Which inequalities are true? Select the three correct answers.

Question

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(3 points)
Responses

0<8–√
\(0 < \sqrt{8}\)

8–√>8
square root of 8 is greater than 8

8–√<2
\(\sqrt{8} < 2\)

8–√>7–√
square root of 8 is greater than square root of 7

8–√<3
\(\sqrt{8} < 3\)

Answer 1

To evaluate the inequalities involving \(\sqrt{8}\), we can calculate the approximate value of \(\sqrt{8}\):

\[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \approx 2.828 \]

Now we can analyze each of the statements:

1. \(0 < \sqrt{8}\): This is true because \(\sqrt{8} \approx 2.828\), which is greater than 0. So this statement is correct.

2. \(\sqrt{8} > 8\): This is false because \(\sqrt{8} \approx 2.828\) which is less than 8.

3. \(\sqrt{8} < 2\): This is also false as \(\sqrt{8} \approx 2.828\) is greater than 2.

4. \(\sqrt{8} > \sqrt{7}\): This is true because \(\sqrt{8} \approx 2.828\) and \(\sqrt{7} \approx 2.6458\), so \(\sqrt{8}\) is indeed greater than \(\sqrt{7}\). Hence, this statement is correct.

5. \(\sqrt{8} < 3\): This is true because \(\sqrt{8} \approx 2.828\) which is indeed less than 3. So this statement is correct.

Therefore, the three correct inequalities are:

1. \(0 < \sqrt{8}\)
2. \(\sqrt{8} > \sqrt{7}\)
3. \(\sqrt{8} < 3\)

So you should select these three options.