Let's analyze the given inequalities step-by-step:
1. For the inequality [tex]\(\sqrt{5} < 2.3 < \sqrt{6}\)[/tex]:
- The square root of 5 ([tex]\(\sqrt{5}\)[/tex]) is approximately 2.236.
- The square root of 6 ([tex]\(\sqrt{6}\)[/tex]) is approximately 2.449.
Checking the inequality:
[tex]\[
\sqrt{5} \approx 2.236 \quad \text{and} \quad \sqrt{6} \approx 2.449
\][/tex]
Comparing with 2.3:
[tex]\[
2.236 < 2.3 < 2.449
\][/tex]
Since the values fit correctly, this inequality is true.
2. For the inequality [tex]\(\sqrt{5} < 2.4 < \sqrt{6}\)[/tex]:
- We already know [tex]\(\sqrt{5} \approx 2.236\)[/tex].
- We already know [tex]\(\sqrt{6} \approx 2.449\)[/tex].
Checking the inequality:
[tex]\[
2.236 < 2.4 < 2.449
\][/tex]
Since these values also fit correctly, this inequality is true.
3. For the inequality [tex]\(\sqrt{4} < \sqrt{5} < \sqrt{5.5}\)[/tex]:
- The square root of 4 ([tex]\(\sqrt{4}\)[/tex]) is exactly 2.
- The square root of 5 ([tex]\(\sqrt{5}\)[/tex]) is approximately 2.236.
- The square root of 5.5 ([tex]\(\sqrt{5.5}\)[/tex]) is approximately 2.345.
Checking the inequality:
[tex]\[
2 < 2.236 < 2.345
\][/tex]
Since these values fit correctly, this inequality is true.
4. For the inequality [tex]\(\sqrt{8} < 3 < \sqrt{9}\)[/tex]:
- The square root of 8 ([tex]\(\sqrt{8}\)[/tex]) is approximately 2.828.
- The square root of 9 ([tex]\(\sqrt{9}\)[/tex]) is exactly 3.
Checking the inequality:
[tex]\[
2.828 < 3 < 3
\][/tex]
Since [tex]\(\sqrt{9}\)[/tex] is exactly 3, the inequality 3 < [tex]\(\sqrt{9}\)[/tex] would not hold true. Therefore, this inequality is false.
5. For the inequality [tex]\(\sqrt{8} < 2.9 < \sqrt{9}\)[/tex]:
- We already know [tex]\(\sqrt{8} \approx 2.828\)[/tex].
- We already know [tex]\(\sqrt{9} = 3\)[/tex].
Checking the inequality:
[tex]\[
2.828 < 2.9 < 3
\][/tex]
Since these values fit correctly, this inequality is true.
6. For the inequality [tex]\(\sqrt{4} < 4.5 < \sqrt{5}\)[/tex]:
- We already know [tex]\(\sqrt{4} = 2\)[/tex].
- We already know [tex]\(\sqrt{5} \approx 2.236\)[/tex].
Checking the inequality:
[tex]\[
2 < 4.5 < 2.236
\][/tex]
Since 4.5 is much greater than [tex]\(\sqrt{5}\)[/tex], this inequality is false.
Thus, the four correct inequalities are:
1. [tex]\(\sqrt{5} < 2.3 < \sqrt{6}\)[/tex]
2. [tex]\(\sqrt{5} < 2.4 < \sqrt{6}\)[/tex]
3. [tex]\(\sqrt{4} < \sqrt{5} < \sqrt{5.5}\)[/tex]
4. [tex]\(\sqrt{8} < 2.9 < \sqrt{9}\)[/tex]
The correct answer selections are: (1), (2), (3), and (5).
