Question

Which inequalities are true? Select the three correct answers.(3 points)
Responses

6–√<2
6–√<2

0<4–√
0<4–√

8–√<5
8–√<5

8–√>3
square root of 8 is greater than 3

8–√>6–√

Answer 1

To determine which of the inequalities are true, let's evaluate each one step by step.

1. **\( 6 - \sqrt{2} < 2 \)**
- First, calculate \( \sqrt{2} \) which is approximately \( 1.414 \).
- Therefore, \( 6 - \sqrt{2} \approx 6 - 1.414 = 4.586 \).
- \( 4.586 < 2 \) is **false**.

2. **\( 0 < 4 - \sqrt{4} \)**
- Calculate \( \sqrt{4} = 2 \).
- Therefore, \( 4 - \sqrt{4} = 4 - 2 = 2 \).
- \( 0 < 2 \) is **true**.

3. **\( \sqrt{8} < 5 \)**
- Calculate \( \sqrt{8} \) which is approximately \( 2.828 \).
- \( 2.828 < 5 \) is **true**.

4. **\( \sqrt{8} > 3 \)**
- We already found \( \sqrt{8} \approx 2.828 \).
- \( 2.828 > 3 \) is **false**.

5. **\( \sqrt{8} > \sqrt{6} \)**
- Calculate \( \sqrt{6} \) which is approximately \( 2.449 \).
- Since \( 2.828 > 2.449 \), this is **true**.

Based on the evaluations above, the three correct true inequalities are:

1. \( 0 < 4 - \sqrt{4} \)
2. \( \sqrt{8} < 5 \)
3. \( \sqrt{8} > \sqrt{6} \)

Therefore, the true inequalities are the second, third, and fifth responses.