Problem 90 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the value of each expression. $$|24-(-16)|-|-51-(-31+2)|$$

Step-by-Step Solution

Answer

The result is 22.

Step 1: Resolving the inner brackets

To begin, you can perform the calculations inside the inner brackets. The operation of subtraction becomes addition when the second number is negative because two minuses make a plus. This simplifies to:\( |24 - (-16)| - | -51 - (-33) | \)

Step 2: Perform Calculations Inside Absolute Values

Next, perform the arithmetic within the absolute value symbols. So it forms: \( |24 + 16| - | -51 + 33| \)

Step 3: Find Absolute Values

Calculate the results inside the absolute value symbols and then find the absolute values:\( |40| - |-18| \)

Step 4: Subtract Results

Now, subtract the second absolute value from the first to get the final answer:\( 40 - 18 \)

Key Concepts

Subtraction: Understanding the Basics

Subtraction is one of the four fundamental arithmetic operations. It helps you find out how much is left when you take away a certain number from another. Think of it like this: if you have an apple and you give it away, you have none left! In mathematical terms, subtraction is expressed as two numbers separated by a minus sign (-). For example, in the expression \(24 - (-16)\), subtraction involves an additional rule which is crucial when negative numbers come into play.

When subtracting a negative number, the operation turns into an addition. This happens because two minus signs together make a plus. Understanding this concept is key to solving complex expressions involving negative numbers. Practice will make perfect here, so try different problems and observe how subtraction behaves.

Negative Numbers: A Friendly Introduction

Negative numbers can seem tricky at first, but once you get to know them, they fit in naturally like any other number. Imagine a thermometer. Zero is the freezing point, and numbers below zero represent temperatures colder than freezing. Similarly, in arithmetic, negative numbers are those less than zero.

When dealing with negative numbers, keep these things in mind:

• A negative sign before a number indicates that it’s below zero.
• Subtracting a negative number changes the operation to addition.
• Addition and subtraction rules apply, even when negatives are involved; it’s all about knowing when the operation changes.

In our expression, understanding negative numbers allows us to simplify the expression \(24 - (-16)\) to \(24 + 16\). Recognizing this pattern will help you manage more complex expressions with confidence.

Basic Arithmetic: Building Blocks of Math

Basic arithmetic involves simple operations like addition, subtraction, multiplication, and division. It forms the foundation for all higher-level math concepts. By mastering these operations, solving complex problems becomes a lot easier.

Let's see how basic arithmetic comes into play:

• The expression \(24 + 16\) is the result of understanding that subtraction of a negative number becomes addition.
• Similarly, resolving the expression \(-51 + 33\) involves straightforward subtraction which results in a negative number.
• Finally, the equation \(40 - 18\) leads us to the answer through plain subtraction.

Learning arithmetic is a gradual process. Start with understanding each operation individually. Practicing different scenarios will strengthen your ability to manage numbers efficiently.