Problem 149 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(147-149,\) perform the indicated operation. \(-6+(-3)\) (Section \(1.7,\) Example 4 )

Step-by-Step Solution

Answer

The result of \(-6 + (-3)\) is \(-9\).

Step 1: Understand the problem

The problem is asking to add two negative numbers: \(-6\) and \(-3\).

Step 2: Add absolute values

Instead of adding \(-6\) and \(-3\), add their absolute values. The absolute value of \(-6\) is \(6\) and the absolute value of \(-3\) is \(3\). Therefore, \(6 + 3 = 9\).

Step 3: Assign negative sign

Since the original numbers were negative, the result from step 2 should also be negative. Therefore, the result is \(-9\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. You will often see them used to show a loss, a decrease, or temperatures below freezing. Unlike positive numbers, negative numbers have a minus sign in front of them, for example,

• In finances: A negative balance might show debt, written as \(-50\), meaning you owe $50.
• In temperature: if it is \(-5\) degrees outside, it’s below zero, and it’s quite chilly!

When you're dealing with negative numbers in mathematical operations, keep in mind that:

• Subtracting more means you get "less" especially in respect to negative numbers, for example \(-3\) minus \(-1\) results in \(-2\) minus \(1\), which gives \(-4\).
• Adding two negative numbers together results in a larger negative number as both numbers contribute to moving away from zero in the negative direction.

Absolute Value

The absolute value of a number is its distance from zero on a number line, always expressed as a non-negative number. To find the absolute value of a number, ignore the negative sign if there is one.

• The absolute value of \(-6\) is \(6\) because \(-6\) is 6 units away from zero on the number line.
• The absolute value of \(3\) is \(3\) because it is exactly 3 units away from zero.

Understanding absolute values helps simplify the addition and subtraction of integers. You will be able to see the size of the numbers we are working with, regardless of their sign. This is especially useful when we have two negative numbers. We add their absolute values to comprehend the combined size of both, and then assign the final result with a negative sign.

Addition of Integers

Adding integers might seem complicated at first, but it becomes straightforward once you understand the rules. Here, we will specifically look at adding negative numbers, as with the exercise of adding \(-6 + (-3)\).1. **Consider Absolute Values:** - Focus on the absolute values of each integer. - For \(-6\) and \(-3\), consider \(6\) and \(3\).2. **Add the Absolute Values:** - Simply add the numbers as if they were all positive: \(6 + 3 = 9\).3. **Apply the Correct Sign:** - Since both original numbers were negative, the sum will also be negative. In simpler terms, when adding two negative numbers,

• Think of each as a debt. Combining two debts means you simply owe a larger sum.
• The answer is the negative of the sum of the absolute values.

For the initial example: Adding \(-6 + (-3)\), you'd result in \(-9\) after applying these steps: take \(9\) from the sum of absolute values, and assign it a negative sign.