between what two disjoint sections we can do a unification in order to get this group of solutions?

Question

between what two disjoint sections we can do a unification in order to get this group of solutions?

$0<|x+6|\leq{0.4}$

in other words, in what values should I fill the blankets:

(____,____) $\cup$ (____ ,___ )

in order to get two ranges which their unification of the group of solutions above. $0<|x+6|\leq{0.4}$

and what kind of brackets should I use? the ones that I wrote are not accurate !

Answer 1

Hint: Take cases depending on whether $x+6>0$ or $x+6 \le 0$. F.e. in the case $x+6>0$ or equivalently $x>-6$ you have that $|x+6|>0$ thus $$0<x+6\le 0.4 \implies -6<x\le-5.6$$ Now, as you see the assumption $-6<x$ and the solution $-6<x\le-5.6$ yield acceptable solutions, so the first interval in your answer should be $$(-6,-5.4]$$

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Work similarly in the case $x+6<0$ or equivalently $x<-6$. Be careful that know $|x+6|=-x-6$. Thus $$0<-x-6\le 0.4 \implies 6<-x \le 6.4 \implies -6.4 \le x < -6$$ and the second interval should be $$[-6.4, -6)$$ since again there is no conflict between the assumption and the answer.