To determine which statements among the given are true, let's analyze each one step-by-step based on the given numerical results:
1. [tex]\(\sqrt{1.8} < 1.8\)[/tex]
- Since the square root of a number less than 1 is always less than the number itself, this statement is true because [tex]\(1.8 \approx 1.3416\)[/tex].
2. [tex]\(\sqrt{1.8} > 1\)[/tex]
- The square root of 1.8 is greater than 1. The value of [tex]\(\sqrt{1.8}\)[/tex] is approximately 1.3416, which is indeed greater than 1. So, this statement is true.
3. [tex]\(\sqrt{1.8} < \(\sqrt{1.9}\)[/tex]
- [tex]\(\sqrt{1.8}\)[/tex] approximates to 1.3416 and [tex]\(\sqrt{1.9}\)[/tex] approximates to 1.3784. Since 1.3416 is less than 1.3784, this statement is true.
4. [tex]\(1.3 < \sqrt{1.8} < 1.4\)[/tex]
- With [tex]\(\sqrt{1.8}\)[/tex] being approximately 1.3416, it lies between 1.3 and 1.4. Thus, [tex]\(1.3 < 1.3416 < 1.4\)[/tex] is a true statement.
5. [tex]\(\sqrt{1.9} + \sqrt{1.8} > 2\)[/tex]
- Summing up, [tex]\(\sqrt{1.9} \approx 1.3784\)[/tex] and [tex]\(\sqrt{1.8} \approx 1.3416\)[/tex]. Their sum is approximately [tex]\(1.3784 + 1.3416 = 2.72\)[/tex], which is greater than 2. Therefore, this statement is true.
6. [tex]\(\sqrt{1.9} - \sqrt{1.8} > 0.1 \)[/tex]
- With the values [tex]\(\sqrt{1.9} \approx 1.3784\)[/tex] and [tex]\(\sqrt{1.8} \approx 1.3416\)[/tex], subtracting them gives [tex]\(1.3784 - 1.3416 = 0.0368\)[/tex], which is less than 0.1. Hence, this statement is false.
Summarizing, the true statements are:
- [tex]\(\sqrt{1.8} < 1.8\)[/tex]
- [tex]\(\sqrt{1.8} > 1\)[/tex]
- [tex]\(\sqrt{1.8} < \(\sqrt{1.9}\)[/tex]
- [tex]\(1.3 < \sqrt{1.8} < 1.4\)[/tex]
- [tex]\(\sqrt{1.9} + \sqrt{1.8} > 2\)[/tex]
The false statement is:
- [tex]\(\sqrt{1.9} - \sqrt{1.8} > 0.1\)[/tex]
