[ Inequalities with absolute values ]

Solve the inequality:

$|x+2000|<|x-2001|$

#
Let's solve the inequality using the number line. The given inequality holds for all points with coordinate \(x\), which are closer to the point with coordinate -2000 than to the point with coordinate 2001. Since 
\(=0.5\), the points we are looking for are all points located to the left of the point with coordinate 0.5 (see the figure).

There are other ways to solve this, in particular, by expanding the absolute values (three cases); squaring both sides of the inequality.

![](https://cdn.mathpix.com/cropped/2024_05_06_83e2b9d825e20bcabe9eg-06.jpg?height=48&width=363&top_left_y=199&top_left_x=848)

## Answer

\((- \infty; 0.5)\).