Inequality-P1

Problem:


Given that x>1
Find the lowest value of x+4/(x-1)


Solution:


x>1
x-1>0
let a=(x-1) b=4/(x-1)
Since (√a-√b)^2>0 and a≠b
a-2√ab+b>0
a+b>2√ab
By plugging in a and b
(x-1)+4/(x-1)>2√4(x-1)/(x-1)=4
(x-1)+4/(x-1)>4
Since (x-1)+4/(x-1)+1=x+4/(x-1)
x+4/(x-1)>5