Problem 44 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ -18+(-6)+(-40) $$

Step-by-Step Solution

Answer

The sum of -18, -6, and -40 is -64.

Step 1: Understand the Problem

We need to add three negative numbers: \(-18\), \(-6\), and \(-40\). Adding negative numbers means moving further in the negative direction on the number line.

Step 2: Add the First Two Numbers

First, add \(-18\) and \(-6\). To do this, simply add their absolute values and keep the negative sign: \((-18) + (-6) = -(18 + 6) = -24\).

Step 3: Add the Result to the Third Number

Now, take the result from Step 2, which is \(-24\), and add \(-40\) to it. Again, add their absolute values and keep the negative sign: \((-24) + (-40) = -(24 + 40) = -64\).

Step 4: Write the Final Answer

The final result of adding all three numbers \(-18\), \(-6\), and \(-40\) is \(-64\).

Key Concepts

Understanding Absolute Value

When dealing with numbers, especially negative ones, it's crucial to understand the concept of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction.
For example:

• The absolute value of \(-18\) is \(18\).
• The absolute value of \(-6\) is \(6\).
• The absolute value of \(-40\) is \(40\).

These values are always positive because they represent distance, not direction. Knowing this can help you understand why we can add the absolute values of negative numbers together when summing them. It simplifies calculations by converting negative numbers to their "pure distance" from zero, making it easier to combine them.

Navigating the Number Line

The number line is a fundamental tool used in mathematics to visualize and understand the position and value of numbers. It extends infinitely in both directions, with zero positioned in the center. As you move to the right, numbers increase positively. Moving to the left denotes decreasing numbers, which are negative.
When adding negative numbers, think of it as moving further to the left on the number line. Each negative addend moves you even further from zero:

• Adding \(-18\) means starting at zero and moving 18 units left.
• Adding another \(-6\) means moving an additional 6 units left.
• Finally, adding \(-40\) continues further 40 units left.

Together, these movements sum up to a total shift in position, representing the combined negative value, such as \(-64\) in the original example.

Moving in the Negative Direction

Adding negative numbers involves understanding movement in the negative direction on the number line. It's akin to taking steps backward or moving in reverse.
Each negative number you add increases your distance backwards from zero:

• Starting with \(-18\) requires stepping 18 units backward.
• Adding \(-6\) requires 6 more backward units, resulting in \(-24\).
• Adding \(-40\) then takes us an additional 40 steps away from zero.

By adding these numbers, you increase your backward movement each time. Thus, when you sum \(-18 + (-6) + (-40)\), you're effectively compounding your movement away from zero, culminating in the negative result of \(-64\). This visualization reinforces the concept of negative numbers behaving like steps in the opposite direction.