2x+1/2x+5>x/2x+5

Question

Solve for x in the following problem.

Answer 1

You probably meant (2x+1)/(2x+5) > x/(2x+5)? You just left

out the parentheses?

if so, then multiply both sides by 2x+5 to get

2x+1 > x but only if 2x+5>0. or

x>-1. That's half the solution, and true when 2x+5>0

if you multiply both sides of an inequality by a negative number

when 2x+5<0 then the inequality reverses from > to <. When x<2.5 and the denominator <0

to have the left side > right side, the numerator needs to be negative and x<-2.5

the only values of x that do not satisfy the original inequality are -2.5<x<-1

the solution is (-infinity,-2.5)U(-1,+infinity), all real numbers except [-2.5,-1]

But maybe you did mean 2x +1/2x + 5> x/2x +5, which seems unlikely

that simplifies to

2x + 1/2x > 1/2

multiply by x when x>0, then

2x^2 - x/2 + 1/2 >0

x^2 -x/4 + 1/4 >0

solve for x with the quadratic formula or complete the square

but if x<0 then when you multiply by x, it reverses the inequality sign to get

2x^2-x/2 + 1/2<0

solve for x again and you get a different solution than when you put in the parentheses