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

Apr 9, 2015

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 $z \ge 0 \implies | z | = z$.
if $z < 0 \implies | z | = - z$.
Applying this definition to our problema, we have:
if $\left(x - 4\right) \ge 0 \implies | x - 4 | > 5 \implies x - 4 > 5 \implies x > 9$.
if $\left(x - 4\right) < 0 \implies | x - 4 | > 5 \implies - \left(x - 4\right) > 5 \implies - x + 4 > 5 \implies - x > 1 \implies x < - 1$

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.