Problem 26 - Prealgebra

Question

Replace each \(\odot\) with \(<,>,\) or \(=\) to make a true statement. $$-3.3 \odot-\sqrt{10}$$

Step-by-Step Solution

Answer

-3.3 < -\sqrt{10}

Step 1: Evaluate \( -3.3 \)

The first number in our expression is \( -3.3 \). This is a negative decimal number.

Step 2: Simplify \( -\sqrt{10} \)

Calculate the approximate value of \( \sqrt{10} \). The square root of 10 is approximately 3.162. Therefore, \(-\sqrt{10} \approx -3.162\).

Step 3: Compare the Two Values

We compare \(-3.3\) and \(-3.162\). Since -3.3 is less than -3.162 (a more negative number is smaller), we replace \(\odot\) with \(<\).

Step 4: Write the Final Statement

The complete and correct inequality is \(-3.3 < -\sqrt{10}\).

Key Concepts

Understanding Negative Decimals

Negative decimals can be tricky, especially when comparing them to other numbers. To understand negative decimals, it first helps to grasp what a decimal number is. A decimal number represents a fraction of a whole number and is denoted by a dot (.). When you see a number like -3.3, it's a decimal number that's negative. The negative sign indicates it's less than zero.
Negative decimals follow the same rules as positive decimals, but their position on the number line shifts to the left side, indicating they are less than zero.
If you have two negative decimals or a negative decimal and another negative number, you compare them just like positive numbers but with a twist: a larger absolute value in the negative number is actually less because it's further left on the number line. For instance, -3.3 is less than -3.0."

• Negative sign indicates value less than zero.
• More negative = smaller number value.
• -3.3 is found further left on the number line than -3.2.

Exploring Square Roots

The square root of a number is another number that, when multiplied by itself, gives the original number. Therefore, the square root of 10 is a number which, when squared, equals 10.
The symbol for square root is \( \sqrt{} \), and sometimes numbers don’t come out as neat whole numbers. For example, \( \sqrt{10} \approx 3.162 \).
When dealing with negative square roots, the negative sign should be considered a reflection across zero.

• \( \sqrt{10} \) evaluates to approximately 3.162.
• -\( \sqrt{10} \) reflects to approximately -3.162.

Taking a negative square root means you effectively flip the positive result across the zero line on the number line.

Comparing Numbers Effectively

When comparing numbers, it's important to consider their placement on the number line. Numbers to the left are always smaller than those on the right.
This can be intuitive with positive numbers, but with negative numbers, it can be less obvious.
Negative numbers that seem larger in absolute terms are actually smaller when comparing values. For instance, comparing -3.3 and -3.162 requires us to realize that even though 3.3 is bigger than 3.162, the negative sign means -3.3 is less than -3.162.

• Note the direction: left on the number line is smaller.
• Inferring value of digits after the decimal can be crucial in accurate comparison.
• Use \(<\), \(>\) or \(=\) to denote less than, greater than or equal.

By mastering the principles of the number line and absolute values, comparing any negative decimals and square roots becomes manageable.