Problem 47 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ |5+(-10)| $$

Step-by-Step Solution

Answer

The result is 5.

Step 1: Evaluate the Expression Inside the Absolute Value

First, calculate the expression inside the absolute value bars. Here, we have:\(5 + (-10) = 5 - 10\).Subtract 10 from 5, resulting in \(-5\).

Step 2: Apply the Absolute Value

Next, apply the absolute value to \(-5\). The absolute value function makes any negative number positive. Therefore:\(|-5| = 5\).

Key Concepts

Evaluating Expressions

When working with mathematical expressions, the goal is to simplify them to a single value. Evaluating an expression inside absolute value bars means performing operations in the correct order. This means handling any additions, subtractions, multiplications, or divisions first, before addressing the absolute value itself.

• Start with any operations inside parentheses or absolute value bars.
• Follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Once all operations are completed, if there's an absolute value, convert the final number within the bars to a positive.

In our example, we first consider the expression inside the absolute value: \(5 + (-10)\). This simplifies the expression by recognizing and solving the addition of opposite integers first, so it becomes \(-5\). Once evaluated, we then apply the absolute value rule by making \(-5\) a positive \(5\).

Integer Operations

Integer operations are foundational in mathematics and involve calculations with whole numbers, which can be positive or negative. Understanding these operations helps in simplifying complex expressions.

• Adding positive integers increases the value.
• Adding a negative integer is equivalent to subtraction.
• Subtracting a negative integer is like adding a positive one.

In the expression \(5 + (-10)\), we treat the negative sign in front of 10 as a signal to subtract. This transforms our problem into a straightforward subtraction: \(5 - 10 = -5\). Understanding integer rules makes operations with positives and negatives straightforward and intuitive.

Subtraction

Subtraction is the arithmetic operation of finding the difference between numbers. It involves reducing or taking away one number from another. Negative and positive numbers interact a bit differently during subtraction from what we might consider 'normal' subtraction.

• When subtracting a larger number from a smaller one, the result is negative.
• When subtracting a smaller number from a larger one, the result is positive.
• Subtracting two identical numbers results in zero.

For example, in \(5 - 10\), because 10 is larger than 5, we conclude that the answer will be negative, specifically \(-5\). Remembering this relationship is essential in evaluating differences correctly, especially when handling absolute value problems.