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

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Answer 1 · 2018-06-26T10:04:57

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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]}