Problem 48 - Beginning and Intermediate Algebra

Question

Find each sum. $$ [-8+(-3)]+[4+(-6)] $$

Step-by-Step Solution

Answer

The sum is -13.

Step 1: Simplify the First Sum

First, simplify the expression inside the first pair of brackets: [-8 + (-3)] Combine the numbers: [-11].

Step 2: Simplify the Second Sum

Next, simplify the expression inside the second pair of brackets: [4 + (-6)] Combine the numbers: [-2].

Step 3: Add Both Results

Now add the results from Step 1 and Step 2: [-11] + [-2] Combine the numbers: [-13].

Key Concepts

Combining Integers

When combining integers, keep the following rules in mind:

• If the integers have the same sign, add their absolute values. The result takes the common sign.
• If the integers have different signs, subtract the smaller absolute value from the larger. The result takes the sign of the number with the larger absolute value.

In our given exercise, you can see these rules in action. First, we combined \([-8 + (-3)]\). Both integers are negative. So we add their absolute values: 8 and 3. This gives us 11, and since both numbers were negative, the result is \(-11\).
Next, we combined \([4 + (-6)]\). One integer is positive, and the other is negative. We subtract the smaller absolute value (4) from the larger absolute value (6), giving us 2. Since -6 has a larger absolute value, the result is \(-2\).
Finally, we add the results from the first two sums: \([-11 + (-2)]\). Both are negative, so we add their absolute values (11 and 2) to get 13, and since both numbers were negative, the final result is \(-13\).

Negative Numbers

Negative numbers can sometimes be confusing, but understanding them is crucial for working with integers. A negative number is any number less than zero, which is expressed with a minus sign (-) in front. When you add two negative numbers, as we did in the example \([-8 + (-3)]\), you're essentially adding their distances from zero, but the result remains in the negative spectrum.
If you add a positive number to a negative number, like \([4 + (-6)]\), think of it as moving on a number line. Moving to the right (positive direction) and then to the left (negative direction), you'll land somewhere based on which 'movement' (positive or negative) was larger.
For negative numbers:

• Adding two negative numbers keeps the result negative.
• Subtracting a negative number is the same as adding its positive equivalent.

Remember these rules, and dealing with negative numbers will become much more intuitive!

Simplifying Expressions

Simplifying expressions means making them as simple as possible. This often involves combining like terms and reducing any unnecessary complexity. In our exercise, we simplified each sum inside its brackets before combining the results.
For the expression \([-8 + (-3)]\), we combined the terms to \(-11\). Similarly, for \([4 + (-6)]\), we simplified it to \(-2\). After that, it's just a matter of adding these two simplified numbers together.
Simplifying is about breaking down the problem into smaller, more manageable parts. Always start by handling operations inside parentheses first, then apply the rules for combining like terms.
Important steps in simplifying:

• Combine like terms - numbers that are alike.
• Use arithmetic rules for addition, subtraction, multiplication, and division.

By following these steps, you ensure your final answer is straightforward and easy to understand.