Problem 119 - Elementary Algebra

Question

Simplify. $$ -(-|-45|) $$

Step-by-Step Solution

Answer

The simplified result is 45.

Step 1: Understand the Absolute Value

The absolute value represents the distance of a number from zero on the number line, ignoring any negative signs. So, for the number -45, the absolute value is calculated as follows: \[ |-45| = 45 \]

Step 2: Simplify the Inside Expression

Replacing the absolute value with its result, rewrite the expression without absolute value brackets: \[ -(-45) \]

Step 3: Remove the Double Negative

The expression now has a double negative, which can be simplified by recognizing that two negatives make a positive. \[ -(-45) = 45 \]

Step 4: Final Result: Simplified Expression

Having simplified through each step, the final result of this expression is: \[ 45 \]

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing expressions into their simplest form, making them easier to understand and work with. When faced with complex expressions, follow these steps:

• Look for absolute values first, which are enclosed in vertical bars (e.g., \(|-45|\)).
• Evaluate the absolute values by finding the distance from zero, ignoring any negative signs.
• Replace these values back into the expression to simplify further.

Carefully removing unnecessary complexities helps in making expressions clearer, aiding in solving equations or further calculations. By practicing simplification, you enhance your ability to deal with more complex algebraic challenges in the future.

Negative Numbers

Negative numbers are numbers less than zero, and they appear frequently in algebra. They can be a bit tricky, but there are some rules to keep in mind that make working with them easier.

When dealing with negative numbers:

• Remember that negative numbers are on the left side of zero on the number line.
• The further left you go, the smaller the number is.
• Using absolute value can temporarily remove the negative sign for calculations, as it represents the number's distance from zero.

Understanding negative numbers helps you grasp concepts like debt in math and allows you to handle complex mathematical problems with confidence.

Double Negatives

A double negative occurs when two negative signs appear next to each other in mathematical expressions. It’s crucial to recognize that two negatives make a positive.

For example:

• If you have \(-(-45)\), you change it to a positive \(45\).
• This is because subtracting a negative is equivalent to adding the positive value.

The concept of double negatives isn’t just limited to mathematics. It's also found in grammar! But in math, it's a handy tool for simplifying expressions, ensuring clarity in calculations, and avoiding unnecessary confusion.