Master Rational Expressions: A Comprehensive Guide for Mechanical Engineering Students

Question 1

Find the least common denominator (LCD) for the rational expressions 1/x and 1/(x - 2) to add them:

Accepted Answer

x(x - 2)

Answer

x

Answer

x - 2

Answer

2(x - 2)

Question 2

Perform long division and simplify: (x^3 - 8) / (x - 2)

Accepted Answer

x^2 + 2x + 4

Answer

x^2 - 2x + 4

Answer

x^2 - 4

Answer

x^2 + 4

Question 3

Determine the domain of the rational expression: sqrt(x - 1) / (x^2 - 4)

Accepted Answer

x > 1, x != 2, x != -2

Answer

x > 1

Answer

x >= 1, x != 2, x != -2

Answer

x < 1, x != 2, x != -2

Question 4

Simplify the following complex fraction: ((x + 1) / (x - 2)) ÷ ((x - 1) / (x + 2))

Accepted Answer

((x + 2) / (x - 2))

Answer

(x + 2)

Answer

(x - 2)

Answer

((x - 2) / (x + 2))

Question 5

Determine the domain of the following rational expression: (x² - 1) / (x - 1)

Accepted Answer

All real numbers except 1

Answer

All real numbers except 0

Answer

All real numbers except -1

Answer

All real numbers

Question 6

Add the rational expressions: (2x+1)/(x-2) + (x-3)/(x-2)

Accepted Answer

3x-5 / (x-2)

Answer

3x-1 / (x-2)

Answer

3x+5 / (x-2)

Answer

(2x+1)(x-3) / (x-2)

Question 7

Solve the equation: (x+2)/(x-3) = 2

Accepted Answer

x=7

Answer

x=-7

Answer

x=5

Answer

x=-5

Question 8

Simplify the rational expression: (x³ - 8) / (x - 2)

Accepted Answer

(x² + 2x + 4)

Answer

(x² - 2x + 4)

Answer

(x + 2)(x - 2)

Answer

(x - 2)³

Question 9

Add the rational expressions: (2x) / (x + 1) + (x) / (x - 2)

Accepted Answer

(3x² + 2x) / (x² - x - 2)

Answer

(3x² - 2x) / (x² - x - 2)

Answer

(3x²) / (x² - x - 2)

Answer

(x² + 2x) / (x² - x - 2)

Question 10

Find the least common denominator (LCD) of the rational expressions (2x) / (x - 1) and (x - 2) / (x + 1)

Accepted Answer

(x - 1)(x + 1)

Answer

x - 1

Answer

x + 1

Answer

(x - 1)²(x + 1)²

Question 11

Multiply (x - 3) / (x + 1) by (x + 2) / (x - 2).

Accepted Answer

x + 5 / x - 1

Answer

x - 5 / x - 1

Answer

-x + 5 / x - 1

Answer

x + 5 / x + 1

Question 12

Simplify the rational expression [(3x - 5) / (2x - 1)] / [(x + 2) / (x - 3)].

Accepted Answer

(3x - 5)(x - 3) / (2x - 1)(x + 2)

Answer

(-3x + 5)(x + 2) / (2x - 1)(x - 3)

Answer

(9x - 15)(x - 3) / (4x^2 - 4x - 2)

Answer

(9x - 15)(x + 2) / (4x^2 - 4x - 2)

Question 13

Find the common denominator for the rational expressions 2 / (x - 1) and 3 / (x + 1).

Accepted Answer

(x - 1)(x + 1)

Answer

x - 1

Answer

x + 1

Answer

x^2 - 1

Question 14

Simplify the rational expression (x^2 - y^2) / (x + y)(x - y).

Accepted Answer

1

Answer

x - y

Answer

x + y

Answer

x^2 + y^2

Question 15

Which of the following rational expressions is equivalent to 1 / (x - 2)?

Accepted Answer

x / (x - 2)

Answer

2 / (x - 2)^2

Answer

(x -2) / x

Answer

x^2 / (x - 2)^2

Question 16

Which of the following is equivalent to (x^2 - 4) / (x + 2)?

Accepted Answer

(x - 2)

Answer

(x + 2)

Answer

(x - 4)

Answer

(x + 4)

Question 17

What is the least common denominator (LCD) of 1 / (x - 3) and 2 / (x^2 - 9)?

Accepted Answer

(x - 3)(x + 3)

Answer

(x - 3)

Answer

(x + 3)

Answer

(x^2 - 9)

Question 18

What is the domain of the rational expression (2x) / (x^2 - 1)?

Accepted Answer

All real numbers except x = 1 and x = -1

Answer

All real numbers except x = 0

Answer

All real numbers

Answer

No real numbers

Question 19

Express the simplified form of the rational expression: (x^2 - 9) / (x + 3)

Accepted Answer

(x - 3)

Answer

(x^2 + 3)

Answer

(x^2 - 3)

Answer

(x + 3)

Question 20

Solve the equation: (x - 2)/(x + 1) = 2

Accepted Answer

x = 4

Answer

x = -1

Answer

x = 1

Answer

x = -4

Question 21

Identify the domain of the provided rational expression: (x - 1) / (x^2 - 4)

Accepted Answer

All real numbers except x = 2 and x = -2

Answer

All real numbers

Answer

Only x < 0

Answer

Only x > 0

Question 22

Find the vertical asymptote(s) of the following rational expression: (x^2 - 9) / (x - 3)

Accepted Answer

x = 3

Answer

None

Answer

x = 9

Answer

x = -3

Question 23

Find the domain of the rational expression: (x^3 - 1) / (x^2 - 1)

Accepted Answer

All real numbers except x = ±1

Answer

x ≠ 0 and x ≠ 1

Answer

x = 0 and x = 1

Answer

All real numbers

Question 24

Determine the horizontal asymptote of the rational function: f(x) = (2x^2 - 3x + 1) / (x^2 + 2x - 3)

Accepted Answer

y = 2

Answer

y = -2

Answer

y = 1

Answer

The function has no horizontal asymptotes.

Question 25

Multiply the expressions: ((x + 3) / (x - 2)) * ((x - 2) / (x + 3)) = ?

Accepted Answer

1

Answer

-1

Answer

(x + 3) / (x - 2)

Answer

x - 2

Question 26

Find the domain of the expression: (x + 2) / (x^2 - 4) = ?

Accepted Answer

R - {-2, 2}

Answer

R - {0}

Answer

R

Answer

R + {0}

Question 27

Solve the equation: (2x + 3) / (x - 1) = 5

Accepted Answer

x = 14 / 3

Answer

No solution

Answer

x = -2

Answer

x = 2

Question 28

Which of the following is a horizontal asymptote of the function f(x) = (x - 2) / (x^2 - 4)?

Accepted Answer

y = 0

Answer

No horizontal asymptote

Answer

y = x

Answer

y = 1

Question 29

Which of the following expressions is equivalent to (x^2 - 9) / (x + 3)?

Accepted Answer

x - 3

Answer

(x - 3) / (x + 9)

Answer

(x + 9) / (x - 3)

Answer

(x + 3) / (x - 3)

Question 30

Simplify the rational expression: (x^2 - 4) / (x - 2)

Accepted Answer

x + 2

Answer

x - 2

Answer

(x + 2)^2

Answer

(x - 2)^2

Question 31

Subtract the rational expressions: (x^2 - 1) / (x + 1) - (x - 1) / (x + 1)

Accepted Answer

x / (x + 1)

Answer

(x + 2) / (x + 1)

Answer

(x - 2) / (x + 1)

Answer

(x^2 - 2x + 1) / (x + 1)

Question 32

Multiply the rational expressions: (2x - 3) / (x + 2) * (x - 1) / (3x + 4)

Accepted Answer

(2x - 3)(x - 1) / (x + 2)(3x + 4)

Answer

(x - 1) / (x + 2)

Answer

(2x - 3) / (3x + 4)

Answer

(2x + 3)(x + 1) / (x - 2)(3x + 4)

Question 33

Rationalize the denominator of the expression: 1 / (sqrt(2) + sqrt(3))

Accepted Answer

(sqrt(2) - sqrt(3)) / (2 - 3)

Answer

(2 - 3) / (sqrt(2) + sqrt(3))

Answer

(sqrt(2) + sqrt(3)) / (2 - 3)

Answer

(sqrt(2) - sqrt(3)) / (2 + 3)

Question 34

Find the least common denominator (LCD) of the rational expressions: (x) / (x - 2) and (2x) / (x^2 - 4)

Accepted Answer

(x - 2)(x + 2)

Answer

(x^2 - 4)

Answer

(x + 2)

Answer

(x)(x - 2)

Question 35

Simplify the complex rational expression: ((x - 2) / (x + 1)) / ((x + 2) / (x - 1))

Accepted Answer

(x - 2)(x - 1) / (x + 1)(x + 2)

Answer

(x + 2)(x + 1) / (x - 1)(x - 2)

Answer

(x^2 - 4) / (x^2 + 1)

Question 36

Solve the rational equation: (x - 1) / (x + 2) = 2 / (x - 3)

Accepted Answer

x = 7

Answer

x = 2

Answer

x = -7

Answer

x = -2

Question 37

Simplify the rational expression: (x^2 - 1) / (x + 1)

Accepted Answer

x - 1

Answer

x^2 - 1

Answer

x + 1

Answer

(x - 1)(x + 1)

Question 38

Add the following rational expressions: (2/x) + (3/x^2)

Accepted Answer

(2x + 3)/x^2

Answer

(3x^2 + 2)/x

Answer

(5/x^2)

Answer

(x^2 + 5)/x

Question 39

Subtract the rational expression: (x-2)/(x+1) from (x+1)/(x-2)

Accepted Answer

(-2x - 3)/(x^2 - 1)

Answer

(2x + 3)/(x^2 - 1)

Answer

(x^2 - 1)/(2x + 3)

Answer

(-2x + 3)/(x^2 - 1)

Question 40

Multiply the rational expressions: (x^2 - 4) / (x + 2) * (x - 2) / (x^2 - 2x)

Accepted Answer

1

Answer

x^2 - 4

Answer

x

Answer

x - 2

Question 41

Divide the rational expression: (x^3 + 2x^2 - x) / (x^2 - 1) by (x^2 + x)

Accepted Answer

x - 2

Answer

x - 1

Answer

x + 1

Answer

x + 2

Question 42

Evaluate the rational expression: (x^2 - 9) / (x + 3) at x = 2

Accepted Answer

1

Answer

0

Answer

Undefined

Answer

-1

Question 43

Determine the domain of the rational expression: (2x + 1) / (x^2 - 4)

Accepted Answer

All real numbers except -2 and 2

Answer

All real numbers except 0

Answer

All real numbers

Answer

Only the numbers -2 and 2

Question 44

Solve the rational equation: (x - 1) / (x + 2) = 2

Accepted Answer

x = 5

Answer

x = -5

Answer

x = 2

Answer

x = 1