Problem 40 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ |-6|+(-61) $$

Step-by-Step Solution

Answer

The result of the expression |-6| + (-61) is -55.

Step 1: Understand Absolute Value

The first part of the expression is the absolute value, \(|-6|\). The absolute value notation \(|x|\) gives the distance of a number from zero on a number line, which is always positive or zero. Thus, \(|-6| = 6\).

Step 2: Write the Expression with the Absolute Value

Replace the absolute value term in the original expression with its value. The expression becomes: \(6 + (-61)\).

Step 3: Perform the Addition

To add \(6 + (-61)\), observe that adding a negative number is equivalent to subtraction. Simplify the expression to \(6 - 61\), which equals \(-55\).

Key Concepts

Understanding Negative Numbers

Negative numbers are those that are less than zero. They are represented with a minus sign in front. For example,

• -1
• -2

When you visualize numbers on a number line, negative numbers appear to the left of zero. These numbers are the opposites of the positive numbers, such as

• -4, which is the opposite of 4
• -7, the opposite of 7

In operations, negative numbers can sometimes be tricky, especially when mixed with absolute values and addition. It is essential to remember that the absolute value of a negative number turns it into its positive counterpart because absolute value measures the distance from zero, which is always positive.
So, when dealing with expressions like

• \(|-6| + (-61)\)

The first step is handling the absolute value of

• -6 which becomes 6.

This understanding sets the foundation for properly managing operations involving these numbers.

Mastering Addition with Negative Numbers

Adding negative numbers can initially appear confusing, but it follows a straightforward principle. When you add a positive number to a negative number, or vice versa, essentially, you're finding the difference between the two numbers and keeping the sign of the larger number. Let's break it down:

• When a positive number is added to a negative, subtract the smaller absolute value from the larger absolute value, and give the result the sign of the number with the larger absolute value. For example,
  • \(6 + (-61)\) becomes \(6 - 61\), which results in
    • -55.

If both numbers are negative, you add them together just like positive numbers, but the result remains negative. For example,

• -4 + (-5) becomes -9

This knowledge helps bring clarity when simplifying expressions that incorporate negative numbers.

Basic Integer Operations

Integer operations include both positive and negative whole numbers, and knowing how to handle these operations is fundamental in mathematics. Here's a quick overview of how they work when dealing with addition and expressions.

• Positive + Positive: Simply add both numbers together. For example, 3 + 5 equals 8.
• Negative + Negative: Add the absolute values, and keep the negative sign. For example, -2 + (-3) equals -5.
• Positive + Negative: Subtract the smaller absolute value from the larger absolute value, and take the sign of the number with the larger absolute value. For example, 6 + (-3) equals 3.

By mastering these basic addition operations with integers, you'll handle increasingly complex math problems with confidence. Whether you're finding the solution to a straightforward equation like \(6 + (-61)\) or tackling more complex integer equations, this structured approach ensures clarity and accuracy.