Problem 22 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-14+7$$

Step-by-Step Solution

Answer

The result of \(-14 + 7\) is \(-7\).

Step 1: Identify the Numbers

We have the numbers \(-14\) and \(+7\). Notice that these are both integers, with one negative and one positive.

Step 2: Apply Addition Rule for Opposite Signs

When adding numbers with opposite signs (one negative and one positive), subtract the smaller absolute value from the larger absolute value, and take the sign of the number with the larger absolute value.

Step 3: Calculate Absolute Values

Find the absolute values: \(|-14| = 14\) and \(|7| = 7\).

Step 4: Subtract Absolute Values

Subtract the smaller absolute value from the larger: \(14 - 7 = 7\).

Step 5: Determine the Sign of the Result

Since the number with the larger absolute value is \(-14\), which is negative, the final result will also be negative.

Step 6: Write the Result

Combine everything to form the final result: \(-14 + 7 = -7\).

Key Concepts

Understanding Absolute Value

When discussing integers, understanding the concept of absolute value is very important. Absolute value refers to the distance a number is from zero on a number line, without considering its direction. Thus, absolute value is always a non-negative number.

Let's look at a couple of examples:

• The absolute value of \(-14\) is \(14\), since it is \(14\) units away from zero on the number line.
• Similarly, the absolute value of \(7\) is \(7\).

By finding the absolute values, you're essentially stripping the numbers of their signs and seeing their 'size' relative to zero. This makes it simpler to work with them when performing addition or subtraction.

Working with Opposite Signs

When dealing with addition where the numbers have opposite signs, the process can feel a bit tricky at first, but it's simpler than it seems! Numbers with opposite signs mean one is positive, and the other is negative.

Here's the trick:

• Determine the absolute value of each integer.
• Subtract the smaller absolute value from the larger one.
• Assign the sign of the number with the larger absolute value to the result.

For instance, in the original exercise \(-14 + 7\), since \(|-14| = 14\) and \(|7| = 7\), you would subtract \(7\) from \(14\), giving the result \(7\). Since the larger absolute value belongs to \(-14\), the result is \(-7\). Understanding this makes it easier to tackle any problem involving integers with different signs.

Applying the Integer Addition Rule

The integer addition rule is essential when you’re adding numbers with either the same or opposite signs.

Here’s a quick breakdown based on sign:

• **Same Sign**: Add their absolute values together, keeping the common sign. For example, \(+3 + (+5) = +8\).
• **Opposite Signs**: Subtract the smaller absolute value from the larger and take the sign of the integer with the larger absolute value, as was done for \(-14 + 7\) to get \(-7\).

By adhering to this rule, any addition problem involving integers can be simplified to a basic subtraction of absolute values, then assigning the correct sign. This clarity allows you to handle more complex calculations with confidence.