Question

lalis
The least integer value of x, which satisfy |x|+|x/x-1|=x^2/|x-1|​

Answer 1

given modulus function is , |x| + |x/(x - 1)| = x²/|x - 1|

or, |x| + |x|/|x - 1| = x²/|x - 1|

case 1 : x > 1

x + x/(x - 1) = x²/(x - 1)

or, x(x - 1) + x = x²

or, x² - x + x = x²

or, 0 = 0, it is true for all real numbers .

so, x > 1 is answer in this case.

case 2 : 0 ≤ x < 1

x + x/(1 - x) = x²/(1 - x)

or, x(1 - x) + x = x²

or, x - x² + x = x²

or, 2x - 2x² = 0

or, x = 1 , 0

but x ≠ 1 because function will be undefined.

so, x = 0 is true for this case.

case 3 : x < 0

-x + (-x)/(1 - x) = x²/(1 - x)

or, x(x - 1) - x = x²

or, x² - x - x = x²

or, x = 0

but x < 0 , so there is no solution in this case.

from all aboves cases we see least integer is x = 0 (see case 2) , which satisfy given function.

hence, answer is 0.