We all know that if we want to solve absolute value inequality with two absolute values on both sides we have to squaring them
Like this ->>>
$\begin{aligned}
|2x+1| &\ge|x-2| \\
(2x+1)^2 &\ge (x-2)^2 \\
\end{aligned}$
But how if i want to solve them "without squaring both sides"
I have read some books and theory about solve them without squaring but i still don't understand. If you know some, please feel free to tell me. Thanks
Best Answer
First, set
$$ 2x+1 = \pm (x-2)$$
to obtain the ‘break’ points. They are $x=-3$ and $x=1/3$. This is an important step for dealing with multiple absolute terms appearing in the inequality.
Then, check whether, in each of the three regions separated by the break points, the inequality holds. This could be easily done with some convenient check points. For example, $-5$ for the left region, $0$ for the middle, and $1$ for the right, as shown below,
$$ |2(-5)+1|\ge |-5-2|, \space \text{true}$$
$$ |2\times 0+1|\ge |0-2|, \space \text{false}$$
$$ |2\times 1+1|\ge |1-2|, \space \text{true}$$
The solutions then follow
$$ x \le -3, \space \space x \ge 1/3$$