Problem 58 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ [4+(-11)]-[2+(-10)] $$

Step-by-Step Solution

Answer

Answer: 1

Step 1: Evaluate the inner parentheses

First, we need to evaluate the terms inside each set of parentheses. We have two sets of parentheses in this exercise, so let's start with the first one and then evaluate the second one. $$ [4+(-11)] = 4 - 11 = -7 $$ Now, let's evaluate the second set of parentheses: $$ [2+(-10)] = 2 - 10 = -8 $$ Now our expression looks like this: $$ (-7) - (-8) $$

Step 2: Perform the subtraction operation

Now, we need to perform the subtraction operation: $$ (-7) - (-8) $$ Remember that when you subtract a negative number, it is the same as adding the absolute value of that number. So this is equivalent to: $$ -7 + 8 $$ Now it's easier to perform the operation: $$ -7 + 8 = 1 $$

Step 3: Write the final answer

At this point, we have completed all the operations in the given exercise. The final answer is: $$ 1 $$

Key Concepts

Addition and Subtraction

Addition and subtraction are fundamental arithmetic operations that you will frequently use in daily math exercises. When dealing with addition, you are essentially bringing quantities together. On the other hand, subtraction involves taking quantities away.

• When you encounter problems involving both operations, tackle them left to right, as you find them in the expression.
• Sometimes, parentheses indicate which operations to do first. For example, in \([4 + (-11)] - [2 + (-10)]\), focus on what's inside the parentheses first.

This careful handling of each operation ensures you keep track of positive and negative shifts in value. Understanding how to manipulate expressions using addition and subtraction lays a strong foundation for handling more complex mathematical tasks.

Negative Numbers

Negative numbers are numbers less than zero, often represented with a minus sign (-). They appear in various contexts, such as temperatures below zero, or when accounting for debts.

• When you add a negative number like \(4 + (-11)\), this is the same as subtracting that number's absolute value: \(4 - 11 = -7\).
• In subtraction involving negative numbers, such as \((-7) - (-8)\), remember that subtracting a negative is equivalent to adding the positive version of that number.

This understanding helps simplify expressions, as knowing the effect of the negative sign will guide you through arithmetic without confusion. Paying attention to negative numbers is crucial as it affects the final outcome of mathematical expressions significantly.

Order of Operations

The order of operations is a set of rules that standardize which calculate first. It ensures consistency in mathematical expressions, avoiding potential chaos. A common acronym for recalling the order is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• First, solve operations within parentheses, like \([4+(-11)]\), which means you need to compute \(4 - 11\) before doing anything else.
• Next, manage any exponents if they exist. In this problem, there are none, so you move directly to any multiplication or division—but not in this context.
• Finally, handle any addition or subtraction from left to right in the expression.

Using the correct order of operations avoids mistakes in evaluation. Applying each step systematically assures you of arriving at the correct result each time.