal to 1 or 1, then the above could be simplified to
= x

x^{2}  4

 = 
x^{2} + 4x  12

Solution
Factor the numerator and denominator in the given expression as follows
x^{2}  4

 = 
x^{2} + 4x  12


(x  2)(x + 2) 
=  
(x  2)(x + 6) 

For all x not equal to 2, we can simplify the above to
= (x + 2) / (x + 6)

Simplify the following rational expression.
x^{2} + 1

 
x^{3} + x

Solution
Factor the denominator in the given expression as follows
x^{2} + 1

 
x^{3} + x


x^{2} + 1

=  
x(x^{2} + 1)


For reall x, x^{2} + 1 is never zero, hence the above could be simplified to
= 1 / x , for all x

x^{2} + 2x  3

 = 
2x^{2} + 3x  5

Solution
Factor the numerator and denominator in the given expression as follows
x^{2} + 2x  3

 = 
2x^{2} + 3x  5


(x  1)(x + 3) 
=  = 
(x  1)(2x + 5) 

For all x not equal to 1, the above could be simplified to
= (x + 3) / (2x + 5)

For all x not equal to 1, which of the following is equivalent to the rational expression below?
x  1 
 
(x^{2}  1)(x + 3)

A) 1 / (x + 3)
B) 1 / (x^{2} + 4 x + 3)
C) 1 / (x + 1)
D) 1 / x
E) 1 / (x  1)
Solution
Factor the term x^{2}  1 in the denominator as follows
x  1 
 
(x^{2}  1)(x + 3)


x  1 
=  
(x  1)(x + 1)(x + 3) 

For all x not equal to 1, we can simplify the above to to
= 1 / [(x + 1)(x + 3)]
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