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

Jun 26, 2018

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

$| 8 n + 4 | = 8 n + 4 \text{ if " 8n+4>0,\quad -8n-4 " if } 8 n + 4 < 0$

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

$8 n + 4 = 0 \setminus \iff 8 n = - 4 \setminus \iff n = - \frac{1}{2}$

So, this line is positive from $- \frac{1}{2}$ on, and negative before $- \frac{1}{2}$.

This means that

$| 8 n + 4 | = 8 n + 4 \text{ if " n> -1/2,\quad -8n-4 " if } n < - \frac{1}{2}$

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