Algebraic Expressions - Fractions

Algebra Level 1

Simplify

1 − 1 1 − 1 1 − x . \large 1 - \frac{1}{1 - \frac{1}{1-x}}.

1 x \frac1x − 1 x -\frac1x 1 1 − x \frac1{1-x} x x

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2 solutions

Mahdi Raza
Aug 4, 2020

1 − 1 1 − 1 1 − x ⟹ 1 − 1 1 − x − 1 1 − x ⟹ 1 − 1 − x − x ⟹ − x − 1 + x − x ⟹ 1 x 1 - \dfrac{1}{1- \dfrac{1}{1-x}} \implies 1 - \dfrac{1}{\dfrac{1-x - 1}{1-x}} \implies 1 - \dfrac{1-x}{-x} \implies \dfrac{-x -1+x}{-x} \implies \boxed{\dfrac{1}{x}}

0 Helpful 0 Interesting 2 Brilliant 1 Confused
Brilliant Mathematics Staff
Aug 1, 2020

We have

1 − 1 1 − 1 1 − x = 1 − 1 1 − x − 1 1 − x = 1 − 1 − x ( − x ) = x + ( 1 − x ) x = 1 x . \begin{aligned} 1 - \frac{1}{1 - \dfrac{1}{1-x}} &= 1 - \frac{1}{\dfrac{1-x-1}{1-x}} \\ \\ &= 1 - \frac{1-x}{(-x)} \\ \\ &= \frac{x + (1-x)}{x} \\ \\ &= \frac{1}{x}. \end{aligned}

0 Helpful 0 Interesting 1 Brilliant 1 Confused

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