This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! For this subtraction, use LCD (x - 2)(x + 2) = x² - 4; the first is 3/(x² - 4), second is 1/(x - 2) = (x + 2)/(x² - 4); subtract numerators 3 - (x + 2) = 1 - x, giving (1 - x)/(x² - 4) or equivalently (-x + 1)/((x - 2)(x + 2)) in lowest terms. Choice C correctly performs the subtraction and simplifies, giving (-x + 1)/((x - 2)(x + 2)) in lowest terms. A common mistake is forgetting the negative in subtraction, leading to positive numerators like in choice D, but apply the minus to the entire second numerator. The rational expression operation hierarchy: Multiplication and division are easier—factor everything, cancel common factors, then multiply (or flip and multiply for division). Addition and subtraction are harder—factor denominators, find LCD as the LCM of denominator factors, rewrite each fraction with LCD, then add/subtract numerators. Always simplify at the end by factoring the numerator and canceling any new common factors with the denominator. This systematic approach prevents errors!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! This complex fraction simplifies by multiplying by the reciprocal: rac{x}{x - 2} cdot rac{x + 1}{3} = rac{x(x + 1)}{3(x - 2)}, with no common factors to cancel further. Choice B correctly simplifies the complex fraction to rac{x(x + 1)}{3(x - 2)}. Choice D fails by perhaps canceling incorrectly across the complex structure, leading to an oversimplified rac{x}{3}. Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In [2/(x - 1)] + [3/(x - 1)], you can't cancel the (x - 1)—you add the numerators: (2 + 3)/(x - 1) = 5/(x - 1). But in [2/(x - 1)] · [3/(x - 1)], you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! To subtract, use LCD (x - 1)(x + 1), rewrite as rac{3(x + 1) - 1(x - 1)}{(x - 1)(x + 1)} = rac{3x + 3 - x + 1}{(x - 1)(x + 1)} = rac{2x + 4}{(x - 1)(x + 1)}. Choice B correctly subtracts and simplifies using the LCD, giving rac{2x + $4}{x^2$ - 1}. Choice C fails by likely miscalculating the numerator signs, such as subtracting incorrectly to get 2x - 4. Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In [2/(x - 1)] + [3/(x - 1)], you can't cancel the (x - 1)—you add the numerators: (2 + 3)/(x - 1) = 5/(x - 1). But in [2/(x - 1)] · [3/(x - 1)], you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! Factor $x^2$ - 4 = (x - 2)(x + 2) for LCD (x - 2)(x + 2), rewrite as rac{x + 1 - 3(x + 2)}{(x - 2)(x + 2)} = rac{x + 1 - 3x - 6}{(x - 2)(x + 2)} = rac{-2x - 5}{(x - 2)(x + 2)}. Choice B correctly subtracts and simplifies using the LCD, giving rac{-2x - 5}{(x - 2)(x + 2)}. Choice A fails by likely miscalculating the numerator as -2x + 7, perhaps from adding instead of subtracting or sign errors. Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In [2/(x - 1)] + [3/(x - 1)], you can't cancel the (x - 1)—you add the numerators: (2 + 3)/(x - 1) = 5/(x - 1). But in [2/(x - 1)] · [3/(x - 1)], you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! To add these, use LCD (x - 1)(x + 1) = x² - 1; rewrite as x(x + 1)/(x² - 1) + 2(x - 1)/(x² - 1); combine numerators x² + x + 2x - 2 = x² + 3x - 2, giving (x² + 3x - 2)/(x² - 1) in lowest terms. Choice B correctly finds the LCD and combines, giving (x² + 3x - 2)/(x² - 1) in lowest terms. A common mistake is incorrectly expanding numerators, like forgetting the -2, leading to something like choice C, but verify the combined terms. Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In [2/(x - 1)] + [3/(x - 1)], you can't cancel the (x - 1)—you add the numerators: (2 + 3)/(x - 1) = 5/(x - 1). But in [2/(x - 1)] · [3/(x - 1)], you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! For this subtraction, factor x² - 4 = (x - 2)(x + 2) as the LCD; rewrite the second as 3(x + 2)/((x - 2)(x + 2)); subtract numerators (x + 1) - 3(x + 2) = x + 1 - 3x - 6 = -2x - 5, giving (-2x - 5)/((x - 2)(x + 2)) in lowest terms. Choice A correctly performs the subtraction and simplifies, giving (-2x - 5)/((x - 2)(x + 2)) in lowest terms. A common mistake is switching signs in subtraction, leading to positive terms like in choice C, but remember subtraction means minus the entire numerator. Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In [2/(x - 1)] + [3/(x - 1)], you can't cancel the (x - 1)—you add the numerators: (2 + 3)/(x - 1) = 5/(x - 1). But in [2/(x - 1)] · [3/(x - 1)], you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! To subtract, factor $x^2-4=(x-2)(x+2)$ for LCD $(x-2)(x+2)$; rewrite second as $\frac{3(x+2)}{(x-2)(x+2)}$, then $\frac{x+1 - (3x+6)}{x^2-4} = \frac{-2x-5}{x^2-4}$, which is simplified. Choice B correctly finds the LCD, subtracts the numerators carefully (noting the negative), and simplifies, giving $\frac{-2x-5}{x^2-4}$ in lowest terms. Choice C fails by getting the signs wrong in the numerator, perhaps forgetting to distribute the negative in subtraction—always subtract the entire numerator! Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In $ \frac{2}{x - 1} + \frac{3}{x - 1} $, you can't cancel the $(x - 1)$—you add the numerators: $ \frac{2 + 3}{x - 1} = \frac{5}{x - 1} $. But in $ \frac{2}{x - 1} \cdot \frac{3}{x - 1} $, you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! To subtract, find LCD $ (x-3)(x+3) $; $ \frac{2(x+3) - 1(x-3)}{x^2-9} = \frac{2x+6 - x + 3}{x^2-9} = \frac{x+9}{x^2-9} $, which doesn't simplify further. Choice B correctly finds the LCD, subtracts the numerators (distributing the negative), and simplifies, giving $ \frac{x+9}{x^2-9} $ in lowest terms. Choice D fails by getting the signs wrong in the numerator, perhaps from subtracting in the wrong order—always distribute the negative carefully! Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In $ [2/(x - 1)] + [3/(x - 1)] $, you can't cancel the (x - 1)—you add the numerators: $ (2 + 3)/(x - 1) = 5/(x - 1) $. But in $ [2/(x - 1)] \cdot[3/(x - 1)] $, you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! To add, find LCD $(x-2)(x+2)$; rewrite as $$\frac{3(x+2) + 2(x-2)}{(x-2)(x+2)} = \frac{3x+6 + 2x-4}{x^2-4} = \frac{5x+2}{x^2-4}$$, which can't be factored further to cancel, so it's simplified. Choice C correctly finds the LCD, combines the numerators, and simplifies, giving $\frac{5x+2}{x^2-4}$ in lowest terms. Choice A fails by incorrectly adding the numerators without distributing, perhaps treating it like same denominators when they're not—always use the LCD for different denominators! The rational expression operation hierarchy: Multiplication and division are easier—factor everything, cancel common factors, then multiply (or flip and multiply for division). Addition and subtraction are harder—factor denominators, find LCD as the LCM of denominator factors, rewrite each fraction with LCD, then add/subtract numerators. Always simplify at the end by factoring the numerator and canceling any new common factors with the denominator. This systematic approach prevents errors!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. The golden rule: factor everything BEFORE operating! Multiplying (x² - 4)/(x + 3) by (x + 3)/(x + 2) looks messy, but factor x² - 4 = (x + 2)(x - 2) first: [(x + 2)(x - 2)/(x + 3)] · [(x + 3)/(x + 2)]. Now (x + 3) and (x + 2) cancel immediately, leaving just (x - 2). Factoring first prevents working with huge expressions! To find the LCD, factor denominators: x²-4 = (x-2)(x+2), x²+4x+4 = $(x+2)^2$; the least common multiple takes the highest powers, so $(x-2)(x+2)^2$. Choice C correctly factors the denominators and takes the LCM with highest powers, giving $(x-2)(x+2)^2$. Choice B fails by not using the highest power of (x+2), perhaps overlooking the square—always check exponents in factoring! The rational expression operation hierarchy: Multiplication and division are easier—factor everything, cancel common factors, then multiply (or flip and multiply for division). Addition and subtraction are harder—factor denominators, find LCD as the LCM of denominator factors, rewrite each fraction with LCD, then add/subtract numerators. Always simplify at the end by factoring the numerator and canceling any new common factors with the denominator. This systematic approach prevents errors!