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Si la ecuación:

[tex]\[
\frac{-2(x-2)}{9}-\frac{3-x}{6}+\frac{2(x-1)}{3}+\frac{x+2}{12}=1
\][/tex]

posee [tex]\( C.S = \left\{ \frac{m}{n} \right\} \)[/tex], calcule el valor de [tex]\( \sqrt{m+n} \)[/tex].


Sagot :

Para resolver la ecuación dada:
[tex]\[ \frac{-2(x-2)}{9} - \frac{3-x}{6} + \frac{2(x-1)}{3} + \frac{x+2}{12} = 1 \][/tex]
sigamos los siguientes pasos.

Paso 1: Simplifiquemos cada fracción.

[tex]\[ \frac{-2(x-2)}{9} = \frac{-2x + 4}{9} \][/tex]

[tex]\[ \frac{3-x}{6} \][/tex]

[tex]\[ \frac{2(x-1)}{3} = \frac{2x - 2}{3} \][/tex]

[tex]\[ \frac{x+2}{12} \][/tex]

Paso 2: Busquemos un denominador común. Todos los denominadores son múltiplos de 36 (el MCM de 9, 6, 3 y 12):

[tex]\[ \frac{-2x + 4}{9} = \frac{-8x + 16}{36} \][/tex]
[tex]\[ \frac{3 - x}{6} = \frac{6(3 - x)}{36} = \frac{18 - 6x}{36} \][/tex]
[tex]\[ \frac{2x - 2}{3} = \frac{12(2x - 2)}{36} = \frac{24x - 24}{36} \][/tex]
[tex]\[ \frac{x + 2}{12} = \frac{3(x + 2)}{36} = \frac{3x + 6}{36} \][/tex]

Paso 3: Rescribimos la ecuación con un denominador común:

[tex]\[ \frac{-8x + 16}{36} - \frac{18 - 6x}{36} + \frac{24x - 24}{36} + \frac{3x + 6}{36} = 1 \][/tex]

Paso 4: Combinamos los numeradores:

[tex]\[ \frac{-8x + 16 - 18 + 6x + 24x - 24 + 3x + 6}{36} = 1 \][/tex]

Simplificamos el numerador:

[tex]\[ \frac{(6x + 24x + 3x - 8x) + (16 - 18 - 24 + 6)}{36} = 1 \][/tex]

[tex]\[ \frac{25x - 20}{36} = 1 \][/tex]

Paso 5: Resolviendo la fracción igualada a 1:

Multiplicamos ambos lados de la ecuación por 36:

[tex]\[ 25x - 20 = 36 \][/tex]

Paso 6: Sumamos 20 a ambos lados:

[tex]\[ 25x = 56 \][/tex]

Paso 7: Dividimos ambos lados entre 25:

[tex]\[ x = \frac{56}{25} \][/tex]

Entonces, [tex]\( C.S. = \left\{\frac{56}{25}\right\} \)[/tex].

Paso 8: Calculamos [tex]\( m \)[/tex] y [tex]\( n \)[/tex]:

[tex]\( m = 56 \)[/tex] y [tex]\( n = 25 \)[/tex].

Paso 9: Finalmente, calculamos [tex]\( \sqrt{m + n} \)[/tex]:

[tex]\[ \sqrt{m + n} = \sqrt{56 + 25} = \sqrt{81} = 9 \][/tex]

Por lo tanto, el valor de [tex]\(\sqrt{m+n}\)[/tex] es [tex]\( \boxed{9} \)[/tex].
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