Problem 61 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$8(-10)+|4(-5)|$$

Step-by-Step Solution

Answer

-60

Step 1: Perform Operations in Parentheses

Simplify expressions inside parentheses/brackets. We start by dealing with the mathematical operations within parentheses, in this case the multiplications. We have \(8(-10)\), which equates to -80 and \(4(-5)\), which equates to -20. Now, the expression will look like \(-80 + |-20|\).

Step 2: Calculate the Absolute Value

We have |-20|, the absolute value of -20 which results in 20. Absolute value turns negative numbers into positive numbers. So, \(-80 + |-20|\) will become \(-80 + 20\).

Step 3: Perform Addition

The remaining operation is the addition operation. \(-80 + 20\) equals \(-60\). So, the expression simplifies to \(-60\).

Key Concepts

Absolute Value

Understanding absolute value is fundamental to evaluating expressions correctly. The absolute value of a number refers to its distance from zero on the number line, without considering its direction.

• For positive numbers, the absolute value is the number itself.
• For negative numbers, the absolute value is the positive counterpart of the number.

For instance, in the expression given, we are asked to evaluate \(|-20|\). Here, even though -20 appears, taking the absolute value converts it into 20. This transformation simplifies our calculations significantly because it removes any negative sign influence. Therefore, understanding absolute value as a measure of magnitude leads us seamlessly into the right solution.

Simplifying Expressions

Simplifying expressions involves breaking them down into their simplest form step-by-step according to mathematical conventions. The goal is to reduce complexity while maintaining equivalence.To simplify an expression efficiently:

• Follow the order of operations, often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
• Simplify any terms inside parentheses first.
• Perform multiplication or division in the order they appear from left to right.
• Finally, carry out addition or subtraction as they appear from left to right.

In our given exercise, by addressing the operations inside the parentheses first—like \(8(-10)\) and \(4(-5)\)—we maintain accuracy in our next steps: calculating the absolute value and performing the remaining operations.

Algebraic Operations

Algebraic operations are the building blocks of handling equations and expressions in mathematics. These include basic arithmetic operations like addition, subtraction, multiplication, and division, extended to include operations involving algebraic expressions.Performing these operations accurately requires:

• Recognizing the role of each operation in simplifying expressions.
• Understanding how each operation interacts, especially when absolute values or equations are involved.

In the exercise at hand, algebraic operations were key to reaching the solution. Here, after simplifying terms \(8(-10)\) and \(4(-5)\), we used addition to combine them effectively: transforming \(-80 + | -20 |\) into \(-80 + 20\). Every step respects the fundamental rules of algebra to systematically arrive at the final answer of \(-60\). By approaching problems with this understanding, complexity is reduced, leading to effective problem-solving.