Using rational approximations, what is an inequality statement comparing π

Question

and 8–√
? Enter <, >, or =

Answer 1

To compare $\pi$ and $8-\sqrt{8}$, we can first evaluate $8-\sqrt{8}$.

Calculating $\sqrt{8}$: $$\sqrt{8}=\sqrt{4\times 2}=2\sqrt{2}\approx 2\times 1.414\approx 2.828$$ Then, we compute $8-\sqrt{8}$: $$8-\sqrt{8}\approx 8-2.828\approx 5.172$$

Now, we will use the approximate value of $\pi$: $$\pi \approx 3.14159$$

Now we can compare the two values: $$\pi \approx 3.14159<5.172\approx 8-\sqrt{8}$$

Thus, the inequality statement is: $$\pi <8-\sqrt{8}$$

So, the answer is: $$<$$