Answer to Question #244365 in Real Analysis for dasuuu

Question

how to get domain in |x-2|/|x-5|<2?

Answer 1

|x - 2| = x - 2, x - 2 >= 0

= - (x - 2), x - 2 < 0

Similarly, we can write for |x - 5|

We can write the given inequality as,

(x - 2)/(x - 5) < 2

(x - 2)/(x - 5) - 2 < 0

(-x + 8)/(x - 5) < 0

x < 5 or x > 8......(1)

(x - 2)/-(x - 5) < 2

(x - 2)/(5 - x) - 2 < 0

(x - 4)/(5 - x) < 0

x > 5 or x < 4........(2)

-(x - 2)/(x - 5) < 2

(2 - x)/(5 - x) - 2 < 0

(-x + 4)/(x - 5) < 0

x < 4 or x > 5.........(3)

Finding intersection in (1), (2) and (3),

required domain is x < 4 or x > 8

Answer

Required domain is x < 4 or x > 8