Master Rational Expressions: Simplify, Solve, and Conquer

Question 1

Multiply the rational expressions: (2x - 1) / (x - 2) * (x + 3) / (x² - 4)

Accepted Answer

(2x - 1)(x + 3) / (x - 2)(x² - 4)

Answer

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

Answer

(x + 3) / (x² - 4)

Answer

(2x - 1)(x - 2) / (x + 3)(x² - 4)

Question 2

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

Accepted Answer

(x + 2) / (x² - 1)

Answer

(x - 2) / (x² - 1)

Answer

(2x + 2) / (x² - 1)

Answer

(2x - 2) / (x² - 1)

Question 3

Solve the rational inequality: (x - 1) / (x + 1) > 0

Accepted Answer

x > -1

Answer

x < -1

Answer

x > 0

Answer

x < 0

Question 4

Simplify the expression: (x^2 - 9) / (x - 3)

Accepted Answer

x - 3

Answer

x + 3

Answer

x^2 + 3

Answer

x^2 - 3

Question 5

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

Accepted Answer

x = 1

Answer

x = 0

Answer

x = 2

Answer

x = 3

Question 6

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

Accepted Answer

(x - 1) / (x + 1)

Answer

x - 1

Answer

x + 1

Answer

(x + 1) / (x - 1)

Question 7

Simplify the expression: (2x^2 + 3x - 2) / (x^2 - 1)

Accepted Answer

2x + 1

Answer

2x - 1

Answer

x + 1

Answer

x - 1

Question 8

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

Accepted Answer

All real numbers except 2

Answer

All real numbers except 0

Answer

All real numbers

Answer

No real numbers

Question 9

Simplify the complex rational expression: ((2x - 3) / (x + 1)) / ((x - 4) / (2x + 5))

Accepted Answer

(4x + 10) / (x - 3)

Answer

(4x + 10) / (3 - x)

Answer

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

Answer

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

Question 10

Which of the following is NOT a valid rational expression?

Accepted Answer

x^2 / 0

Answer

2x / (x - 3)

Answer

(x - 1) / (x + 1)

Answer

2 / x

Question 11

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

Accepted Answer

{x | x ≠ 2, x ≠ -2}

Answer

{x | x ≠ 0}

Answer

{x | x ≠ 1}

Answer

{x | x ∈ R}

Question 12

Find the value of the rational expression: (2x + 1) / (x - 2) when x = 3

Accepted Answer

1

Answer

-1

Answer

2

Answer

-2

Question 13

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

Accepted Answer

A rational expression is defined at all real numbers.

Answer

The product of two rational expressions is a rational expression.

Answer

The sum of two rational expressions with the same denominator is a rational expression.

Answer

The quotient of two rational expressions is a rational expression.

Question 14

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 15

Which of the following is the least common multiple of (x - 2) and (x + 1)?

Accepted Answer

(x - 2)(x + 1)

Answer

(x - 2)

Answer

(x + 1)

Answer

x

Question 16

What is the range of the function  f(x) = (x² - 1) / (x - 2)?

Accepted Answer

All real numbers except 2

Answer

All real numbers

Answer

All real numbers including 2

Answer

Only the number 2

Question 17

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

Accepted Answer

x = 3

Answer

x = 5

Answer

x = -3

Answer

x = -5

Question 18

Which of the following expressions is equivalent to (x² - 4) / (x - 2)?

Accepted Answer

x + 2

Answer

x - 2

Answer

2x + 4

Answer

2x - 4

Question 19

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

Accepted Answer

x² + 2x + 4

Answer

x² - 2x + 4

Answer

x² + 4

Answer

x² - 4

Question 20

Multiply the rational expressions: (a/b) × (c/d)

Accepted Answer

(a × c) / (b × d)

Answer

(a + c) / (b + d)

Answer

(a + d) / (b + c)

Answer

(a - c) / (b - d)

Question 21

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

Accepted Answer

2x / (x² - 1)

Answer

2 / (x² - 1)

Answer

2x² / (x² - 1)

Answer

2(x² + 1) / (x² - 1)

Question 22

Which expression is undefined?

Accepted Answer

(x - 1) / (x² - 1)

Answer

(x² - 1) / (x + 1)

Answer

(x + 1) / (x - 1)

Answer

(x² - 1) / (x - 2)

Question 23

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

Accepted Answer

x - 2

Answer

x + 2

Answer

x^2 + 2x - 4

Answer

x - 4

Question 24

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 25

Which of the following is NOT a valid method for simplifying a rational expression?

Accepted Answer

Cross-multiplying

Answer

Factoring

Answer

Dividing the numerator and denominator by a common factor

Answer

Combining like terms

Question 26

In a rational expression, the denominator can be zero if and only if:

Accepted Answer

The numerator is also zero

Answer

The numerator is a constant

Answer

The coefficient of x in the numerator is zero

Answer

The coefficient of x in the denominator is zero

Question 27

A rational expression is said to be in lowest terms if:

Accepted Answer

The numerator and denominator have no common factors

Answer

The numerator and denominator are both perfect squares

Answer

The numerator is less than the denominator

Answer

The denominator is less than the numerator

Question 28

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

Accepted Answer

(x - 3)

Answer

(x - 5)

Answer

(x - 4)

Answer

(x + 2)

Question 29

Solve the inequality: (x² - 1) / (x - 1) >= 0

Accepted Answer

x < 0 or x > 1

Answer

x >= 1

Answer

x <= -1

Answer

x > 0 or x < -1

Question 30

Add the rational expressions: $\frac{x + 1}{x - 1} + \frac{x - 1}{x + 1}$

Accepted Answer

$\frac{2}{x^2 - 1}$

Answer

$\frac{x}{2(x^2 - 1)}$

Answer

$\frac{x^2}{x^2 - 1}$

Answer

$\frac{2x}{x^2 - 1}$

Question 31

Rationalize the denominator of the expression: $\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}}$

Accepted Answer

$\frac{\sqrt{x} + \sqrt{y}}{x + \sqrt{xy} + y}$

Answer

$\frac{\sqrt{x} - \sqrt{y}}{x - \sqrt{xy} + y}$

Answer

$\frac{\sqrt{x} - \sqrt{y}}{x + \sqrt{xy} + y}$

Answer

$\frac{\sqrt{x} + \sqrt{y}}{x - \sqrt{xy} + y}$

Question 32

Find the domain of the rational expression: $\frac{x^2 - 1}{x + 2}$

Accepted Answer

All real numbers except $x = -2$

Answer

All real numbers

Answer

All real numbers except $x = 1$

Answer

All real numbers except $x = 0$

Question 33

Determine whether the rational expression: $\frac{x - a}{x + a}$ is an even or odd function.

Accepted Answer

Odd

Answer

Cannot be determined

Answer

Neither even nor odd

Answer

Even

Question 34

Simplify the rational expression:  (3x² - 6x) / (9x)

Accepted Answer

(x - 2) / 3

Answer

(x - 6) / 3

Answer

3x - 2

Answer

(x² - 2x) / (3x)

Question 35

Divide the following rational expressions: (2x² - 8) / (x² - 5x + 6) ÷ (x + 2) / (x - 3)

Accepted Answer

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

Answer

2(x + 2) / (x - 2)²

Answer

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

Answer

2

Question 36

What is the least common denominator (LCD) of the rational expressions: 1 / (x + 2) and 3 / (2x - 4)?

Accepted Answer

2(x + 2)(x - 2)

Answer

2(x + 2)

Answer

2(x - 2)

Answer

(x + 2)(x - 2)

Question 37

Which value(s) of x are excluded from the domain of the rational expression: (x² + 5x) / (x² - 2x)?

Accepted Answer

x = 0 and x = 2

Answer

x = 0 only

Answer

x = -5 and x = 0

Answer

x = 2 only

Question 38

A boat travels 10 miles downstream in the same amount of time it takes to travel 6 miles upstream.  If the current of the river is 2 mph, what is the boat's speed in still water?

Accepted Answer

8 mph

Answer

12 mph

Answer

6 mph

Answer

10 mph

Question 39

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

Accepted Answer

(y + x) / (x² - y²)

Answer

(x + y) / (xy)

Answer

(xy) / (x² - y²)

Answer

1 / (x - y)

Question 40

What are the restrictions on the variable in the expression 5x / (x² - 1)?

Accepted Answer

x ≠ 1, x ≠ -1

Answer

x ≠ 0

Answer

There are no restrictions.

Answer

x ≠ 1

Question 41

Which expression is equivalent to (x³ - 8) / (x - 2)?

Accepted Answer

x² + 2x + 4

Answer

x² + 4

Answer

x² - 4

Answer

x² - 2x + 4

Question 42

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

Accepted Answer

x - 2

Answer

(x - 2)(x + 2)

Answer

x² - 2

Answer

x + 2

Question 43

Find the Least Common Multiple (LCM) of the denominators for the expressions: 1 / (x + 2) and 3 / (x² - 4)

Accepted Answer

(x + 2)(x - 2)

Answer

x² - 4

Answer

(x - 2)

Answer

(x + 2)

Question 44

Simplify the expression: (x + 2) / (x - 3) - (x - 1) / (x + 3)

Accepted Answer

(8x + 3) / (x^2 - 9)

Answer

(8x - 3) / (x^2 - 9)

Answer

(6x + 3) / (x^2 - 9)

Answer

(6x - 3) / (x^2 - 9)

Question 45

A rectangular garden has a length of (x + 5) meters and a width of (x - 2) meters. Its area is 30 square meters. What is the value of x?

Accepted Answer

x = 7

Answer

x = 10

Answer

x = 5

Answer

x = 3

Question 46

A car travels 240 km at a speed of (x + 10) km/h. The journey takes 4 hours. Find the value of x.

Accepted Answer

x = 50

Answer

x = 30

Answer

x = 40

Answer

x = 60

Question 47

Simplify the rational expression: (x^2 - 9) / (x - 3)

Accepted Answer

x + 3

Answer

x^2 + 3

Answer

x - 3

Answer

(x + 3)(x - 3)

Question 48

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

Accepted Answer

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

Answer

(x - 2) / (x - 4)

Answer

(x + 3) / (x + 1)

Answer

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

Question 49

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

Accepted Answer

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

Answer

(x - 2)/(x + 3)

Answer

(x + 2)/(x - 3)

Answer

(x - 3)/(x + 2)

Question 50

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

Accepted Answer

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

Answer

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

Answer

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

Answer

2x / (x^2 - 4)

Question 51

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

Accepted Answer

5x / (x^2 - 1)

Answer

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

Answer

-5 / (x^2 - 1)

Answer

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

Question 52

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

Accepted Answer

x = 5

Answer

x = -5

Answer

x = -2

Answer

x = 2

Question 53

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

Accepted Answer

1

Answer

x - 2

Answer

x

Answer

x + 2

Question 54

Identify the domain of the following rational function: f(x) = (x - 3) / (x^2 - 9)

Accepted Answer

All real numbers except x = ±3

Answer

All real numbers

Answer

x > 3 or x < -3

Answer

x ≠ 0

Question 55

Find the vertical asymptotes of the following rational function: f(x) = (x + 1) / (x - 2)

Accepted Answer

x = 2

Answer

y = -1

Answer

x = -1

Answer

y = 2

Question 56

Determine if the following rational function has any horizontal asymptotes: f(x) = (2x + 1) / (x - 1)

Accepted Answer

Yes, y = 2

Answer

Yes, x = 1

Answer

No, it has no horizontal asymptotes

Answer

Yes, y = -2