x+45/x

Question

x+45/x<14 written in interval notation

amy2012 · Answer

$$x+\frac{ 45 }{x }<14$$

mww · Answer

These can be tricky when you first encounter them. Be aware that non-linear inequalities have solution spans that can involve multiple directions.

There are two ways to deal with these.
1) Convert it to a more familiar polynomial, say a quadratic or cubic. The trick is to multiple everything on LHS and RHS by x^2 (or lowest possible even power of x). This is to ensure than you are multiplying by a positive number so the sign of the inequality remains the same. If you multiply both sides by x only, we don't know whether x is positive or negative so the sign of the inequality is in doubt. 

$$x + \frac{ 45 }{ x } < 14 $$
$$x^2(x+\frac{ 45 }{ x }) < 14x^2 ~so~x^3 - 14x^2 + 45x < 0 \rightarrow  x(x^2 - 14x + 45) < 0$$

Factorise further to get this
$$x(x^2 -14x +45)  = x(x-9)(x-5) < 0$$

Now sketch the graph and determine the regions which are below 0:
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