Question 1

Which of the following best describes the domain of the rational expression $$\frac{5x+1}{x^2$-4}$$?

Accepted Answer

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

Question 2

Simplify the rational expression $$\frac{6x^2$}{9x}.$$

Accepted Answer

$$\frac{2x}{3}$$

Question 3

Which value(s) of $$x$$ make the denominator of $$\frac{2x+1}{x^2$-5x+6}$$ equal to zero?

Accepted Answer

$$x = 2$$ and $$x = 3$$

Question 4

Multiply and simplify: $$\frac{x+2}{x-3} 	imes \frac{x-3}{x+5}.$$

Accepted Answer

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

Question 5

Divide and simplify: $$\frac{4x}{x^2$-9} \div \frac{2}{x+3}.$$

Accepted Answer

$$\frac{2x}{x-3}$$

Question 6

Add the following rational expressions and simplify: $$\frac{2}{x} + \frac{3}{x}.$$

Accepted Answer

$$\frac{5}{x}$$

Question 7

Subtract and simplify: $$\frac{x+4}{x^2$-16} - \frac{2}{x+4}.$$

Accepted Answer

$$\frac{x+4-2(x-4)}{x^2$-16}$$

Question 8

What is the least common denominator (LCD) for $$\frac{1}{x^2$-1}$$ and $$\frac{1}{x-1}$$?

Accepted Answer

$$x^2-1$

Question 9

A chemical solution is mixed such that its concentration is modeled by $$C(x) = \frac{3x}{x^2$-4}. $$For which value(s) of $$x is $$the concentration undefined?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 10

Simplify the complex rational expression $$\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x}}.$$

Accepted Answer

$$\frac{y+2x}{3y}$$

Question 11

A medical dosage formula is given by $$D(x) = \frac{5x}{x^2$-25}. $$What is the domain of $$D(x)$$?

Accepted Answer

All real numbers except $$x = 5$$ and $$x = -5$$

Question 12

Which of the following is a correct step in simplifying $$\frac{9x^2$+6x-3}{12x^2$-12}$$?

Accepted Answer

Factor numerator as $$3(3x^2+2x-1$$)$$ and denominator as 12(x^2$-1)$$

Question 13

If $$f(x) = \frac{x^2$-9}{x-3}$$, what is $$f(3)$$?

Accepted Answer

Undefined

Question 14

Which of the following is a complex rational expression?

Accepted Answer

$$\frac{\frac{a}{b}}{\frac{c}{d}}$$

Question 15

A rational expression models the speed of a car as $$S(t) = \frac{2t+1}{t^2$-4t+3}. $$For which value(s) of $$t is $$the speed undefined?

Accepted Answer

$$t = 1$$ and $$t = 3$$

Question 16

Which of the following is the correct LCD for $$\frac{1}{x^2$-4}$$ and $$\frac{1}{x+2}$$?

Accepted Answer

$$(x-2)(x+2)$$

Question 17

Simplify the expression $$\frac{x^2$-4x+4}{2x^2$-3x+1} + \frac{x+4}{2x-2}$$ using the LCD.

Accepted Answer

$$\frac{4x^2$-x+4}{(2x-1)(x-1)}$$

Question 18

Which of the following is NOT a valid step in simplifying a complex rational expression?

Accepted Answer

Multiplying only the numerator by the LCD and leaving the denominator unchanged

Question 19

A rational expression for the rate of a chemical reaction is $$R(x) = \frac{2x-1}{x^2$-1}. $$What is the domain of $$R(x)$$?

Accepted Answer

All real numbers except $$x = 1$$ and $$x = -1$$

Rational Expressions: Domains, Operations, and Complex Forms

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