x+1/2*x-1<0 inequation

Given the inequality:
$$\left(\frac{x}{2} + x\right) - 1 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{x}{2} + x\right) - 1 = 0$$
Solve:
Given the linear equation:

x+1/2*x-1 = 0

Looking for similar summands in the left part:

-1 + 3*x/2 = 0

Move free summands (without x)
from left part to right part, we given:
$$\frac{3 x}{2} = 1$$
Divide both parts of the equation by 3/2

x = 1 / (3/2)

$$x_{1} = \frac{2}{3}$$
$$x_{1} = \frac{2}{3}$$
This roots
$$x_{1} = \frac{2}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{2}{3}$$
=
$$\frac{17}{30}$$
substitute to the expression
$$\left(\frac{x}{2} + x\right) - 1 < 0$$
$$-1 + \left(\frac{17}{2 \cdot 30} + \frac{17}{30}\right) < 0$$

-3/20 < 0

the solution of our inequality is:
$$x < \frac{2}{3}$$

 _____          
      \    
-------ο-------
       x1