Problem 70 - Algebra A Combined Function

Question

Simplify each of the following. See Example 17. $$ -(-14) $$

Step-by-Step Solution

Answer

The expression simplifies to 14.

Step 1: Understand the Problem

The problem asks us to simplify the expression \(-(-14)\). This expression involves two negative signs, which can be simplified by understanding the rules of arithmetic involving negative numbers.

Step 2: Apply the Rule of Double Negatives

According to arithmetic rules, a negative sign applied to a negative number results in a positive number. Therefore, the expression \(-(-14)\) becomes the positive of 14.

Step 3: Simplify the Expression

By applying the rule of double negatives, the expression \(-(-14)\) simplifies directly to 14.

Key Concepts

Double Negatives

When dealing with negative numbers, sometimes you encounter situations where there are two negative signs together. This scenario is known as "double negatives." Understanding how double negatives work is crucial in arithmetic because it helps us simplify expressions more easily.

So, what happens when there are two negative signs, as in \(-(-x)\)? Simple! The two negative signs cancel each other out and turn into a positive sign. It's like saying the opposite of the opposite. In our example, the expression \(-(-14)\) transforms into \(+14\).

The concept is similar to language: if you say "I am not unhappy," you actually mean "I am happy." Double negatives in math work in much the same way by changing the outcome into a positive.

Arithmetic Rules

To simplify mathematical expressions, understanding arithmetic rules is essential. These rules are like the grammar of numbers. They ensure that we all interpret numbers in the same way.

One of the primary rules in arithmetic is that multiplying or dividing two negative numbers results in a positive outcome. This is because a negative times a negative or a negative divided by a negative flips the sign twice, making it positive. In our example, the expression \(-(-14)\) doesn't involve multiplication, but the same rule of converting double negatives to positive applies.

• Negative + Negative = More Negative
• Positive + Positive = More Positive
• Negative + Positive (or vice versa) = A Negative or Positive depending on larger magnitude
• Negative * Negative = Positive

Remembering these simple rules helps greatly in operations involving negative numbers.

Positive Numbers

Positive numbers are straightforward and make up the set of numbers greater than zero. They represent things like counts, measurements, and generally accepted increases. Positive numbers in arithmetic are largely intuitive because we use them most frequently in everyday life.

In our scenario, by simplifying the expression \(-(-14)\), we ended with a positive number: \(+14\). Positive numbers, unlike negative ones, don't have any additional rules for basic addition, subtraction, multiplication, or division. You simply add or subtract them as needed without worrying about changing the sign.

• Positive numbers are additive in nature.
• Any number greater than zero is positive.
• They form the right side of a number line starting from zero.

Understanding how positive numbers function in math makes it much easier to grasp the outcomes when solving equations that involve negatives, like in our example.