Problem 8 - Algebra 2 and Trigonometry

Question

Find the value of each given expression. \(|-12+(-(-5))|\)

Step-by-Step Solution

Answer

The value of the expression is 7.

Step 1: Simplify the Double Negative

The expression \(-(-5)\) indicates a double negative, which simplifies to \(+5\). Hence, the expression becomes \(-12 + 5\).

Step 2: Perform the Addition

Now calculate \(-12 + 5\) by adding the numbers. When a negative and a positive number are added, you actually subtract the smaller number from the bigger number and keep the sign of the bigger. So, \(-12 + 5 = -7\).

Step 3: Calculate the Absolute Value

The absolute value \(|x|\) represents the distance of number \(x\) from zero on a number line and is always non-negative. Thus, \(|-7|\) becomes \(|7|\), which is \(7\).

Key Concepts

Addition with Negative Numbers

When you add a negative number to a positive number, it's like subtracting. Imagine you're moving along a number line. Positive steps go to the right, and negative steps go to the left. In the expression \(-12 + 5\), you're starting at \(-12\) and taking 5 steps to the right. This leads you to \(-7\).

Here's a tip:

• When adding a positive number to a negative number, subtract the smaller number from the larger number.
• Keep the sign of the larger number.

This rule helps simplify problems and avoid mistakes when dealing with negative values.

Double Negative Simplification

Double negatives can often be tricky, but they have a simple rule.

In math, two negatives together become a positive. It's similar to saying, "I don't not like it," which means you do like it. For the expression \(-(-5)\), the two negatives cancel each other out, transforming it into \(+5\).

By understanding this, you'll find it easier to handle problems that involve negation. Just remember:

• A double negative becomes a positive.
• Apply this transformation early in problem-solving to simplify the steps.

Number Line Distance

The concept of absolute value revolves around distance on a number line. Absolute value, denoted \(|x|\), measures how far \(x\) is from zero without considering direction. So, negative or positive doesn't matter in terms of distance.

For example, \(|-7|\) means you check how far \(-7\) is from zero. The answer? It is 7 units away.

Practically speaking:

• The absolute value changes only the negative sign to positive or leaves it unchanged if it's already positive.
• It helps in identifying size or magnitude without focusing on the direction.

Understanding this can clear up any confusion when dealing with negative and positive values.