Problem 24 - College Algebra

Question

rewrite each expression without absolute value bars. $$ \|-5|-|-13|| $$

Step-by-Step Solution

Answer

The final answer, after rewriting the expression without absolute value bars, is -8.

Step 1: Determine the Absolute Value of -5

The absolute value of a negative number is the number made positive. Therefore, the absolute value of -5 is 5. The expression now reads as: 5 - |-13|.

Step 2: Determine the Absolute Value of -13

Following the same logic, the absolute value of -13 is 13. Substituting into the previous result gives: 5 - 13.

Step 3: Perform subtraction

Subtracting 13 from 5 gives a result of -8.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are often represented with a minus sign (\(-\)). For example, \(-5\) means you have 5 units less than zero. Negative numbers are a fundamental concept in math and are crucial for understanding real-world situations involving loss or debt.

When dealing with negative numbers, remember these key points:

• Negative numbers are always less than positive numbers.
• Adding two negative numbers results in a more negative number.
• When you subtract a larger positive number from a smaller positive number, the result is negative.

In operations, pay close attention to their placement, especially within expressions. Knowing how negative numbers interact is critical for simplifying mathematical expressions.

Mastering Subtraction

Subtraction is the process of taking one number away from another. It is signified by the minus sign (\(-\)). When we perform subtraction with absolute values, it's crucial to determine the values inside the bars first.

Consider the expression simplified to \(5 - 13\) from the solution. It reflects typical subtraction where you subtract 13 from 5.

• Visualize subtraction on a number line: move left when you subtract a positive number.
• In our example, you start at 5 and move 13 steps left to reach -8.
• Remember, subtracting a larger number from a smaller number yields a negative result.

Subtraction involving negative numbers might look tricky, but with regular practice, these steps will become intuitive.

Expression Simplification Essentials

Expression simplification involves reducing complex mathematical expressions into simpler forms. This is often necessary for solving equations and understanding their meaning. Let's look at how you can simplify expressions by removing absolute value bars.

Start by determining the absolute values of each operand in the expression, like \(|-5|\) resulting in 5 and \(|-13|\) resulting in 13. Removing the bars changes the focus from magnitude to actual subtraction operation. Simplification steps include:

• First, calculate or simplify the values within the absolute value bars.
• Then place those positive results back into the arithmetic operation.
• Finally, complete the operation according to the original math operations involved, such as addition or subtraction.

By following this method, you can systematically reduce the complexity of expressions, making them more manageable to solve.