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I have this set $E=\{x\in\mathbb R:|2x+1|>|x+2|\}$.

I want to decompose this inequality in something like this: \begin{align} |x+1|>0\Leftrightarrow \end{align} \begin{array}{col1col2col3col4col5} x & + & 1 & > & 0\\ -x & - & 1 & > & 0 \end{array}

But I don't what steps to take.

1 Answers 1

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One easy way out in dealing with inequalities of the form $\vert x \vert > \vert y \vert$ is to square them since $\vert x \vert > \vert y \vert \iff x^2 > y^2$ In your case, we get that $(2x+1)^2 > (x+2)^2$ Rearranging, we get that $4x^2 + 4x + 1 > x^2 + 4x + 4 \implies 3x^2 > 3 \implies x^2 > 1 \implies x \in (-\infty,-1) \cup (1, \infty)$

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    I see, but don't we have to do \sqrt{(2x+1)^2} > \sqrt{(x+2)^2}?2012-11-06