Problem 24 - Intermediate Algebra

Question

Perform the operations. See Example 2 . $$ -11-(-17) $$

Step-by-Step Solution

Answer

The result is 6.

Step 1: Identify the Problem

You have an expression \[-11 - (-17)\]that involves subtraction of a negative number.

Step 2: Simplify the Expression

Subtraction of a negative number is equivalent to addition. So, rewrite the expression:\[-11 - (-17) = -11 + 17\].

Step 3: Perform the Addition

Add the numbers by starting with the larger absolute value, 17, and then subtract 11:\[17 - 11 = 6\].

Key Concepts

Negative Numbers

Negative numbers are an important part of arithmetic, often representing values below zero.
In everyday life, negative numbers can be used to describe things like temperature below freezing or debt in finance.

A negative number is typically written with a minus sign (−) in front of it, such as \(-11\). The presence of the minus sign before a number tells us that it is less than zero.

• Negative numbers are less than zero.
• They are opposite to positive numbers, which are greater than zero.
• When comparing two negative numbers, the number with the smaller absolute value is considered greater (e.g., \(-3\) is greater than \(-7\)).

Understanding negative numbers is crucial for learning how certain operations affect them, especially when we subtract or add them.

Subtraction and Addition

Operations involving subtraction and addition of numbers come naturally once you understand the relationship between these operations.

In mathematics, subtraction of a number is fundamentally about finding the difference and can be transformed into an addition, especially when dealing with negative numbers.

Turning Subtraction into Addition

When we see an expression where a negative is being subtracted, such as \(-11 - (-17)\), we can transform it. Subtracting a negative is the same as adding the positive equivalent:

• In the expression \[-11 - (-17)\], you change it to \[-11 + 17\].
• This change simplifies calculations and makes it easier to handle more complex expressions.

Understanding The Process

When adding numbers like \(-11\) and \(17\), consider their absolute values to help:

• The absolute value tells how far a number is from zero.
• Perform the operation based on these absolute values, \(17 - 11 = 6\).

Operations become straightforward once these basic concepts are internalized.

Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction.
Bringing this concept into play can simplify solving equations that involve negative numbers.

• The absolute value is denoted by vertical bars, such as \(|-11| = 11\). It's always a non-negative number.
• Absolute values ignore the sign of the original number, focusing on magnitude.

Utilizing Absolute Values

When performing operations like \(-11 + 17\), looking at the absolute values first (\(|-11| = 11\) and \(|17| = 17\)) helps understand that:

• 17 has a greater absolute value than 11, influencing the outcome of the computation.
• The subtraction process is simplified since \(17 - 11 = 6\), retaining the sign of the number with the larger absolute value.

Understanding absolute values deepens comprehension of other number operations, especially when negatives are involved. This foundational skill will be extremely beneficial across various mathematical problems.