Question

Plot √2 and √3 on the number line.

Which statements are true? Check all that apply.
StartRoot 2 EndRoot < 2
StartRoot 2 EndRoot < StartRoot 3 EndRoot
StartRoot 2 EndRoot > 1.5
StartRoot 3 EndRoot < 3
StartRoot 2 EndRoot is about 0.5 less than StartRoot 3 EndRoot.
StartRoot 3 EndRoot is about 0.3 greater than StartRoot 2 EndRoot.

Answer 1

To analyze the statements, let's first approximate the values of \(\sqrt{2}\) and \(\sqrt{3}\):

- \(\sqrt{2} \approx 1.414\)
- \(\sqrt{3} \approx 1.732\)

Now, we can evaluate each statement:

1. \(\sqrt{2} < 2\): **TRUE**
- \(1.414 < 2\)

2. \(\sqrt{2} < \sqrt{3}\): **TRUE**
- \(1.414 < 1.732\)

3. \(\sqrt{2} > 1.5\): **FALSE**
- \(1.414 < 1.5\)

4. \(\sqrt{3} < 3\): **TRUE**
- \(1.732 < 3\)

5. \(\sqrt{2}\) is about 0.5 less than \(\sqrt{3}\): **FALSE**
- \(\sqrt{3} - \sqrt{2} \approx 1.732 - 1.414 = 0.318\), which is not about 0.5.

6. \(\sqrt{3}\) is about 0.3 greater than \(\sqrt{2}\): **TRUE**
- As calculated above, \(\sqrt{3} - \sqrt{2} \approx 0.318\), which is approximately 0.3.

Based on this analysis, the true statements are:
- \(\sqrt{2} < 2\)
- \(\sqrt{2} < \sqrt{3}\)
- \(\sqrt{3} < 3\)
- \(\sqrt{3}\) is about 0.3 greater than \(\sqrt{2}\)