Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Place the correct symbol \((<,>\), or \(=)\) between the two real numbers.\(-(-2) \quad-2\)

Step-by-Step Solution

Answer

The correct symbol to place between \(-(-2)\) and \(-2\) is \(>\). Therefore, the full equation is \(2 > -2\).

Step 1: Simplify the Double Negative

Simplify the number to the left side of the comparison which is \(-(-2)\). In mathematics, a double negative equates to a positive. Therefore, \(-(-2)\) will be equal to \(2\).

Step 2: Compare the Numbers

Compare the simplified number (\(2\)) to the number on the right (\(-2\)). We already know that \(2 > -2\). So, the correct symbol to place between \(2\) and \(-2\) is \(>\).

Step 3: Write the Final Equation

Write the final equation, including the comparison symbol that was determined in step 2. The full equation will then be \(2 > -2\).

Key Concepts

Double Negative

In mathematics, encountering a double negative often leads to confusion, but it is a straightforward concept. A double negative occurs when there are two negative signs together, such as \[ -(-2) \].When we come across this situation, it's essential to remember that two negatives cancel each other out, similar to multiplying two negative numbers, resulting in a positive.

• This means that -(-2) turns into a positive 2.
• To visualize it, think of removing a negative sign, leaving what was inside, positive.

This simplification helps us approach comparisons with clarity, as we are dealing with clear positive and negative numbers rather than added layers of negativity.

Inequalities

Understanding inequalities is crucial when comparing real numbers. Inequalities are mathematical expressions used to determine the relative size or order of two values by using signs like <, >, or =. For example, comparing 2 and -2:

• The symbol '<' indicates less than. This shows when one number is smaller.
• The symbol '>' indicates greater than, displaying that one number is bigger than another.
• If two numbers are the same, we use '=', meaning they are equal.

Inequalities simplify the understanding of numbers by providing a clear view of their relationship in context, such as with 2 > -2.

Mathematics

In the realm of mathematics, logical thinking and methods help us solve problems like this one. The process of comparing numbers or variables unfolds in clear steps:

• Identify the problem: Determine if numbers need simplification such as with double negatives.
• Apply necessary changes: Simplify where necessary, as we did from -(-2) to a simple 2.
• Make the comparison: Once simplified, use inequalities to clarify the relationship, whether '<', '>', or '='.

These steps, backed by consistent mathematical principles, ensure understanding and correct conclusions, evolving skills in numerical comparison and broader mathematical comprehension.