Question

How do you graph the inequalities `2abs(x-4)>10` on a number line?

Answer 1

There are two solutions: `x < -1` and `x > 9`.

The reasoning is the following:
First, you can simplify both members of the inequality by 2, obtaining `|x-4| > 5`.
Then, we must apply the definition of the absolute value that is:
if `z >=0 => |z| = z`.
if `z < 0 => |z| = -z`.
Applying this definition to our problema, we have:
if `(x-4) >=0 => |x-4| > 5 => x-4 > 5 => x > 9`.
if `(x-4) < 0 => |x-4| > 5 => -(x-4) > 5 => -x+4 > 5 => -x > 1 => x < -1`

Sorry but I don't know how to insert the graph. Anyway, it is very easy to represent it when you know the solution: you only have to draw a horizontal line, mark the point (-1) on the left side, and the point (+9) on the right side (with a regular distance between both), and then drawing thicker the portion of the line from the left extreme until the point (-1), and also drawing thicker the portion of the line from the point (+9) until the right extreme.