Simple inequalities question I promise

[smashbrohamme]

(x+2)/(x+4) greater or equal to 1.

I got two different answers here.

X is greater than 4.

Or a interval notation (-Infinite, 4) - which doesn't make sense but wouldn't the correct answer just be X is greater than 4?

which would mean (4, infinite)?

Homework Equations

The Attempt at a Solution

 

[rock.freak667]

if x = 5 then you'd get 7/9 which is not greater than or equal to one, so I think you messed up your simplification.

Post your working and see if you can find the error.

 

[HallsofIvy]

I think you just have a silly problem! For all x, x+ 2< x+ 4 so you have a fraction in which the denominator is larger than the numerator.

 

[smashbrohamme]

lol shoot! I made a silly typo...its (x+2)/(x-4) greater then or equal to 1

 

[smashbrohamme]

so how would you write that in interval notation, (4, Infinite)?

 

[I like Serena]

I think you just have a silly problem! For all x, x+ 2< x+ 4 so you have a fraction in which the denominator is larger than the numerator.

If x < -4, the inequality changes its direction, so that would be the solution...

 

[Dickfore]

$$\begin{align*}<br /> \frac{x+2}{x+4} \ge 1 \Leftrightarrow \\<br /> <br /> \frac{x+2}{x+4}-1 \ge 0 \Leftrightarrow \\<br /> <br /> \frac{(x+2)-(x+4)}{x+4} \ge 0 \Leftrightarrow \\<br /> <br /> \frac{x+2-x-4}{x+4} \ge 0 \Leftrightarrow \\<br /> <br /> \frac{-2}{x+4} \ge 0<br /> \end{align*}$$

When does this inequality hold, i.e. when is the ratio of a negative number and another one nonnegative?