Master Rational Expressions: Simplify, Multiply, Divide, and Solve Equations

Question 1

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

Accepted Answer

x - 2

Answer

x / 2

Answer

x² - 4

Answer

x² - 2x - 4

Question 2

Determine the domain of the expression: (x² - 9) / (x + 3)

Accepted Answer

All real numbers except -3

Answer

All real numbers except 0 and -3

Answer

All real numbers

Answer

All real numbers except 3 and 9

Question 3

Find the least common denominator (LCD) of the expressions: (x - 1) / x and (x + 1) / x²

Accepted Answer

x²

Answer

x

Answer

(x + 1)

Answer

(x - 1)

Question 4

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

Accepted Answer

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

Answer

(x - 3) / (x + 4)

Answer

(x + 1) / (x + 2)

Answer

1

Question 5

Multiply the expressions: [(x - y) / (x + y)] * [(x + y) / (x - y)]

Accepted Answer

1

Answer

0

Answer

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

Answer

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

Question 6

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

Accepted Answer

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

Answer

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

Answer

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

Question 7

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

Accepted Answer

x = 5/3

Answer

x = -5/3

Answer

x = 0

Answer

x = 3

Question 8

Which of the following expressions is equivalent to 1 / (x^2 - 4)?

Accepted Answer

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

Answer

1 / (x + 4)^2

Answer

1 / x^2 - 4

Answer

1 / (x - 4)^2

Question 9

Which of the following is a valid restriction on the variable x for the rational expression (x - 3) / (x^2 - 9)?

Accepted Answer

x ≠ 3 and x ≠ -3

Answer

x ≠ 9

Answer

No restrictions

Answer

x ≠ 0

Question 10

Which of the following rational expressions represents the area of a triangle with base b and height h?

Accepted Answer

(1/2)bh

Answer

bh

Answer

b/h

Answer

h/b

Question 11

Divide the rational expressions: (x^3 - 27) / (x - 3)

Accepted Answer

x^2 + 3x + 9

Answer

x^2 - 3x + 9

Answer

x^2 - 2x - 3

Answer

x^3 + 3x + 9

Question 12

Find the vertical asymptote of the rational expression: (x^2 - 4) / (x - 2)

Accepted Answer

x = 2

Answer

x = -2

Answer

y = -2

Answer

y = 2

Question 13

Which of the following is the least common denominator (LCD) of the rational expressions: (x / (x - 2)) and (2 / (x^2 - 4))?

Accepted Answer

(x - 2)(x + 2)

Answer

(x + 2)

Answer

x^2 - 4

Answer

(x - 2)

Question 14

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

Accepted Answer

x/z

Answer

x*y/z

Answer

z/x

Answer

y/x

Question 15

Which of the following is a condition that must be met when simplifying or solving rational expressions?

Accepted Answer

The denominator cannot be 0.

Answer

The expression must be in lowest terms.

Answer

The operations used must be valid.

Answer

The numerator must be greater than 0.

Question 16

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

Accepted Answer

x - 2

Answer

x + 2

Answer

(x - 2)(x + 2)

Answer

x² - 2

Question 17

What value of *x* makes the expression 1/(x - 3) undefined?

Accepted Answer

x = 3

Answer

x = -3

Answer

x = 1

Answer

x = 0

Question 18

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

Accepted Answer

(x + 2) / (x - 1)

Answer

(x + 1)(x + 2)

Answer

(x² + 2x) / (x² - 1)

Answer

(x - 2) / (x + 1)

Question 19

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

Accepted Answer

x + 3

Answer

1

Answer

x² - 9

Answer

x - 3

Question 20

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

Accepted Answer

x = 5

Answer

x = -5

Answer

x = -1

Answer

x = 1

Question 21

What is the Least Common Multiple (LCM) of the denominators for the expressions: 1/(x - 2) and 2/(x² - 4)?

Accepted Answer

(x - 2)(x + 2)

Answer

x² - 4

Answer

x - 2

Answer

x + 2

Question 22

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

Accepted Answer

(x + y) / (x - y)

Answer

(x - y) / (x + y)

Answer

x/y

Answer

y/x

Question 23

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

Accepted Answer

x - 2

Answer

(x - 2)(x + 2)

Answer

x² - 2

Answer

x + 2

Question 24

Multiply the rational expressions: (x² + 3x) / (x - 2) * (x - 2) / (x² + 6x + 9)

Accepted Answer

x / (x + 3)

Answer

x / (x - 3)

Answer

x² / (x + 3)

Answer

(x² + 3x) / (x + 3)

Question 25

Divide the rational expressions: (x² - 9) / (x + 4) ÷ (x - 3) / (x² - 16)

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 26

Find the Least Common Multiple (LCM) of the expressions 2x + 4 and x^2 - 4.

Accepted Answer

2(x + 2)(x - 2)

Answer

(x + 2)(x - 2)

Answer

2(x + 2)

Answer

4(x + 2)(x - 2)

Question 27

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

Accepted Answer

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

Answer

5x/(x^2 - 1)

Answer

5/(x^2 - 1)

Question 28

For what value(s) of x is the rational expression (x^2 - 1)/(x^2 - 4x + 3) undefined?

Accepted Answer

x = 1, x = 3

Answer

x = -1, x = 3

Answer

x = -1, x = -3

Answer

x = 1, x = -3

Question 29

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

Accepted Answer

2x

Answer

1/2x

Answer

2/x

Answer

x/2

Question 30

A rectangular garden has a length of (x + 5) meters and a width of (x - 2) meters. What is the area of the garden in terms of x?

Accepted Answer

x^2 + 3x - 10

Answer

x^2 + 3x + 10

Answer

x^2 - 3x - 10

Answer

x^2 - 3x + 10

Question 31

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

Accepted Answer

x = 5

Answer

x = 2

Answer

x = -2

Answer

x = -5

Question 32

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

Accepted Answer

2x/(x^2 - 4)

Answer

(x - 2)/(x + 2)

Answer

(x + 2)/(x - 2)

Answer

2/(x^2 + 4)

Question 33

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

Accepted Answer

All real numbers except x = 2

Answer

All real numbers

Answer

All real numbers except x = 0

Answer

All real numbers except x = -2

Question 34

Identify the vertical asymptote of the rational function: y = (x - 3)/(x^2 - 9)

Accepted Answer

x = 3

Answer

y = -3

Answer

x = -3

Answer

y = 3

Question 35

Rationalize the denominator of the expression: 1/√(x + 2)

Accepted Answer

√(x + 2)/(x + 2)

Answer

1/(x + 2)

Answer

x/√(x + 2)

Answer

2/√(x + 2)