Problem 66 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ [-8+(-5+3)]-[9-(-3-5)] $$

Step-by-Step Solution

Answer

Question: Simplify the expression: [-8 + (-5 + 3)] - [9 - (-3 - 5)] Answer: -27

Step 1: Simplify inside the parentheses

Start by computing the expressions inside the parentheses. Remember that in addition, subtracting a number is equivalent to adding its negative. $$ [-8 + (-5 + 3)] - [9 - (-3 - 5)] \\ [-8 + (-2)] - [9 + 8] $$

Step 2: Combine numbers inside parentheses

Now, combine the numbers inside the parentheses by performing the addition operations. $$ [-10] - [17] $$

Step 3: Subtract the numbers

Lastly, subtract the second number from the first as indicated by the remaining operation. $$ -10 - 17 = -27 $$ The final answer is -27.

Key Concepts

Understanding Addition and Subtraction

When it comes to basic math, addition and subtraction are some of the first operations we learn. They are similar yet distinct processes. Addition involves combining numbers to get a larger number. For example:

• Adding 3 and 5 gives 8, because we start at 3 and count up 5 steps to reach 8.

Subtraction, on the other hand, involves taking away. We find out how much is left when one number is removed from another.

• Subtracting 5 from 8 gives 3, since we start at 8 and count back 5 steps to reach 3.

In operations involving both addition and subtraction, it's crucial to follow the rules of arithmetic accurately to ensure correct answers. Thus, problems with combined operations require keen attention to detail as we perform the steps.

The Role of Parentheses in Arithmetic

Parentheses in arithmetic are like traffic signals for numbers—they guide the order of operations, ensuring calculations are carried out properly. When you spot parentheses in an expression, it means you need to handle the calculations inside them first. Consider the expression \([-8 + (-5 + 3)]\):

• You would start by solving \((-5 + 3)\), resulting in \(-2\).
• Then, incorporate this result back into the expression to get \([-8 + (-2)]\).

Ignoring parentheses can lead to mistakes. They help prioritize operations just like road signs prioritize the flow of traffic. Remember, when operations like multiplication and division are also in play, the rules of BIDMAS/BODMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) tell us parentheses (or brackets) come first!

Demystifying Negative Numbers

Negative numbers are basically the opposite of positive numbers, lying on the left side of the number line. They represent values less than zero and follow the rules of arithmetic, particularly during addition and subtraction, which can seem tricky at first.

• Addition with a negative number: Think of it as moving left on the number line. For instance, adding \(-3\) to 5 lands you on 2.
• Subtraction with a negative number: This is like adding its absolute value. For example, subtracting \(-5\) from 3 actually means adding 5, resulting in 8.

Understanding how negative numbers interact with positive ones, especially within parentheses, aids in solving complex expressions easily. Try visualizing these on a number line for better clarity.