please solve the question:

Asked by manasd | 17th Sep, 2015, 11:04: PM

Expert Answer:

T h e space g i v e n space i n e q u a l i t y space i s space open vertical bar x minus 6 close vertical bar greater than open vertical bar x squared minus 5 x plus 9 close vertical bar T h e space q u a d r a t i c space e x p r e s s i o n space i n space t h e space r i g h t space h a n d space s i d e space x squared minus 5 x plus 9 space h a d space D equals open parentheses negative 5 close parentheses squared minus 4 open parentheses 1 close parentheses open parentheses 9 close parentheses equals 25 minus 36 equals negative 11 space left parenthesis n e g a t i v e right parenthesis comma space a equals 1 space left parenthesis p o s i t i v e right parenthesis H e n c e comma space i t space i s space a l w a y s space g r e a t e r space t h a n space z e r o. space H e n c e comma space open vertical bar x squared minus 5 x plus 9 close vertical bar equals x squared minus 5 x plus 9 space f o r space a l l space r e a l space v a l u e s space o f space apostrophe x apostrophe. N o w space t h e space l e f t space h a n d space s i d e space open vertical bar x minus 6 close vertical bar equals x minus 6 comma space f o r space x greater or equal than 6 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 6 minus x comma space f o r space x less than 6 C a s e space 1 colon space x greater or equal than 6 T h e space i n e q u a l i t i y space b e c o m e s x minus 6 greater than space x squared minus 5 x plus 9 rightwards double arrow x squared minus 6 x plus 15 less than 0... left parenthesis 1 right parenthesis D space f o r space t h e space a b o v e space q u a d r a t i c space i s space open parentheses negative 6 close parentheses squared minus 4 open parentheses 1 close parentheses open parentheses 15 close parentheses equals 36 minus 60 equals negative 24 space w h i c h space i s space n e g a t i v e. space a equals 1 space left parenthesis p o s i t i v e right parenthesis H e n c e comma space t h e space a b o v e space q u a d r a t i c space x squared minus 6 x plus 15 greater than 0 comma space f o r space a l l space v a l u e s space o f space apostrophe x apostrophe. space S o comma space x squared minus 6 x plus 15 less than 0 space i s space n o t space p o s s i b l e space f o r space a n y space v a l u e space o f space apostrophe x apostrophe.  C a s e space 2 colon space x less than 6 T h e space i n e q u a l i t y space b e c o m e s space 6 minus x greater than x squared minus 5 x plus 9 rightwards double arrow x squared minus 4 x plus 3 less than 0 rightwards double arrow open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses less than 0 rightwards double arrow x element of open parentheses 1 comma 3 close parentheses S o space f r o m space c a s e space 1 space a n d space c a s e space 2 space comma space t h e space v a l u e s space o f space x space s a t i s f y i n g space t h e space i n e q u a l i t y space x element of open parentheses 1 comma 3 close parentheses space

Answered by satyajit samal | 18th Sep, 2015, 01:17: PM