Question

Solve the following system of equation in R. 

\[\frac{2x + 1}{7x - 1} > 5, \frac{x + 7}{x - 8} > 2\] 

Answer 1

`(2x + 1)/(7x - 1) - 5 > 0`

⇒ `((2x + 1) - 5(7x - 1))/(7x - 1)>0`

⇒ `(2x + 1 - 35x + 5)/(7x - 1)>0`

⇒ `(-33x + 6)/(7x - 1)>0`

⇒ `(11x - 2)/(7x - 1)<0`

⇒ 11x − 2 < 0 and 7x − 1 > 0 or 11x − 2 > 0 and 7x − 1 < 0

⇒ x < 2/11 and x > 1/7 or x > 2/11 and x < 1/7

⇒ x ∈ (1/7, 2/11)     ...(i)

Also `(x + 7)/(x - 8)>2`

⇒ `(x + 7)/(x - 8) - 2 > 0`

⇒ `(x + 7 - 2 (x - 8))/(x - 8)>0`

⇒ `(x + 7 - 2x + 16)/(x - 8)>0`

⇒ `(-x + 23)/(x - 8)>0`

⇒ `(x - 23)/(x - 8)<0`

⇒ x − 23 < 0 and x − 8 > 0 or x − 23 > 0 and x − 8 < 0

⇒ x < 23 and x > 8 or x > 23 and x < 8

⇒ x ∈ (8, 23)     ...(ii)

From (i) and (ii), we can find that there is no common set of values of x. So, the given system of equation has no solution