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The question is as follows: $y=x+4, y=|x^2 - 16|$

Currently, I understand how to solve basic modulus functions (as per IGCSE Additional Mathematics). However, I'm stumped on this question because I do not know how to solve simultaneous equations of this kind, and there are no examples for me to refer to.

I would appreciate if anyone answers this! It would really help my summer studies. This is also my first post on any Stack forum, and it definitely won't be my last.

RiverX15
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Mutsuki
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  • Draw graph or solve by using cases. – S.M. Jul 05 '21 at 06:17
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    This would be easier to solve if you split your abs function in 2 parts. for -4 < x < 4, the second equation would be (16 - x^2) else (x^2-16) Now you can try solving between these ranges. – sudhackar Jul 05 '21 at 06:40
  • For the graph approach see https://www.desmos.com/calculator/ekxljhyvuw – sudhackar Jul 05 '21 at 06:41
  • For a shortcut, start with $,x+4 = |x+4|,|x-4|,$. – dxiv Jul 05 '21 at 06:48
  • The Exercise that I'm doing that involves this question comes before all the graphing part, so I have to do it with just calculation. One thing I've noticed thanks to the comments and after looking at the question for a fair bit is that it is actually a trick question. Everything is just the same but the simultaneous part is there to screw with me. – Mutsuki Jul 05 '21 at 07:13

1 Answers1

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It schould be clear that $x=-4$ is a solution of

$$(*) \quad \,x+4 = |x+4|\,|x-4|\,.$$

If $x \ne -4$ we get

$$ \frac{x+4}{|x+4|}=|x-4|.$$

The RHS is $ \ge 0$, hence we see that $x+4 >0.$ Thus we derive

$$|x-4|=1.$$

Conclusion: the equation $(*)$ has the solutions

$$-4,3,5.$$

Fred
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