Question

Tell the correct answer with explaination​

Answer 1

$\large\text{\underline{Let's begin.}}$

After rationalization, we get $\dfrac{\sqrt{5}-1}{4}$.

We need to check if the number lies between (a), (b), (c), or (d).

$\large\text{\underline{Point of the Question}}$

If we multiply a positive number on each term of inequality, the relative amount of the number is kept.

So, let's multiply a positive number 4, which removes the denominator.

We can check the inequality is true by squaring all terms of the inequality.

$\large\text{\underline{Solution}}$

(a) $\dfrac{4}{3}<\sqrt{5}-1<2\implies\dfrac{7}{3}<\sqrt{5}<3$

(b) $2<\sqrt{5}-1<2\sqrt{2}\implies3<\sqrt{5}<1+2\sqrt{2}$

(c) $1<\sqrt{5}-1<\dfrac{4}{3}\implies2<\sqrt{5}<\dfrac{7}{3}$

(d) $\dfrac{4}{5}<\sqrt{5}-1<1\implies\dfrac{9}{5}<\sqrt{5}<2$

Now it's time to check the interval is whether true or false.

(a) $\dfrac{49}{9}<5<9$, which is false.

(b) $9<5<9+4\sqrt{2}$, which is false.

(c) $4<5<\dfrac{49}{9}$, which is true.

(d) $\dfrac{81}{25}<5<4$, which is false.

$\large\text{\underline{Conclusion}}$

Hence, the answer is option (c).