Master Rational Expressions: Enhance Your College Algebra Skills

Question 1

Subtract the rational expressions (x / y) and (z / w). Which of the following is the result?

Accepted Answer

(xw - yz) / (yw)

Answer

(x - z) / (y - w)

Answer

(x / y) - (z / w)

Answer

Cannot be determined from the given information

Question 2

Divide (6x²y) / (2xy) and express the result in its simplest form.

Accepted Answer

3x

Answer

3xy

Answer

6xy

Answer

2x

Question 3

Subtract the rational expressions [(2x) / (x + 1)] and [(x) / (x - 1)]. Simplify your answer.

Accepted Answer

(x - 1) / (x² - 1)

Answer

(x) / (x² - 1)

Answer

(x - 2) / (x² - 1)

Answer

(x + 2) / (x² - 1)

Question 4

Find the least common denominator (LCD) of the rational expressions: (2x) / (x² - 4) and (x) / (x - 2)

Accepted Answer

(2x)(x - 2)

Answer

(x)(x² - 4)

Answer

(2x)(x)

Answer

(x)(x - 2)

Question 5

Which of the following is equivalent to the expression: (x + 2) / (x - 1)?

Accepted Answer

x + 3

Answer

x - 3

Answer

-(x + 3)

Answer

-(x - 3)

Question 6

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

Accepted Answer

(x - 2) / (x + 1)

Answer

(x + 2) / (x - 1)

Answer

(x + 1) / (x - 2)

Answer

(x - 1) / (x + 2)

Question 7

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

Accepted Answer

x = -1

Answer

x = 1

Answer

x = 2

Answer

x = -2

Question 8

Which of the following rational expressions is undefined?

Accepted Answer

(x - 2) / (x² - 4)

Answer

(x + 2) / (x - 3)

Answer

(x - 3) / (x + 2)

Answer

(x² - 4) / (x - 2)

Question 9

Which of the following expressions is NOT a rational expression?

Accepted Answer

x^2+1

Answer

3/x

Answer

x/3

Answer

2x/(x-1)

Question 10

Divide the following rational expressions: (x^2-9)/(x+3) / (x-3)/(x^2-9)

Accepted Answer

1

Answer

x+3

Answer

x-3

Answer

x^2-9

Question 11

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

Accepted Answer

x=3

Answer

x=5

Answer

x=-3

Answer

x=-5

Question 12

Which of the following rational expressions represents the area of a rectangle with length x and width (x-3)?

Accepted Answer

x(x-3)

Answer

x^2

Answer

x^2-3

Answer

x/(x-3)

Question 13

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

Accepted Answer

x

Answer

-x

Answer

x^2

Answer

-x^2

Question 14

Which of the following is a rational expression?

Accepted Answer

(x + 1)/(x - 2)

Answer

x^2 + 2x + 1

Answer

sqrt(x + 2)

Answer

ln(x + 1)

Question 15

Simplify the expression: (2x)/(x - 3) + (x)/(x - 3)

Accepted Answer

(3x)/(x - 3)

Answer

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

Answer

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

Answer

3x

Question 16

Which of the following is a factor of the rational expression (x^2 - 4)/(x - 2)?

Accepted Answer

(x - 2)

Answer

(x + 2)

Answer

(x - 4)

Answer

(x + 4)

Question 17

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

Accepted Answer

x = 3

Answer

x = 2

Answer

x = 1

Answer

x = -1

Question 18

A rectangular garden has a length of (x + 3) feet and a width of (x - 2) feet. Write an expression for the area of the garden.

Accepted Answer

(x + 3)(x - 2)

Answer

x^2 + x - 6

Answer

x^2 - x - 6

Answer

(x - 3)(x + 2)

Question 19

A car travels a distance of (2x + 5) miles in (x - 1) hours. Find an expression for the car's average speed.

Accepted Answer

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

Answer

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

Answer

2x + 4

Answer

x - 2

Question 20

To simplify a rational expression, which step is essential?

Accepted Answer

Divide both the numerator and denominator by their greatest common factor (GCF).

Answer

Multiply the numerator and denominator by each other.

Answer

Add the numerator and the denominator.

Answer

Subtract the numerator from the denominator.

Question 21

Which of the following is NOT a property of rational expressions?

Accepted Answer

The denominator of a rational expression is allowed to be zero.

Answer

The product of two rational expressions is always a rational expression.

Answer

The sum of two rational expressions with similar denominators results in a rational expression.

Question 22

Determine whether the rational expression (x^2 - 4)/(x - 2) is undefined for any value of x.

Accepted Answer

Yes, x = 2

Answer

Yes, x = -2

Answer

No, it is undefined for x = 0.

Answer

No, it is defined for all real numbers.

Question 23

Which of the following is a factor of the polynomial (x^3 - y^3)?

Accepted Answer

x - y

Answer

x^2 + y^2 + xy

Answer

x + y

Question 24

Add the rational expressions: (3x)/(2y) and (x)/(y²)

Accepted Answer

(7xy + 2y²)/2y²

Answer

(6x)/(y³)

Answer

(4xy)/(y²)

Answer

(4x)/(2y²)

Question 25

Multiply the rational expressions: (x+2)/(3x) * (6x-4)/(2x+4)

Accepted Answer

2(x-2)/3

Answer

3(x+2)

Answer

1

Answer

6x-4

Question 26

Find the domain of the rational expression: (x - 2)/(x² - 4)

Accepted Answer

All real numbers except -2 and 2

Answer

All real numbers except 2

Answer

All real numbers including -2 and 2

Answer

All real numbers except -2

Question 27

Which of the following is a vertical asymptote of the rational expression: (x+1)/(x-3)?

Accepted Answer

x = 3

Answer

x = 1

Answer

x = -1

Answer

The expression has no vertical asymptotes

Question 28

Which of the following is NOT a rational expression?

Accepted Answer

x^2 - 4

Answer

3x - 2

Answer

y/x^2

Answer

x/(x + 5)

Question 29

Simplify the rational expression: (x - 4)/(x - 5)

Accepted Answer

1

Answer

x - 4

Answer

x - 1

Answer

x + 1

Question 30

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

Accepted Answer

(5x - 6)/(x^2 + x - 6)

Answer

(5x - 6)/(x^2 - x + 6)

Answer

(5x + 6)/(x^2 - x - 6)

Answer

(5x + 6)/(x^2 + x + 6)

Question 31

Find the least common denominator of the rational expressions: (x - 1)/(x + 2) and (x + 3)/(2x - 1)

Accepted Answer

2x(x + 2)

Answer

(x - 1)(2x + 1)

Answer

2x(x - 2)

Answer

(x + 2)(2x - 1)

Question 32

Find the least common denominator (LCD) for the expressions: (x+2)/(x-2) and (x-3)/(x-1)

Accepted Answer

(x-2)(x-1)

Answer

x-2

Answer

(x+2)(x-3)

Answer

x-1

Question 33

Identify the value(s) of x for which the rational expression (x-3)/(x^2-9) is undefined:

Accepted Answer

x = 3, x = -3

Answer

x = 1

Answer

x = 0

Answer

x = 2

Question 34

Simplify the rational expression (x² - 4) / (x + 2) to its simplest form.

Accepted Answer

x - 2

Answer

x² - 2

Answer

x² + 2

Answer

x + 2

Question 35

Find the sum of the rational expressions (3/x) and (2/y).

Accepted Answer

(3y + 2x) / xy

Answer

(5xy) / (x + y)

Answer

(3 + 2) / xy

Answer

(3 + 2) / (x + y)

Question 36

Simplify the rational expression (x² - 1) / (x + 1).

Accepted Answer

x - 1

Answer

x + 1

Answer

x² - 1

Answer

x² + 1

Question 37

Find the excluded values for the rational expression 4 / (x² - 25).

Accepted Answer

x = 5 and x = -5

Answer

x = 0

Answer

x = -5

Answer

x = 5

Question 38

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

Accepted Answer

(2y + 3x) / (xy)

Answer

(2/x + 3/y)

Answer

(5xy) / (x + y)

Answer

(5/xy)

Question 39

Multiply the rational expressions: (x/y) * (y/z)

Accepted Answer

x/z

Answer

y/z

Answer

y/x

Answer

z/x

Question 40

Divide the rational expression: (x^2 - 4) / (x + 2) by (x + 2)

Accepted Answer

x - 2

Answer

x + 2

Answer

(x + 2)^2

Answer

(x - 2)^2

Question 41

Simplify the complex fraction: (x/y) / ((x - y) / (y^2))

Accepted Answer

y / (x - y)

Answer

x / (y - x)

Answer

x / (x - y)

Answer

y / (y - x)

Question 42

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

Accepted Answer

x = 5

Answer

x = -5

Answer

x = 3

Answer

No solution

Question 43

Identify the vertical asymptote of the rational function: f(x) = (x^2 - 1) / (x - 1)

Accepted Answer

x = 1

Answer

x = -1

Answer

No vertical asymptote

Answer

x = 0

Question 44

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

Accepted Answer

y = 0

Answer

y = 1

Answer

No horizontal asymptote

Answer

y = 2