Master Rational Expressions: A Comprehensive Guide for High School Juniors

Question 1

Multiply the rational expressions: ((x - 2) / (x + 3)) * ((x + 3) / (x - 5))

Accepted Answer

1

Answer

(x - 2) / (x - 5)

Answer

(x + 3) / (x + 3)

Answer

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

Question 2

Divide the rational expressions: ((x^2 - 9) / (x - 3)) / ((x + 3) / (x - 1))

Accepted Answer

x - 1

Answer

x + 1

Answer

x - 3

Answer

x + 3

Question 3

Find the domain of the rational expression: (x² - 1) / (x - 3)

Accepted Answer

x ≠ 3

Answer

All real numbers

Answer

x > 3

Answer

x < 3

Question 4

Simplify the complex rational expression [(2x - 6) / (x^2 - 9)] / [(x + 3) / (2x - 6)] and express it in its simplest form.

Accepted Answer

(2 / x)

Answer

(x / 2)

Answer

(2x / x)

Answer

(x / 2x)

Question 5

Determine which of the following rational expressions is equivalent to (x^2 + 2x) / (x^2 - 4)?

Accepted Answer

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

Answer

(x / (x - 2))

Answer

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

Answer

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

Question 6

Find the value of the rational expression (x^2 - 4) / (x + 2) when the value of x is 3.

Accepted Answer

1

Answer

0

Answer

2

Answer

3

Question 7

Identify the rational expression that is undefined.

Accepted Answer

(x / (x^2 - 1))

Answer

(x / (x - 1))

Answer

(x / (x + 1))

Answer

(x / (x - 2))

Question 8

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

Accepted Answer

x - 2

Answer

-2

Answer

x + 2

Answer

2x

Question 9

Calculate: (x + 3) / (x - 1) * (x - 1) / (x + 2)

Accepted Answer

1

Answer

(x + 3) / (x + 2)

Answer

(x - 1) / (x + 3)

Answer

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

Question 10

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

Accepted Answer

x = 5

Answer

x = -5

Answer

x = 1

Answer

x = -1

Question 11

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

Accepted Answer

2x / 3y

Answer

2 / 3y

Answer

2x / y

Answer

2x

Question 12

Simplify the rational expression: (3x^2y) / (6xy^2)

Accepted Answer

x / (2y)

Answer

2x / (3y)

Answer

(3x) / (2y)

Answer

1 / (2xy)

Question 13

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

Accepted Answer

1

Answer

x - 2

Answer

x + 3

Answer

x - 1

Question 14

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

Accepted Answer

2x / (x^2 - 4)

Answer

2 / (x - 2)

Answer

4 / (x^2 - 4)

Answer

2 / (x + 2)

Question 15

Simplify the rational expression: ((x^2 - 25) / (x + 5))

Accepted Answer

x - 5

Answer

x + 5

Answer

(x - 5)(x + 5)

Answer

Cannot be simplified

Question 16

Which of the following expressions is already in its simplest form?

Accepted Answer

1

Answer

x + 2

Answer

(x - 4) / (x + 2)

Answer

2 / (x - 2)

Question 17

Multiply and simplify: (x + 3) / (x - 1) * (x - 1) / (x + 2)

Accepted Answer

(x + 3) / (x + 2)

Answer

(x - 3) / (x + 2)

Answer

(x - 3) / (x - 2)

Answer

(x + 3) / (x - 2)

Question 18

Simplify the expression: [(x^2 - 4) / (x - 2)] / [(x + 2) / (x - 2)]

Accepted Answer

x - 2

Answer

-x - 2

Answer

x + 2

Answer

-x + 2

Question 19

Which expression below is NOT a rational expression?

Accepted Answer

x^2 + 1

Answer

(x + 1) / x

Answer

x / (x - 1)

Answer

1 / x

Question 20

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

Accepted Answer

x + 1

Answer

(x + 1)^2

Answer

x - 1

Answer

x^2

Question 21

Subtract the rational expression 3/(x - 2) from 2/(x + 1).

Accepted Answer

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

Answer

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

Answer

1 / (x^2 - x - 2)

Answer

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

Question 22

Determine the domain of the rational expression (x - 3) / (x^2 - 9).

Accepted Answer

All real numbers except -3 and 3

Answer

All real numbers

Answer

All real numbers except 0

Answer

All real numbers except -3

Question 23

Subtract the rational expressions: (3x^2 - 2x)/(x + 2) - (x^2 + 4x)/(x + 2)

Accepted Answer

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

Answer

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

Answer

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

Answer

3x - 2

Question 24

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

Accepted Answer

x = 5

Answer

x = 3

Answer

x = -5

Answer

x = -3

Question 25

Find the vertical asymptote of the rational expression: (2x)/(x - 3)

Accepted Answer

x = 3

Answer

y = 3

Answer

y = 2

Answer

x = 2

Question 26

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

Accepted Answer

1

Answer

x - 1

Answer

x + 1

Answer

(x + 1)/(x - 1)

Question 27

Simplify the rational expression: $\frac{x^2 - 9}{x + 3}$

Accepted Answer

$\frac{x - 3}{1}$

Answer

$\frac{x - 9}{1}$

Answer

$\frac{x^2 + 3}{1}$

Answer

$\frac{x + 3}{x - 3}$

Question 28

Divide the rational expressions: $\frac{\frac{x^2 - 4}{x + 2}}{\frac{x + 2}{x - 2}}$

Accepted Answer

$x - 2$

Answer

$x^2 + 4$

Answer

$x + 2$

Answer

$x^2 - 4$

Question 29

Add the rational expressions: $\frac{3}{x - 2} + \frac{2}{x + 2}$

Accepted Answer

$\frac{5x}{x^2 - 4}$

Answer

$\frac{5}{x - 2}$

Answer

$\frac{x - 2}{x + 2}$

Answer

$\frac{x + 2}{x - 2}$

Question 30

Solve for x: $\frac{x + 3}{x - 2} = 2$

Accepted Answer

$x = 7$

Answer

$x = 2$

Answer

$x = -7$

Answer

$x = -2$

Question 31

Simplify the expression: (2x^2 + 3x - 5) / (x - 1)

Accepted Answer

2x + 5

Answer

x^2 + 5

Answer

2x - 5

Answer

2x^2 + 5

Question 32

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

Accepted Answer

3x - 5 / (x - 3)

Answer

3x + 5 / (x - 3)

Answer

x - 5 / (x - 3)

Answer

x + 1 / (x - 3)

Question 33

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

Accepted Answer

x + 3

Answer

x^2 + 3

Answer

x^2 - 3

Answer

x - 3

Question 34

Which expression is equivalent to (x - 2) / (x + 2) when x ≠ -2?

Accepted Answer

1 - 4 / (x + 2)

Answer

1 / (x - 2)

Answer

1 + 4 / (x + 2)

Question 35

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

Accepted Answer

(x - 2) / (x + 2)

Answer

(x^2 - 4) / 1

Answer

1 / (x^2 - 4)

Answer

(x + 2) / (x - 2)

Question 36

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

Accepted Answer

x + 2

Answer

x^2 + 2x + 4

Answer

(x - 2)^2

Answer

x - 2

Question 37

Multiply the following rational expressions: (2x + 3) / (x - 1) * (x - 1) / (4x + 6)

Accepted Answer

1 / 2

Answer

(x - 1)^2 / (2x + 3)(4x + 6)

Answer

(2x + 3) / (4x + 6)

Answer

2 / x

Question 38

Divide the following rational expressions: (x^2 - 9) / (x + 3) ÷ (x - 3) / (x^2 + 6x + 9)

Accepted Answer

x + 3

Answer

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

Answer

x - 3

Answer

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

Question 39

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

Accepted Answer

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

Answer

(x + 3) / (x - 1)

Answer

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

Answer

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

Question 40

Subtract the following rational expressions: (5x + 2) / (x - 3) - (2x - 1) / (x + 2)

Accepted Answer

(x^2 + 11x - 5) / ((x - 3)(x + 2))

Answer

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

Answer

(x^2 - 11x + 5) / ((x - 3)(x + 2))

Answer

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

Question 41

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

Accepted Answer

x = 5

Answer

x = -5

Answer

x = 2

Answer

x = 1

Question 42

Solve the following rational inequality: (x + 3) / (x - 2) > 0

Accepted Answer

x < -3 or x > 2

Answer

x < -2 or x > 3

Answer

x > -3 and x < 2

Answer

-3 < x < 2

Question 43

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

Accepted Answer

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

Answer

x > 0

Answer

All real numbers except x = 0

Answer

All real numbers

Question 44

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

Accepted Answer

2/(2 - x)

Answer

(2 + x) / (2 - x)

Answer

1 / (2 - x)

Answer

(2 - x) / (2 + x)

Question 45

Rationalize the denominator of the following expression: 1 / (√5 - 1)

Accepted Answer

√5 + 1

Answer

1 / (5 - 1)

Answer

5

Answer

√5 - 1