Question

order these numbers from least to greatest. 12\frac{1}{10}, -12\overline{1}, -\sqrt{145}, -\frac{86}{7} note that for this question you can use your mouse to drag the numbers into their positions.

Explanation:

Step1: Convert to decimal

• \(12\frac{1}{10} = 12.1\)
• \(-12.\overline{1} = -12.111\cdots\)
• \(-\sqrt{145} \approx -12.0416\) (since \(12^2 = 144\), so \(\sqrt{145}\approx12.0416\))
• \(-\frac{86}{7} \approx -12.2857\) (dividing 86 by 7)

Step2: Compare negative decimals

For negative numbers, the larger the absolute value, the smaller the number.

• \(|-12.2857| = 12.2857\)
• \(|-12.111\cdots| = 12.111\cdots\)
• \(|-12.0416| = 12.0416\)

So, \(-\frac{86}{7} < -12.\overline{1} < -\sqrt{145}\)

Step3: Include positive number

The positive number \(12.1\) is greater than all negative numbers.

Answer:

\(-\frac{86}{7}\) < \(-12.\overline{1}\) < \(-\sqrt{145}\) < \(12\frac{1}{10}\)