Master Rational Expressions: Online Quizzes and Practice

Question 1

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

Accepted Answer

x(x - 1)

Answer

x(x + 1)

Answer

x(x^2 - 1)

Answer

x^2(x - 1)

Question 2

Subtract the rational expressions: 3x/2 - x/4

Accepted Answer

(5x) / 4

Answer

(2x) / 2

Answer

(x) / 8

Answer

(-x) / 4

Question 3

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

Accepted Answer

2

Answer

2x - 6

Answer

2 + 4/(x - 2)

Answer

cannot be simplified

Question 4

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

Accepted Answer

1

Answer

x

Answer

x - 2

Answer

x + 3

Question 5

Identify the excluded value(s) of the rational expression: (x - 2) / (x^2 - 4)

Accepted Answer

2 and -2

Answer

0

Answer

2

Answer

-2

Question 6

Find the domain of the rational function: f(x) = (x + 1) / (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

Empty set

Question 7

Identify the least common denominator (LCD) of the following rational expressions: (2) / (x - 5) and (x) / (x² - 25)

Accepted Answer

(x - 5)(x² - 25)

Answer

x - 5

Answer

x² - 25

Answer

x(x - 5)

Question 8

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

Accepted Answer

1

Answer

x + 3

Answer

x - 1

Answer

x² + 2x - 3

Question 9

Divide the expressions: ((x^2 - 4)/(x^2 + 4x + 4)) / ((x - 2)/(x + 2))

Accepted Answer

1

Answer

-1

Answer

x - 2

Answer

x + 2

Question 10

Divide the rational expressions: (x² - y²) / (x + y)

Accepted Answer

x - y

Answer

x + y

Answer

x² - xy + y²

Answer

x² + xy + y²

Question 11

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

Accepted Answer

1 - (5 / x + 3)

Answer

1 + (5 / x + 3)

Answer

1 - (3 / x + 5)

Answer

1 + (3 / x + 5)

Question 12

What is the least common denominator of the rational expressions (2 / x) and (x / 3)?

Accepted Answer

6x

Answer

2x

Answer

3x

Answer

6

Question 13

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

Accepted Answer

x(x+1)

Answer

x

Question 14

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

Accepted Answer

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

Answer

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

Question 15

Divide the following rational expressions: (x²-9) / (x+3) ÷ (x-3) / (x+3)

Accepted Answer

x+3

Answer

x-3

Question 16

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

Accepted Answer

x = 3

Answer

x = -3

Question 17

Which of the following statements about rational expressions is NOT true?

Accepted Answer

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

Answer

The product of two rational expressions is a rational expression.

Question 18

The rational expression 1/(x-2) is undefined when x is equal to:

Accepted Answer

2

Answer

1

Question 19

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

Accepted Answer

(x^2 - 6x + 8) / (x^2 + 2x - 3)

Answer

(x - 6) / (x - 1)

Answer

(x + 6) / (x + 1)

Answer

(x^2 + 6x - 8) / (x^2 - 2x + 3)

Question 20

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

Accepted Answer

(x + 2) / (x - 2)

Answer

(x - 2) / (x + 2)

Answer

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

Answer

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

Question 21

Determine whether the rational expression: ((x^2 - 1) / (x - 1)) - ((x^2 + 1) / (x + 1)) is undefined for any value of x.

Accepted Answer

Yes, undefined at x = ±1

Answer

Yes, undefined at x = 0

Answer

No, defined for all values of x

Answer

Yes, undefined at x = 1

Question 22

Solve the rational equation for x:

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

Accepted Answer

x = 5

Answer

x = 2

Answer

x = -3

Answer

No solution

Question 23

Find the least common multiple (LCM) of the denominators of the following rational expressions:

(x-2)/(x^2+2x-3) and (x+3)/(x^2-9)

Accepted Answer

(x-2)(x^2+2x-3)(x^2-9)

Answer

(x^2+2x-3)(x^2-9)

Answer

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

Answer

Cannot be found

Question 24

Rationalize the denominator of the following expression:

1/√(x-1)

Accepted Answer

1/((x-1)√(x-1))

Answer

√(x-1)

Answer

1/(√(x-1))

Answer

Cannot be rationalized

Question 25

If the product of two rational expressions is zero, what can you conclude?

Accepted Answer

At least one of the expressions is zero.

Answer

Both expressions are zero.

Answer

Neither expression is zero.

Answer

The expressions are equal.

Question 26

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

Accepted Answer

2x(x^2 - 1)

Answer

2x(x + 1)

Answer

2x(x - 1)

Answer

(x^2 - 1)

Question 27

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

Accepted Answer

x(x - 1)

Answer

x

Answer

x + (x - 1)

Answer

(x - 1)

Question 28

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

Accepted Answer

All real numbers except 1

Answer

All real numbers except 0

Answer

All real numbers

Answer

All real numbers except -1

Question 29

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

Accepted Answer

x(x + 3)

Answer

(x + 3)

Answer

x

Answer

(x - 3)

Question 30

Which of the following rational expressions is undefined?

Accepted Answer

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

Answer

(x - 2) / (x + 2)

Answer

(x + 1) / (x - 1)

Answer

(x + 3) / (x - 3)

Question 31

Simplify the rational expression: (2x² - 5x + 3) / (x - 3)

Accepted Answer

2x - 1

Answer

x + 3

Answer

2x + 1

Answer

x - 3

Question 32

What is the least common denominator (LCD) of the expressions x/y and 2/(3y²)?

Accepted Answer

3y²

Answer

6y²

Answer

3y

Answer

xy

Question 33

What is the reciprocal of the rational expression a/b?

Accepted Answer

b/a

Answer

b - a

Answer

a/b

Answer

a - b

Question 34

Simplify the rational expression: (x² - 4) / (x + 2)

Accepted Answer

x - 2

Answer

x² - 2

Answer

(x - 2)(x + 2)

Answer

x + 2

Question 35

What is the least common multiple (LCM) of the denominators of the expressions 2x / (x - 3) and 5 / (x² - 9)?

Accepted Answer

(x - 3)(x + 3)

Answer

x + 3

Answer

x² - 9

Answer

x - 3

Question 36

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

Accepted Answer

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

Answer

5 / (x + 1)

Answer

5x / (x + 1)

Answer

3x² + 2 / (x + 1)

Question 37

Subtract the rational expressions: (4x / (x - 2)) - (1 / (x - 2))

Accepted Answer

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

Answer

3 / (x - 2)

Answer

3x / (x - 2)

Answer

4x - 1

Question 38

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

Accepted Answer

(x - 1) / (x - 2)

Answer

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

Answer

x² + 2x - 3 / x² - 4

Answer

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

Question 39

Divide the rational expressions: ((x² - 1) / (x + 4)) / ((x - 1) / (x + 2))

Accepted Answer

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

Answer

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

Answer

(x + 1) / (x + 4)

Answer

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

Question 40

What are the excluded values for the rational expression: (3x / (x² - 4))?

Accepted Answer

x = 2, x = -2

Answer

x = 0, x = 4

Answer

x = -2

Answer

x = 2

Question 41

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

Accepted Answer

(x + y) / (x - y)

Answer

1 / (x + y)

Answer

(x - y) / (x + y)

Answer

1 / (x - y)

Question 42

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

Accepted Answer

x = 5

Answer

x = 0

Answer

x = -5

Answer

x = 1

Question 43

A rectangular garden has a length of (x + 5) feet and a width of (x - 3) feet. What is the area of the garden expressed as a rational expression?

Accepted Answer

(x^2 + 2x - 15) square feet

Answer

(x + 5)(x - 3) square feet

Answer

(2x + 2) square feet

Answer

(x^2 - 15) square feet

Question 44

What is the simplified form of the expression (6x² - 12x) / (3x - 6)?

Accepted Answer

2x

Answer

2x - 2

Answer

3x

Answer

6x

Question 45

Which value of x is excluded from the domain of the expression (x + 5) / (x - 2)?

Accepted Answer

2

Answer

0

Answer

All real numbers

Answer

-5

Question 46

What is the sum of the rational expressions (3x / (x + 2)) + (4 / (x + 2))?

Accepted Answer

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

Answer

7 / (2x + 4)

Answer

3x + 4

Answer

7x / (x + 2)

Question 47

Find the difference of the rational expressions (5x / (x - 1)) - (2 / (x - 1)).

Accepted Answer

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

Answer

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

Answer

3x / (x - 1)

Answer

3

Question 48

What is the least common denominator (LCD) of the expressions (2 / (x + 3)) and (5 / (x - 1))?

Accepted Answer

(x + 3)(x - 1)

Answer

(x + 3)²

Answer

x + 3

Answer

x - 1

Question 49

Multiply the rational expressions: (x² - 4) / (x + 2) * 3 / (x - 2)

Accepted Answer

3

Answer

3x

Answer

3 / (x + 2)

Question 50

Divide the rational expressions: (x² - 9) / (x - 3) ÷ (x + 3) / 5

Accepted Answer

5

Answer

5(x - 3)

Answer

5x

Answer

5 / (x - 3)

Question 51

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

Accepted Answer

8

Answer

2

Answer

-2

Answer

-8

Question 52

A boat travels 10 miles upstream against a current of 2 mph, then returns downstream. If the total trip takes 3 hours, which equation represents the situation? Let x be the boat's speed in still water.

Accepted Answer

10/(x-2) + 10/(x+2) = 3

Answer

10/x + 10/(x-2) = 3

Answer

10/(x+2) + 10/(x-2) = 3

Question 53

What is the restricted domain of the expression √((x + 5) / (x - 2))?

Accepted Answer

x < -5 or x > 2

Answer

-5 < x < 2

Answer

All real numbers

Answer

x > 2