Question

How do you graph and solve `|8n+4| > -4`?

Answer 1

See below

Explanation:

The solving part is quite easy: the absolute value of a certain quantity is always positive (or zero), so the inequality is always true.

In fact, you're basically asking: "When is a non-negative quantity greater than something negative?"

Well, by definition, the answer is "always", otherwise we would have a positive number which is smaller than a negative number.

As for the graphing part, recall what we said above to see that

`|8n+4| = 8n+4 " if " 8n+4>0,\quad -8n-4 " if " 8n+4<0`

The point where `8n+4` swithces sign is when it equals zero:

`8n+4=0 \iff 8n=-4 \iff n=-1/2`

So, this line is positive from `-1/2` on, and negative before `-1/2`.

This means that

`|8n+4| = 8n+4 " if " n> -1/2,\quad -8n-4 " if " n< -1/2`

graph{|8x+4| [-1 0.5 -0.7 4]}