Question 1

Which of the following values must be excluded from the domain of the rational expression $$\displaystyle \frac{2x}{x^2$-9}$$?

Accepted Answer

$$x = 3,\ -3$$

Question 2

Simplify the rational expression $$\displaystyle \frac{x^2$-1}{x^2$+2x+1}.$$

Accepted Answer

$$\displaystyle \frac{x-1}{x+1}$$

Question 3

Multiply and simplify: $$\displaystyle \frac{x^2$-4}{x+2} \cdot \frac{x+3}{x-2}.$$

Accepted Answer

$$x+3$$

Question 4

Divide and simplify: $$\displaystyle \frac{x^2$-9}{x+3} \div \frac{x-3}{x-1}.$$

Accepted Answer

$$x+1$$

Question 5

Add the rational expressions and simplify: $$\displaystyle \frac{2}{x-1} + \frac{3}{x-1}.$$

Accepted Answer

$$\displaystyle \frac{5}{x-1}$$

Question 6

Subtract and simplify: $$\displaystyle \frac{5}{x+2} - \frac{3}{x+2}.$$

Accepted Answer

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

Question 7

Find the least common denominator (LCD) for the rational expressions $$\displaystyle \frac{1}{x^2$-4}$$ and $$\displaystyle \frac{1}{x+2}.$$

Accepted Answer

$$x^2-4$

Question 8

Add and simplify: $$\displaystyle \frac{1}{x} + \frac{2}{x^2$}.$$

Accepted Answer

$$\displaystyle \frac{x+2}{x^2$}$$

Question 9

Subtract and simplify: $$\displaystyle \frac{3}{x+1} - \frac{2}{x-1}.$$

Accepted Answer

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

Question 10

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

Accepted Answer

$$1 + \frac{x}{y}$$

Question 11

A chemical solution is mixed such that the concentration is modeled by $$\displaystyle \frac{c_1v_1 + c_2v_2}{v_1$ + v_2$}$$, where $$c_1$$ and $$c_2$$ are concentrations and $$v_1$$ and $$v_2$$ are volumes. If $$c_1 = 5$, v_1$ = 2$$, $$c_2 = 10$, v_2$ = 3$$, what is the resulting concentration?

Accepted Answer

$$8$$

Question 12

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

Accepted Answer

Add the numerator and denominator together.

Question 13

A medical dosage formula is given by $$\displaystyle D = \frac{mg}{kg}$$, where $$mg is $$the total milligrams of medication and $$kg is $$the patient's weight in kilograms. If a patient weighs $$70 kg $$and receives $$1400 mg$$, what is the dosage per kilogram?

Accepted Answer

$$20$$

Question 14

Simplify the expression: $$\displaystyle \frac{x^2$-4x+4}{x^2$-2x}.$$

Accepted Answer

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

Question 15

Subtract and simplify: $$\displaystyle \frac{2}{x} - \frac{1}{x+1}.$$

Accepted Answer

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

Question 16

Which of the following is the correct process for dividing two rational expressions $$\displaystyle \frac{A}{B} \div \frac{C}{D}$$?

Accepted Answer

$$\displaystyle \frac{A}{B} \cdot \frac{D}{C}$$

Question 17

A rational expression is undefined for which of the following values of $$x in$$ $$\displaystyle \frac{3x+2}{x^2$-5x+6}$$?

Accepted Answer

$$x = 2,\ 3$$

Question 18

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

Accepted Answer

$$\displaystyle \frac{1}{x y (x-y)}$$

Question 19

A real-world scenario: A car travels $$d$$ miles in $$t$$ hours at a constant speed. The speed is given by $$\displaystyle \frac{d}{t}. If $$the car travels $$180$$ miles in $$3$$ hours, what is its speed?

Accepted Answer

$$60$$

Rational Expressions: Operations and Simplification

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