Problem 18 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$4+(-11)$$

Step-by-Step Solution

Answer

The result of combining \(4 + (-11)\) is \(-7\).

Step 1: Identify the Numbers

The expression given is \( 4 + (-11) \). There is one positive number, \(4\), and one negative number, \(-11\).

Step 2: Apply the Rule for Addition

When combining a positive number and a negative number, subtract the smaller absolute value from the larger absolute value. In this case, subtract \(4\) from \(11\), because \(11\) is larger.

Step 3: Calculate the Absolute Difference

Calculate \(11 - 4 = 7\). This gives us the absolute value of the result.

Step 4: Determine the Sign

Since the larger absolute value is from the negative number \(-11\), the result will take the negative sign. Thus, the result is \(-7\).

Key Concepts

Positive and Negative Numbers

When dealing with positive and negative numbers, it's helpful to understand how they are positioned on the number line. Positive numbers are found to the right of zero; they increase as you move further right. Negative numbers, on the other hand, are located to the left of zero and become more negative as they extend leftwards.

When you add a positive number and a negative number together, the numbers essentially "balance" each other out. It's similar to taking a step forward with a positive number and then stepping backward with a negative number. The direction you end up going in (positive or negative) depends on which number has a larger absolute value. This balance is a crucial concept to grasp for interior operations.

Absolute Value

Absolute value is a concept that helps in understanding the true distance of a number from zero on the number line, without considering its direction. In mathematics, the absolute value of a number is denoted with vertical bars, like this: \(|x|\).

- For a positive number or zero, the absolute value is the number itself.- For a negative number, the absolute value is its opposite positive number.
For example, the absolute value of \(4\) is \(|4| = 4\), and for \(-11\), it's \(|-11| = 11\).

When combining numbers such as in addition, we often compare their absolute values to determine which has a more substantial effect on the overall result.

Integer Operations

Getting comfortable with integer operations is foundational for understanding math concepts. Here's how you add positive and negative integers:

• Identify if the numbers are positive or negative.
• Find the absolute values of both numbers to see which is greater.
• Subtract the smaller absolute value from the larger absolute value.
• The result takes the sign of the number with the larger absolute value.

In the exercise, with \(4 + (-11)\), you have a positive \(4\) and a negative \(-11\).

Subtract the absolute values: \(11 - 4 = 7\). Here, the absolute value \(11\) (from \(-11\)) is bigger, making the result \(-7\). This rule of thumb helps in getting accurate results in integer operations.