| |x-4|-2|<3 I need help with this absolute value

  -1 < x < 9

Step-by-step explanation:

Eliminate the outer absolute value symbol.

  -3 < |x -4| -2 < 3

Add 2

  -1 < |x -4| < 5

We know that no absolute value can be negative, so this is fully equivalent to ...

  |x -4| < 5

Eliminate the absolute value symbol.

  -5 < x -4 < 5

Add 4

  -1 < x < 9

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For the graph, we graph the inequality ...

  ||x -4| -2| -3 < 0

The zero crossings are easy to find in this graphing calculator.

it would rotate about 180 degrees.

step-by-step explanation:

points a and b are equidistant from the desired line, so the line you find by using them will go through point r.

using points other than a and b will give the desired line only if they are equidistant from r. if they are randomly located, the perpendicular line you create will not necessarily go through point r, hence will not be a solution to the problem.