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Find the solution set for the inequality:

$|x-2| > |x+6|$

I'm just not sure how to work this out?

I've found the answer is $(-\infty,-2)$ using wolframAlpha, however there isn't a step by step.

If you could explain how I would go about trying to find the answer for myself?

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Since both sides of the inequality are nonnegative, it is safe to square both sides, yielding: $$ x^2 - 4x + 4 > x^2 + 12x + 36 \iff -32 > 16x \iff x < -2 $$

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Separate the cases:

  1. $x \leq-6$ (so that $x-2$ is negative and $x+6$ is negative)
  2. $-6\leq x\leq 2$ (so that $x-2$ is negative and $x+6$ is positive)
  3. $x\geq 2$ (so that both expressions are positive)
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