To complete the inequality by placing the given numbers in the correct order, let's first understand each expression's value:
1. [tex]\(4 \sqrt{2}\)[/tex]
2. [tex]\(\sqrt{8}\)[/tex]
3. [tex]\(\sqrt{20}\)[/tex]
4. [tex]\(\frac{\sqrt{90}}{3}\)[/tex]
5. [tex]\(5 \sqrt{3}\)[/tex]
Here are their estimated values based on the provided sorted array:
- [tex]\(4 \sqrt{2} \approx 5.656854249492381\)[/tex]
- [tex]\(\sqrt{8} \approx 2.8284271247461903\)[/tex]
- [tex]\(\sqrt{20} \approx 4.47213595499958\)[/tex]
- [tex]\(\frac{\sqrt{90}}{3} \approx 3.1622776601683795\)[/tex]
- [tex]\(5 \sqrt{3} \approx 8.660254037844386\)[/tex]
Arrange these values in ascending order:
[tex]\( \sqrt{8} \approx 2.8284271247461903 \)[/tex]
[tex]\( \frac{\sqrt{90}}{3} \approx 3.1622776601683795 \)[/tex]
[tex]\( \sqrt{20} \approx 4.47213595499958 \)[/tex]
[tex]\( 4 \sqrt{2} \approx 5.656854249492381 \)[/tex]
[tex]\( 5 \sqrt{3} \approx 8.660254037844386 \)[/tex]
Thus, the inequality completed correctly is:
[tex]\[ 2 < \sqrt{8} < \frac{\sqrt{90}}{3} < \sqrt{20} < 4 < 4 \sqrt{2} < 5 < 5 \sqrt{3} \][/tex]
