Problem 71 - Algebra A Combined Function

Question

Simplify each of the following. See Example 17. $$ -(-7.9) $$

Step-by-Step Solution

Answer

\(7.9\)

Step 1: Identify Double Negative

The exercise involves the expression \(-(-7.9)\). Notice that there is a double negative here because there is a negative sign outside and another negative sign inside the brackets.

Step 2: Apply the Rules of Negation

When you negate a negative number, it becomes positive. This is based on the rule that \(-(-a) = a\) for any number \(a\). So, when you apply this rule to \(-(-7.9)\), it simplifies to \(7.9\).

Key Concepts

Double Negation

When we talk about double negation, we're dealing with two negative signs in a row. Imagine them like two barriers that cancel each other out. It's like saying "not not happy"—which simply means "happy." In mathematics, this concept is neat and clean:

• Whenever you see two negative signs together,
• They cancel each other out,
• And turn the number positive.

With the example \( -(-7.9), \) we have two negatives: one inside, one outside. The first negative flips the sign of 7.9, and then the outside negative flips it back to positive, resulting in 7.9. This clear, simple rule can save you from the confusion when you encounter multiple negatives.

Rules of Negation

The rules of negation are fundamental. Negation refers to turning a positive number into a negative and vice versa. This is key in solving many mathematical problems involving negative numbers:

• A negative sign in front of a positive number makes it negative, for instance, \( -a = -a,\) where \( a \) is positive.
• A negative sign in front of a negative number makes it positive, according to the rule \( -(-a) = a. \)

Think of negation like flipping a switch: one switch is ON (positive), and one switch is OFF (negative). If both switches are OFF, it’s as if neither was—thus, leaving the light ON (positive). Understanding these shifts and flips ensures that we accurately simplify expressions like the ones seen in our original exercise.

Simplification of Expressions

Simplifying expressions is all about making them easier to work with. By applying mathematical rules, including the rules of negation, we can turn complex-looking problems into something manageable. When simplifying:

• First, look for any double negatives and resolve them.
• Apply rules like \( -(-a) = a, \)
• This process turns negatives, where appropriate, into positives.

For instance, simplifying \( -(-7.9) \) involved identifying and canceling out the double negation to achieve a simple positive number \( 7.9. \) It may seem small, but this simplification reduces errors and makes calculations cleaner. The goal is always clarity and accuracy, making your math journey smoother.