solving pair of inequality involving question.$0.0004<\dfrac{4,000,000}{d^2}<0.01$

From Stewart Precalculus, P86.

This question is from the chapter about inequalities and we are supposed to set up a model using inequalities to solve this problem.

my working out was

$$0.0004<\dfrac{4,000,000}{d^2}<0.01$$

In this chapter, the author did not show us how to solve a pair of simultaneous inequalities involving a quotient and I don't know how to do it, the only way I can think of is to solve the problem by separating the above pair of inequality into two separate single inequality, is this what I am suppose to do here?

$$0.0004<\dfrac{4,000,000}{d^2},\\\dfrac{4,000,000}{d^2}<0.01$$

Thanks.

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1 Answer

$$0.0004<\dfrac{4,000,000}{d^2}<0.01$$ when we reciprocal a fraction then sign of inequality changes:$x<y\implies\dfrac1x>\dfrac1y$ $$\dfrac{1}{0.0004}>\dfrac{d^2}{4,000,000}>\dfrac{1}{0.01}$$ multiply wach term by $4,000,000$ $$\dfrac{4,000,000}{0.0004}>\dfrac{d^2}{1}>\dfrac{4,000,000}{0.01}$$ $$10^{10}>d^2>4\times10^8$$ $$4\times10^8<d^2<10^{10}$$ $$\sqrt{4\times10^8}<d<\sqrt{10^{10}}$$ since d=distance it can't be negative so we take only positive value after square root $${2\times10^4}<d<{10^{5}}$$ $$20000<d<100000$$

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