Problem 48 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ |7+(-17)| $$

Step-by-Step Solution

Answer

The absolute value is 10.

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. Therefore, the absolute value of a negative or positive number is always positive.

Step 2: Perform the Addition Inside the Absolute Value

Calculate the expression inside the absolute value first. Here, you have to add \(7\) and \(-17\). Performing this addition gives: \(7 + (-17) = -10\).

Step 3: Apply the Absolute Value

Now, apply the absolute value to the result of the addition. Since the result \(-10\) is a negative number, take the positive of \(-10\) to get its absolute value: \(|-10| = 10\).

Key Concepts

Integer Addition

Adding integers involves understanding how positive and negative numbers interact. The key is to consider their direction on the number line.
For instance:

• When adding a positive integer and a negative integer, you essentially subtract the smaller from the larger and keep the sign of the larger number.
• When both integers are positive, your result is straightforward: simply add the values together.
• Conversely, if both integers are negative, you add their absolute values and then assign a negative sign to the result, because you moved left on the number line.

The operation we see in the example is adding 7 and -17. Here, you effectively subtract 7 from 17, resulting in -10, because 17 (the negative part) is larger.

Negative Numbers

Negative numbers represent values less than zero. These numbers can be thought of as the opposites of their positive counterparts.
On a number line:

• Negative numbers are located to the left side of zero.
• They decrease in value as you move further left.

Understanding negative numbers is crucial when performing integer operations, such as addition. When including negative numbers, you are simply adding in reverse. For example, with -17, you're starting from zero and moving 17 units leftwards.

Number Line

A number line is a visual representation that helps us understand numbers and their operations relative to zero.
Key characteristics include:

• Numbers increase in value as you move to the right.
• Numbers decrease in value when moving to the left.
• Zero is the central reference point that separates positive numbers from negative numbers.

When adding integers on a number line, you start at the first number's position, then move horizontally according to the second number’s magnitude and direction. For our example of adding 7 and -17, start at 7, and move 17 units left to arrive at -10.

Distance from Zero

The absolute value is defined as a number's distance from zero on a number line, without considering which side of zero the number lies. This distance is always a non-negative figure.
Considerations when interpreting absolute value:

• Positive numbers maintain their value in absolute terms because they are counted right of zero.
• Negative numbers lose their sign, turning into positive numbers since distance cannot be negative.

In the example, the result of 7 + (-17) was -10. Thus, the absolute value of -10 is simply 10, indicating it is 10 units away from zero on the number line.