(2x+1)/(2x+5)>(x)/(2x+5)

Question

Solve the nonlinear inequality and find the x.

Answer 1

(2x+1) / (2x+5) > x / (2x+5)

The first I'd do is eliminate the (2x+5) factor on both sides. This leads to two cases, depending on whether 2x+5 is positive or negative. (If 2x+5=0, both sides are meaningless.)

1. If 2x+5>0, multiply both sides by (2x+5) to get 2x+1>x. Subtracting x+1 on both sides, we get x > -1. This agrees with 2x+5>0,
2. If 2x+5<0, multiply both sides by (2x+5) to get 2x+1<x. Subtracting x+1 on both sides, we get x < -1. This is not enough, though. We need 2x+5<0, which is the same as x<-5/2.

So the answer is x>-1 or x<-5/2.

Answer 2

This problem may appear straightforward: just see if the numerator of the first expression is greater than the numerator of the second. However, remember that when you multiply by a negative number, inequalities reverse.

Thus,

2x+5 > 0 when 2x > -5 or x > -5/2

(2x+1)/(2x+5) > (x)/(2x+5)

Multiplying both sides by 2x+5,

2x+ 1 > x

2x +1 - 1 - x > x - 1 - x

x > -1; Note that this is a subset of x > -5/2, so this entire interval is part of the solution.

2x+5 < 0 when x < -5/2

(2x+1)/(2x+5) > (x)/(2x+5)

Multiplying both sides by 2x+5,

2x + 1 < x; note that the inequality reverses.

Then, x <-1. However, this is only true for x <-5/2, which is a subset of x <-1, so x < -5/2

Thus, our solution is

(-∞,-5/2) ∪ (-1, ∞)