Long Washington
03/30/2024 · Senior High School
\( \left(\frac{x+2}{4}-\frac{x+1}{2}\right) \frac{x+1}{5} \) b) \( \left(\frac{1}{4}-\frac{x+1}{2}\right)\left(x-\frac{x+1}{3}\right)-\left(\frac{1}{4}-\frac{x+1}{2}\right): \frac{1}{3} \) c) \( \frac{1}{2}\left(\frac{x}{2}-\frac{x+1}{3}\right)-\frac{x}{3}\left(3-\frac{x+1}{4}\right) \)
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Step-by-step Solution
Vamos a simplificar cada una de las expresiones que has proporcionado.
### a)
\[
\left(\frac{x+2}{4}-\frac{x+1}{2}\right) \frac{x+1}{5}
\]
Primero, simplificamos la parte dentro del paréntesis:
\[
\frac{x+2}{4} - \frac{x+1}{2} = \frac{x+2}{4} - \frac{2(x+1)}{4} = \frac{x+2 - 2(x+1)}{4} = \frac{x+2 - 2x - 2}{4} = \frac{-x}{4}
\]
Ahora, multiplicamos por \(\frac{x+1}{5}\):
\[
\frac{-x}{4} \cdot \frac{x+1}{5} = \frac{-x(x+1)}{20}
\]
### b)
\[
\left(\frac{1}{4}-\frac{x+1}{2}\right)\left(x-\frac{x+1}{3}\right)-\left(\frac{1}{4}-\frac{x+1}{2}\right): \frac{1}{3}
\]
Primero, simplificamos \(\frac{1}{4} - \frac{x+1}{2}\):
\[
\frac{1}{4} - \frac{x+1}{2} = \frac{1}{4} - \frac{2(x+1)}{4} = \frac{1 - 2(x+1)}{4} = \frac{1 - 2x - 2}{4} = \frac{-2x - 1}{4}
\]
Ahora simplificamos \(x - \frac{x+1}{3}\):
\[
x - \frac{x+1}{3} = \frac{3x - (x+1)}{3} = \frac{3x - x - 1}{3} = \frac{2x - 1}{3}
\]
Multiplicamos las dos partes:
\[
\left(\frac{-2x - 1}{4}\right)\left(\frac{2x - 1}{3}\right) = \frac{(-2x - 1)(2x - 1)}{12}
\]
Ahora, simplificamos la segunda parte:
\[
\left(\frac{1}{4} - \frac{x+1}{2}\right) : \frac{1}{3} = \left(\frac{-2x - 1}{4}\right) \cdot 3 = \frac{-3(2x + 1)}{4} = \frac{-6x - 3}{4}
\]
Finalmente, combinamos ambas partes:
\[
\frac{(-2x - 1)(2x - 1)}{12} - \frac{-6x - 3}{4}
\]
Para restar, necesitamos un denominador común. El común es 12:
\[
\frac{(-2x - 1)(2x - 1)}{12} + \frac{6x + 3}{12} = \frac{(-2x - 1)(2x - 1) + (6x + 3)}{12}
\]
### c)
\[
\frac{1}{2}\left(\frac{x}{2}-\frac{x+1}{3}\right)-\frac{x}{3}\left(3-\frac{x+1}{4}\right)
\]
Primero simplificamos \(\frac{x}{2} - \frac{x+1}{3}\):
\[
\frac{x}{2} - \frac{x+1}{3} = \frac{3x - 2(x+1)}{6} = \frac{3x - 2x - 2}{6} = \frac{x - 2}{6}
\]
Multiplicamos por \(\frac{1}{2}\):
\[
\frac{1}{2} \cdot \frac{x - 2}{6} = \frac{x - 2}{12}
\]
Ahora simplificamos \(3 - \frac{x+1}{4}\):
\[
3 - \frac{x+1}{4} = \frac{12 - (x + 1)}{4} = \frac{11 - x}{4}
\]
Multiplicamos por \(-\frac{x}{3}\):
\[
-\frac{x}{3} \cdot \frac{11 - x}{4} = -\frac{x(11 - x)}{12}
\]
Finalmente, combinamos ambas partes:
\[
\frac{x - 2}{12} - \left(-\frac{x(11 - x)}{12}\right) = \frac{x - 2 + x(11 - x)}{12}
\]
Ahora tenemos las expresiones simplificadas:
### Resultados finales:
a) \(\frac{-x(x+1)}{20}\)
b) \(\frac{(-2x - 1)(2x - 1) + (6x + 3)}{12}\)
c) \(\frac{x - 2 + x(11 - x)}{12}\)
Quick Answer
a) \(\frac{-x(x+1)}{20}\)
b) \(\frac{(-2x - 1)(2x - 1) + (6x + 3)}{12}\)
c) \(\frac{x - 2 + x(11 - x)}{12}\)
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