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Veámoslo así
![E=\left|\dfrac{x+1}{x+3}\right|\\ \\ \\
E=\left|1-\dfrac{2}{x+3}\right|\\ \\ E=\left|\dfrac{x+1}{x+3}\right|\\ \\ \\
E=\left|1-\dfrac{2}{x+3}\right|\\ \\](https://tex.z-dn.net/?f=E%3D%5Cleft%7C%5Cdfrac%7Bx%2B1%7D%7Bx%2B3%7D%5Cright%7C%5C%5C+%5C%5C+%5C%5C%0AE%3D%5Cleft%7C1-%5Cdfrac%7B2%7D%7Bx%2B3%7D%5Cright%7C%5C%5C+%5C%5C)
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![\#1: -2\leqx\ \textless \ 0\\ \\
\#2: -2+3\leqx+3\ \textless \ 0+3\\ \\
\#3: 1\leq x+3\ \textless \ 3\\ \\ \\
\#4: \dfrac{1}{3}\ \textless \ \dfrac{1}{x+3}\leq1\\ \\ \\
\#5: -1\leq-\dfrac{1}{x+3}\ \textless \ -\dfrac{1}{3}\\ \\ \\
\#6: -2\leq-\dfrac{2}{x+3}\ \textless \ -\dfrac{2}{3}\\ \\ \\
\#7: 1-2\leq1-\dfrac{2}{x+3}\ \textless \ 1-\dfrac{2}{3}\\ \\ \\
\#8: -1\leq\dfrac{x+1}{x+3}\ \textless \ \dfrac{1}{3}\\ \\ \\
\#9: -1\leq\dfrac{x+1}{x+3}\ \textless \ \dfrac{1}{3}\leq 1\\ \\ \\
\#10: \left|\dfrac{x+1}{x+3}\right|\leq 1
\#1: -2\leqx\ \textless \ 0\\ \\
\#2: -2+3\leqx+3\ \textless \ 0+3\\ \\
\#3: 1\leq x+3\ \textless \ 3\\ \\ \\
\#4: \dfrac{1}{3}\ \textless \ \dfrac{1}{x+3}\leq1\\ \\ \\
\#5: -1\leq-\dfrac{1}{x+3}\ \textless \ -\dfrac{1}{3}\\ \\ \\
\#6: -2\leq-\dfrac{2}{x+3}\ \textless \ -\dfrac{2}{3}\\ \\ \\
\#7: 1-2\leq1-\dfrac{2}{x+3}\ \textless \ 1-\dfrac{2}{3}\\ \\ \\
\#8: -1\leq\dfrac{x+1}{x+3}\ \textless \ \dfrac{1}{3}\\ \\ \\
\#9: -1\leq\dfrac{x+1}{x+3}\ \textless \ \dfrac{1}{3}\leq 1\\ \\ \\
\#10: \left|\dfrac{x+1}{x+3}\right|\leq 1](https://tex.z-dn.net/?f=%5C%231%3A+-2%5Cleqx%5C+%5Ctextless+%5C+0%5C%5C+%5C%5C%0A%5C%232%3A+-2%2B3%5Cleqx%2B3%5C+%5Ctextless+%5C+0%2B3%5C%5C+%5C%5C%0A%5C%233%3A+1%5Cleq+x%2B3%5C+%5Ctextless+%5C+3%5C%5C+%5C%5C+%5C%5C%0A%5C%234%3A+%5Cdfrac%7B1%7D%7B3%7D%5C+%5Ctextless+%5C+%5Cdfrac%7B1%7D%7Bx%2B3%7D%5Cleq1%5C%5C+%5C%5C+%5C%5C%0A%5C%235%3A+-1%5Cleq-%5Cdfrac%7B1%7D%7Bx%2B3%7D%5C+%5Ctextless+%5C+-%5Cdfrac%7B1%7D%7B3%7D%5C%5C+%5C%5C+%5C%5C%0A%5C%236%3A+-2%5Cleq-%5Cdfrac%7B2%7D%7Bx%2B3%7D%5C+%5Ctextless+%5C+-%5Cdfrac%7B2%7D%7B3%7D%5C%5C+%5C%5C+%5C%5C%0A%5C%237%3A+1-2%5Cleq1-%5Cdfrac%7B2%7D%7Bx%2B3%7D%5C+%5Ctextless+%5C+1-%5Cdfrac%7B2%7D%7B3%7D%5C%5C+%5C%5C+%5C%5C%0A%5C%238%3A+-1%5Cleq%5Cdfrac%7Bx%2B1%7D%7Bx%2B3%7D%5C+%5Ctextless+%5C+%5Cdfrac%7B1%7D%7B3%7D%5C%5C+%5C%5C+%5C%5C%0A%5C%239%3A+-1%5Cleq%5Cdfrac%7Bx%2B1%7D%7Bx%2B3%7D%5C+%5Ctextless+%5C+%5Cdfrac%7B1%7D%7B3%7D%5Cleq+1%5C%5C+%5C%5C+%5C%5C%0A%5C%2310%3A+%5Cleft%7C%5Cdfrac%7Bx%2B1%7D%7Bx%2B3%7D%5Cright%7C%5Cleq+1%0A)
Entonces el menor valor es 0 y el mayor es 1
entonces partamos de...
Entonces el menor valor es 0 y el mayor es 1
PascualDavid:
Creo que en el #9 te equivocaste, debió ser -1/3≤(x+1)/(x+3)<1/3 y te hubiera quedado |(x+1)/(x+3)|<1/3 y el mayor valor no sería 1
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