Solving the Inequality Problem: Finding the Solution Set for x/(2-x)<4 | Skook"

[skook]

Could someone tell me how to find the solution set for the following, please.

$$\frac{x}{2-x}<4$$

thanks
skook

 

[HallsofIvy]

A good way to solve non-linear inequalities such as this is to solve the corresponding equation,
[tex]\frac{x}{2-x}= 4[/itex]
The function
$$f(x)= \frac{x}{2-x}- 4$$
is continuous every where except at x= 2 and so can only change sign at x= 2 or where it is equal to 0. In other words, x= 2 and the solution to the equation divide the number line into intervals on which f(x) is always positive or always negative. By checking one point in each interval you can decide which.

 

[Steve10]

Start by multiplying both sides by (2-x).

However you must bear in mind that, depending upon the possible values of x, that the term (2-x) could be either positive or negative.
And, when you divide an inequality by a negative number, then you change the direction of the inequality symbol.

 

[skook]

Got it

Multiply both sides by $$(2-x)^2$$ and then factorise to get solution $$x \in (- \infty, \frac{8}{5}) \bigcup (2,\infty)$$.

thanks
skook