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?
|
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
-
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 - 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
-
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
-
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)
-
|
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
-
|
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)
-
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
-
|
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)
-
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)]
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