Evaluate the Expression

Algebra Level 3

1 1 + a − 1 − 1 a 1 + 1 a \large \frac{1}{1+a}-\frac{1-\frac{1}{a}}{1+\frac{1}{a}}

Simplify the expression above.

a a + 1 \frac{a}{a+1} 3 a + 1 − 1 \frac{3}{a+1}-1 − a a + 1 \frac{-a}{a+1} 2 + a a + 1 \frac{2+a}{a+1} a − 2 a + 1 \frac{a-2}{a+1}

Omar Sehlouli by Omar Sehlouli , 21, Morocco

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1 solution

1 1 + a − 1 − 1 a 1 + 1 a = 1 1 + a − a − 1 a a + 1 a = 1 1 + a − a − 1 a + 1 = 2 − a a + 1 = 3 − a − 1 a + 1 = 3 a + 1 − 1 \begin{aligned} \frac 1{1+a} - \frac {1-\frac 1a}{1+\frac 1a} & = \frac 1{1+a} - \frac {\frac{a-1}a}{\frac {a+1}a} \\ & = \frac 1{1+a} - \frac {a-1}{a+1} \\ & = \frac {2-a}{a+1} \\ & = \frac {3-a-1}{a+1} \\ & = \boxed{\dfrac 3{a+1}-1} \end{aligned}

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