Simplify.
\[
\frac{1+\frac{x}{x+1}}{1+\frac{x-1}{x+2}}
\]

Answer

Expert–verified

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Answer

\[\boxed{1}\]

Steps

Step 1 ：\[\frac{1+\frac{x}{x+1}}{1+\frac{x-1}{x+2}} = \frac{(x+1)+x}{(x+1)(x+2)} \div \frac{(x+2)+(x-1)}{(x+1)(x+2)}\]

Step 2 ：\[= \frac{2x+1}{(x+1)(x+2)} \div \frac{2x+1}{(x+1)(x+2)}\]

Step 3 ：\[= \frac{2x+1}{(x+1)(x+2)} \times \frac{(x+1)(x+2)}{2x+1}\]

Step 4 ：\[= \frac{1}{1} = 1\]

Step 5 ：\[\boxed{1}\]