# How to manipulate absolute values when shifting parts in an inequality

I have the following inequality... $|4x - 2| \le 0.5$

I want to manipulate this so it is just $|x|$ on one side, and everything else on the other, but I'm not sure how the absolute value complicates things. How would I go about this?

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Can we agree that, from the definition of absolute values, $|4x - 2| \le 0.5 \implies -0.5 \leq 4x-2 \leq 0.5$ ? – Galois Group Jan 25 '12 at 23:24
You cannot get $|x|$ on one side and numbers on the other. The nearest you can get is $|x-0.5|\le 0.125$. – André Nicolas Jan 25 '12 at 23:26

$$|4x-2| \leq 0.5 \Leftrightarrow -0.5 \leq 4x-2 \leq 0.5 \Leftrightarrow 1.5 \leq 4x \leq 2.5 \Leftrightarrow \frac{3}{8} \leq x \leq \frac{5}{8}$$