If &(a)/(10^x-1)+(b)/(10^x+2) & =2×10^x+3 (10^x-1)(10^x+2) is an identity for positive rational values of x, then the value of a-b is:

1967 AMC 12 Problem 10

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10.

\begin{aligned} &\frac{a}{10^x-1}+\frac{b}{10^x+2}\\ &\qquad=\frac{2\cdot10^x+3} {(10^x-1)(10^x+2)} \end{aligned}

is an identity for positive rational values of

x,

then the value of

a-b

is:

\dfrac43

\dfrac53

2

\dfrac{11}{4}

3

Multiply by

(10^x-1)(10^x+2)

Compare the coefficient of

10^x

and the constant term

Clearing denominators gives

\begin{aligned} &a(10^x+2)+b(10^x-1)\\ &\qquad=2\cdot10^x+3. \end{aligned}

Therefore

a+b=2

and

2a-b=3.

Solving gives

a=\frac{5}{3},\ b=\frac{1}{3},

a-b=\frac{4}{3}.

Therefore, the correct answer is A.