Question 1

Which of the following best describes the domain of the rational expression $$\displaystyle \frac{5x+1}{x^2$-9}$$?

Accepted Answer

All real numbers except $$x = 3$$ and $$x = -3$$

Question 2

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

Accepted Answer

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

Question 3

A chemical solution is mixed such that its concentration is modeled by $$\displaystyle \frac{4x}{x^2$-16}. $$For which value(s) of $$x is $$the concentration undefined?

Accepted Answer

$$x = 4$$ and $$x = -4$$

Question 4

Which of the following is the correct simplification of $$\displaystyle \frac{9x^2$ + 6x - 3}{12x^2$ - 12}$$?

Accepted Answer

$$\displaystyle \frac{3x-1}{4(x-1)}$$

Question 5

A medical dosage formula is given by $$\displaystyle \frac{2x+8}{x^2$-4}. $$What is the least common denominator (LCD) if you need to add this to $$\displaystyle \frac{x-2}{x+2}$$?

Accepted Answer

$$x^2-4$

Question 6

Which of the following is the correct result of multiplying $$\displaystyle \frac{a}{b}$$ and $$\displaystyle \frac{c}{d}$$?

Accepted Answer

$$\displaystyle \frac{ac}{bd}$$

Question 7

A rational expression is undefined for certain values of the variable. What is the main reason for this?

Accepted Answer

The denominator becomes zero

Question 8

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

Accepted Answer

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

Question 9

A rational expression models the rate of a chemical reaction: $$\displaystyle \frac{t^2$-9}{t-3}. $$What is the simplified form, and for which value of $$t is $$the expression undefined?

Accepted Answer

$$\displaystyle t+3$$, undefined at $$t=3$$

Question 10

Which of the following is the correct way to add $$\displaystyle \frac{2}{x-1}$$ and $$\displaystyle \frac{3}{x+2}$$?

Accepted Answer

$$\displaystyle \frac{2(x+2)+3(x-1)}{(x-1)(x+2)}$$

Question 11

A rational expression for the speed of a car is $$\displaystyle \frac{v^2$-4v+4}{v-2}. $$What is the simplified form?

Accepted Answer

$$\displaystyle v-2$$

Question 12

Which of the following is the correct way to divide $$\displaystyle \frac{a}{b} by$$ $$\displaystyle \frac{c}{d}$$?

Accepted Answer

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

Question 13

A rational expression is given by $$\displaystyle \frac{x^2$-8x+8}{2x^2$+7x-4}. $$What is the domain of this expression?

Accepted Answer

All real numbers except the values of $$x$$ that make $$2x^2+7x-4=0$

Question 14

Which of the following is the correct least common denominator (LCD) for $$\displaystyle \frac{1}{x^2$-1}$$ and $$\displaystyle \frac{1}{x-1}$$?

Accepted Answer

$$x^2-1$

Question 15

A complex rational expression is given by $$\displaystyle \frac{\frac{a}{b} + \frac{c}{d}}{\frac{e}{f}}. $$What is the first step to simplify it?

Accepted Answer

Find the LCD of all denominators in the numerator and denominator

Question 16

Which of the following is the correct simplification of $$\displaystyle \frac{x^2$-4x+4}{2x^2$-3x+1} + \frac{x+4}{2x-2}$$?

Accepted Answer

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

Question 17

A rational expression for the cost per item in a factory is $$\displaystyle \frac{100}{x-5}. $$For which value of $$x is $$the cost per item undefined?

Accepted Answer

$$x=5$$

Question 18

Which of the following is a complex rational expression?

Accepted Answer

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

Rational Expressions: Domains, Operations, and Complex Forms

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