Free Compass Math Practice Questions on Reducing Rational Expressions with Solutions and Explanations - Sample 7

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

1. For all x not equal to 1, which of the following is equivalent to the rational expression below?

 x2 + 5x - 6 ---------------- x - 1

A) x - 6
B) x - 1
C) x + 6
D) - x - 6
E) 6 - x

Solution

Factor the numerator
 x2 + 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

2. Which of the following is a simplified expression equal to

 5 - x --------- 2x - 10

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 --------- 2x - 10
 5 - x = --------- -2(5 - x)
For x not equal to 5, the above could be simplified to

= - 1 / 2

3. For all x not equal to - 4, which of the given expressions is equivalent to the expression below?

 16 - x2 ------------ 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 - x2 ------------ x + 4
 (4 - x)(4 + x) = ------------------- x + 4
For x not equal to -4, the above could be simplified to

= 4 - x

4. Simplify the following rational expression.

 x + 2 ------------ = x2 + 2x

Solution

Factor the denominator in the given expression as follows
 x + 2 ------------ = x2 + 2x
 x + 2 ------------------ x(x + 2)
For x not equal to -2, the above could be simplified to

= 1/x

5. For all x not equal to 3, which of the given expressions is equivalent to the expression below?

 3 - x ------------ x2- 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 ------------ x2- x - 6
 -(x - 3) = ------------ (x + 2)(x - 3)
For x not equal to 3, the above could be simplified to

= - 1 / (x + 2)

6.  x3 - x ------------ = x2 - 1

Solution

Factor the numerator and denominator in the given expression as follows
 x3 - x ------------ = x2 - 1
 x(x2 - 1) ------------------ x2 - 1
Since x2 - 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

7.  x2 - 4 ---------------- = x2 + 4x - 12

Solution

Factor the numerator and denominator in the given expression as follows
 x2 - 4 ---------------- = x2 + 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)

8. Simplify the following rational expression.

 x2 + 1 --------------- x3 + x

Solution

Factor the denominator in the given expression as follows
 x2 + 1 --------------- x3 + x
 x2 + 1 = --------------- x(x2 + 1)
For reall x, x2 + 1 is never zero, hence the above could be simplified to

= 1 / x , for all x

9.  x2 + 2x - 3 ---------------- = 2x2 + 3x - 5

Solution

Factor the numerator and denominator in the given expression as follows
 x2 + 2x - 3 ---------------- = 2x2 + 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)

10. For all x not equal to 1, which of the following is equivalent to the rational expression below?

 x - 1 ---------------- (x2 - 1)(x + 3)

A) 1 / (x + 3)
B) 1 / (x2 + 4 x + 3)
C) 1 / (x + 1)
D) 1 / x
E) 1 / (x - 1)

Solution

Factor the term x2 - 1 in the denominator as follows
 x - 1 ---------------- (x2 - 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)]

More ACT, SAT and Compass practice