The equality 1x-1=2x-2 is satisfied by:

1955 AMC 12 Problem 4

Attempt Problem 4 of the 1955 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1955 AMC 12 solutions, or check the answer key.

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The equality

\dfrac1{x-1}=\dfrac2{x-2}

is satisfied by:

no real values of

x

either

x=1

x=2

only

x=1

only

x=2

only

x=0

Cross-multiply, while remembering that

x\ne1,2

Solve

x-2=2x-2

For

x\ne1,2,

cross-multiplication gives

x-2=2(x-1)=2x-2,

x=0.

This value makes both original denominators nonzero and satisfies the equality.

Thus, the correct answer is E.