Problem 73 - Introductory Algebra for College Students

Question

Simplify: \(-10+16 \div 2(-4) .\) (Section 1.8, Example 4)

Step-by-Step Solution

Answer

-42

Step 1: Perform Division

Divide 16 by 2 which gives 8. The expression becomes \(-10 + 8(-4)\)

Step 2: Calculate Multiplication

Next, complete the multiplication operation. Multiply 8 and -4 to get -32. Substitute this result back into the expression which should now be \(-10 + (-32)\)

Step 3: Perform Addition

Finally, add -10 and -32 together which yields -42. Hence, the resulting simplification of the given expression is -42.

Key Concepts

Understanding the Order of Operations

When simplifying expressions like \(-10 + 16 \div 2(-4)\), it's crucial to use the order of operations. This set of rules dictates the sequence in which operations should be performed to ensure consistent results. Remember the acronym PEMDAS:

• Parentheses: Solve expressions within parentheses first.
• Exponents: Address any exponents or powers next.
• Multiplication and Division: Perform these operations from left to right. They are of equal precedence, so tackle them in the order they appear.
• Addition and Subtraction: Like multiplication and division, these are done from left to right.

In the original exercise, we first handle the division: \(16 \div 2\), before moving on to multiplication and addition. This ensures the expression is simplified correctly, giving the final result of \(-42\).

Navigating Negative Numbers

Negative numbers can be tricky but are essential in algebraic simplification. They often appear in expressions, requiring careful handling to avoid mistakes.

When you see a negative sign before a number, it indicates the number’s direction on the number line is opposite to positive numbers. Here are some rules to help out:

• Addition with Negatives: When adding, combine their absolute values and keep the sign of the number with the higher absolute value. For example, adding \(-10\) and \(-32\) gives \(-42\) since they are both negative.
• Multiplication with Negatives: Negative times a positive equals a negative. And negative times negative equals a positive. In the exercise, \(8(-4)\) gives \(-32\) because the product of a positive and a negative is negative.

Using these principles is essential in achieving the correct result in algebraic expressions.

The Basics of Arithmetic

Arithmetic provides the foundation for understanding more complex mathematical operations. In algebra, basic arithmetic involves simple operations like addition, subtraction, multiplication, and division. Here’s a quick recap:

• Addition: The process of combining numbers together to find their total.
• Subtraction: Involves finding the difference between numbers by "removing" a quantity from another.
• Multiplication: Simplifies repeated addition. For example, \(8 \times (-4)\) involves adding \(-4\), eight times, resulting in \(-32\).
• Division: Opposite of multiplication, it splits a number into equal parts. \(16 \div 2\) divides 16 into 2 equal parts, resulting in 8.

By mastering these basic operations, students can tackle expressions effortlessly, ensuring each step in the process is performed accurately.