Problem 69 - Elementary Algebra

Question

Find the sums for the the following problems. \([4+(-12)]+[12+(-3)]\)

Step-by-Step Solution

Answer

Answer: The final sum is \(1\).

Step 1: Group the numbers within the parentheses

To solve this problem, start by adding the numbers within each set of parentheses: \((4 + (-12))\) and \((12 + (-3))\).

Step 2: Add the numbers within the first parentheses

For the first set of parentheses, we have \(4 + (-12)\). When adding a negative number, we can rewrite it as subtraction: \(4 - 12\). This gives us \(-8\) as the result of the sum within the first parentheses.

Step 3: Add the numbers within the second parentheses

Now, let's turn our attention to the second set of parentheses: \((12 + (-3))\). We can rewrite this as subtraction, so we have \(12 - 3\). This gives us \(+9\) as the result of the sum within the second parentheses.

Step 4: Add together the sums from the parentheses

Now we have two sums: \(-8\) and \(+9\). When summing these two numbers, we have \(-8 + 9\). Since these numbers have opposite signs, we subtract the smaller absolute value from the larger one and keep the sign of the number with the largest absolute value. In this case, \(|-8| = 8 < 9\). Thus, we have \(9 - 8 = 1\) as the final sum. The final answer is \(1\).

Key Concepts

Addition of Integers

Understanding the addition of integers involves combining positive and negative whole numbers. Integers include both positive numbers (like 5 or 10), negative numbers (like -3 or -11), and zero. When adding integers:

• If the signs are the same, add the absolute values and keep the common sign.
• If the signs are different, subtract the smaller absolute value from the larger one, and keep the sign of the number with the larger absolute value.

For example, in the problem given, adding \(4+(-12)\) involves two different signs. We treat this as a subtraction (4 - 12), resulting in -8.
Similarly, for \(12 + (-3)\), the calculation becomes 12 - 3 with a result of +9. This gives insight into handling such problems systematically.

Parentheses in Math

In mathematics, parentheses are used to indicate operations that should be performed first in calculations. They help specify the order of operations, often denoted by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
For instance, in the expression \[ (4 + (-12)) + (12 + (-3)) \], we must first resolve the operations inside parentheses:

• Calculate \(4 + (-12)\) to get -8.
• Calculate \(12 + (-3)\) to get +9.

Executing operations inside parentheses first ensures clarity and prevents errors in more extended and complex calculations.

Signed Numbers

Signed numbers are numbers that have either a positive (+) or a negative (-) sign. They represent values above and below zero, respectively, and are crucial in calculations involving changes or directional shifts.
When dealing with signed numbers, it is essential to:

• Understand the rules for adding, subtracting, multiplying, and dividing them.
• Recognize that a positive sign often remains unspoken but assumed (e.g., 5 = +5).

In our example, \(4 + (-12)\) and \(12 + (-3)\) both involve understanding how signs affect operations. In \(-8 + 9\), the numbers have different signs, so their absolute values are compared and subtracted (resulting in +1). Recognizing signed numbers simplifies real-life scenarios such as credit (positive) and debt (negative).