# Question 78b61

Apr 26, 2017

$x < 3$

#### Explanation:

color(blue)((x+3)/x<2

This is an inequality problem. Solve it by balancing both sides by the same operation

Simplify the left hand side

$\rightarrow \frac{x}{x} + \frac{3}{x} < 2$

$\rightarrow 1 + \frac{3}{x} < 2$

Subtract $1$ from both sides

$\rightarrow \frac{3}{x} < 1$

Divide both sides by $3$

$\rightarrow \frac{1}{x} < \frac{1}{3}$

Multiply both sides by $3 x$ (Reverse the sides)

color(green)(rArrx<3#

Apr 26, 2017

The solution is $x \in \left(- \infty , 0\right) \cup \left(3 , + \infty\right)$

#### Explanation:

We cannot do crossing over.

So,

$\frac{x + 3}{x} < 2$

$\frac{x + 3}{x} - 2 < 0$

$\frac{x + 3 - 2 x}{x} < 0$

$\frac{3 - x}{x} < 0$

Let,

$f \left(x\right) = \frac{3 - x}{x}$

We build a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$3 - x$$\textcolor{w h i t e}{a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

$f \left(x\right) < 0$ when $x \in \left(- \infty , 0\right) \cup \left(3 , + \infty\right)$
graph{(3-x)/x [-16.02, 16.02, -8.01, 8.01]}