Question

Using rational approximations, what is an inequality statement comparing 120−−−√
120
and 121−−−√
121
? Enter <, >, or =.(1 point)
The inequality is: 120−−−√
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120
121−−−√
Answer
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Answered by GPT-4o mini
To compare \( \sqrt{120} \) and \( \sqrt{121} \), we can use the fact that \( \sqrt{121} = 11 \) since \( 121 = 11^2 \).

Now, let's compute \( \sqrt{120} \) using rational approximations. We know:

\[
\sqrt{120} \approx 11 \text{ (since } 11^2 = 121 \text{ and } 120 \text{ is slightly less than } 121\text{)}
\]

To be more precise, we can find \( \sqrt{120} \) by realizing:

\[
\sqrt{120} \approx \sqrt{121 - 1} \approx \sqrt{121} - \frac{1}{2\sqrt{121}} \text{ (using the binomial approximation for small changes)}
\]
\[
\sqrt{120} \approx 11 - \frac{1}{2 \times 11} \approx 11 - 0.0455 \approx 10.9545
\]

Thus:

1. \( \sqrt{120} \approx 10.95 \)
2. \( \sqrt{121} = 11 \)

Since \( 10.954 < 11 \), we have:

\[
\sqrt{120} < \sqrt{121}
\]

Therefore, the inequality is:

\[
\sqrt{120} < \sqrt{121}
\]

The final answer is \( \sqrt{120} < \sqrt{121} \).
Answered by GPT-4o mini
The inequality is:

\(\sqrt{120} < \sqrt{121}\)
Answered by GPT-4o mini
Sure! Here are the steps to compare \(\sqrt{120}\) and \(\sqrt{121}\):

1. **Identify the known values**:
- We know \(\sqrt{121} = 11\) because \(11^2 = 121\).

2. **Estimate \(\sqrt{120}\)**:
- To approximate \(\sqrt{120}\), we note that \(120\) is close to \(121\).
- We can use rational approximations. Let's compare \(120\) with \(121\).

3. **Compare \(120\) and \(121\)**:
- Notice that \(120 < 121\).
- Since the square root function is increasing, we can say if \(a < b\), then \(\sqrt{a} < \sqrt{b}\).

4. **Apply the comparison**:
- Since \(120 < 121\), it follows that:
\[
\sqrt{120} < \sqrt{121}
\]

5. **Conclusion**:
- Therefore, the inequality statement is:
\(\sqrt{120} < \sqrt{121}\) or simply \(120 < 121\) when referring to their square roots.

So, the inequality is:

\[
\sqrt{120} < \sqrt{121}
\]
Answered by GPT-4o mini
1. \(\sqrt{121} = 11\) (since \(11^2 = 121\)).
2. \(120 < 121\).
3. If \(a < b\), then \(\sqrt{a} < \sqrt{b}\).
4. Thus, \(\sqrt{120} < \sqrt{121}\).

Final answer: \(\sqrt{120} < \sqrt{121}\).