hello madam / sir
 
please explain me how to solve the following linear inequality
 
x-3 /x-5 > 0
 
x-2/x+5 > 2
 
modulus of 3x-2 > 1/2
 
 
 

Asked by kamalnayana | 29th Aug, 2015, 06:57: PM

Expert Answer:

(i)
C o n s i d e r space t h e space g i v e n space i n e q u a l i t y fraction numerator x minus 3 over denominator x minus 5 end fraction greater than 0 E q u a t i n g space t h e space n u m e r a t o r space a n d space t h e space d e n o m i n a t o r space s e p a r a t e l y comma space w e space h a v e comma space x minus 3 equals 0 space a n d space x minus 5 equals 0. T h u s comma space t h e space c r i t i c a l space p o i n t s space a r e space x equals 3 space a n d space x equals 5. D r a w space a space n u m b e r space l i n e space a n d space m a r k space t h e space p o i n t s space 3 space a n d space 5.
N o w space t h e space r e a l space l i n e space h a s space b e e n space d i v i d e d space i n space t o space t h r e e space r e g i o n s. I f space x space l i e s space i n space t h e space i n t e r v a l comma space left parenthesis negative infinity comma 3 right parenthesis comma space t h e n space t h e space f u n c t i o n comma space f left parenthesis x right parenthesis equals fraction numerator x minus 3 over denominator x minus 5 end fraction greater than 0 I f space x space l i e s space i n space t h e space i n t e r v a l comma space left parenthesis 3 comma 5 right parenthesis comma space t h e n space t h e space f u n c t i o n comma space f left parenthesis x right parenthesis equals fraction numerator x minus 3 over denominator x minus 5 end fraction less than 0 I f space x space l i e s space i n space t h e space i n t e r v a l comma space left parenthesis 5 comma infinity right parenthesis comma space t h e n space t h e space f u n c t i o n comma space f left parenthesis x right parenthesis equals fraction numerator x minus 3 over denominator x minus 5 end fraction greater than 0 S i n c e space t h e space e x p r e s s i o n space i s space p o s i t i v e comma space t h e space s o l u t i o n space s e t space o f space i s space t h e space u n i o n space o f space t h e space r e g i o n s space c o n t a i n i n g space p o s i t i v e space s i g n s. T h e r e f o r e comma space t h e space s o l u t i o n space s e t space i s space open parentheses negative infinity comma 3 close parentheses union open parentheses 5 comma infinity close parentheses
 
(ii)
 
C o n s i d e r space t h e space g i v e n space i n e q u a l i t y fraction numerator x minus 2 over denominator x plus 5 end fraction greater than 2 rightwards double arrow fraction numerator x minus 2 over denominator x plus 5 end fraction minus 2 greater than 0 rightwards double arrow fraction numerator x minus 2 minus 2 open parentheses x plus 5 close parentheses over denominator x plus 5 end fraction greater than 0 rightwards double arrow fraction numerator x minus 2 minus 2 x minus 10 over denominator x plus 5 end fraction greater than 0 rightwards double arrow fraction numerator negative x minus 12 over denominator x plus 5 end fraction greater than 0 rightwards double arrow fraction numerator x plus 12 over denominator x plus 5 end fraction less than 0 E q u a t i n g space t h e space n u m e r a t o r space a n d space t h e space d e n o m i n a t o r space s e p a r a t e l y comma space w e space h a v e comma space x plus 12 equals 0 space a n d space x plus 5 equals 0. T h u s comma space t h e space c r i t i c a l space p o i n t s space a r e space x equals negative 12 space a n d space x equals negative 5. D r a w space a space n u m b e r space l i n e space a n d space m a r k space t h e space p o i n t s space minus 12 space a n d space minus 5.
 
N o w space t h e space r e a l space l i n e space h a s space b e e n space d i v i d e d space i n space t o space t h r e e space r e g i o n s. I f space x space l i e s space i n space t h e space i n t e r v a l comma space left parenthesis negative infinity comma negative 12 right parenthesis comma space t h e n space t h e space f u n c t i o n comma space f left parenthesis x right parenthesis equals fraction numerator x plus 12 over denominator x plus 5 end fraction greater than 0 I f space x space l i e s space i n space t h e space i n t e r v a l comma space left parenthesis negative 12 comma negative 5 right parenthesis comma space t h e n space t h e space f u n c t i o n comma space f left parenthesis x right parenthesis equals fraction numerator x plus 12 over denominator x plus 5 end fraction less than 0 I f space x space l i e s space i n space t h e space i n t e r v a l comma space left parenthesis negative 5 comma infinity right parenthesis comma space t h e n space t h e space f u n c t i o n comma space f left parenthesis x right parenthesis equals fraction numerator x plus 12 over denominator x plus 5 end fraction greater than 0 S i n c e space t h e space e x p r e s s i o n space i s space n e g a t i v e comma space t h e space s o l u t i o n space s e t space o f space i s space t h e space u n i o n space o f space t h e space r e g i o n s space c o n t a i n i n g space n e g a t i v e space s i g n s. T h e r e f o r e comma space t h e space s o l u t i o n space s e t space i s space open parentheses negative 12 comma negative 5 close parentheses

Answered by Vimala Ramamurthy | 31st Aug, 2015, 10:22: AM