Problem 64 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ (-4+7)-(2-5) $$

Step-by-Step Solution

Answer

Answer: The result of the expression is 6.

Step 1: Solve the operations inside the parentheses

First, we will solve the two expressions inside the parentheses: 1. The first expression is: (-4 + 7) 2. The second expression is: (2 - 5)

Step 2: Calculate the first expression (-4 + 7)

To find the result of the first expression, we add -4 and 7: $$ -4 + 7 = 3 $$

Step 3: Calculate the second expression (2 - 5)

To find the result of the second expression, we subtract 5 from 2: $$ 2 - 5 = -3 $$

Step 4: Subtract the two results

Now, we have the results of the two expressions inside the parentheses (3 and -3). We need to subtract the second result from the first result: $$ 3 - (-3) = 3 + 3 = 6 $$ So, the final result of the given expression is 6.

Key Concepts

Expression Evaluation

Expression evaluation is the process of finding the value of a mathematical expression. In algebra, an expression is a combination of numbers, variables, and operation symbols. Evaluating an expression means performing all the necessary operations to simplify the expression to a single value.

Expressions can be complex or simple. They often include operations such as addition, subtraction, multiplication, and division. The order in which we perform these operations is crucial to getting the correct result. In our exercise, the expression \(-4 + 7 - (2 - 5)\) needs to be evaluated. This involves not only performing operations but also understanding how different parts of the expression interact.

Integer Operations

Integer operations involve mathematical tasks performed on whole numbers. Whole numbers include negative numbers, positive numbers, and zero. Let's go through some key points:

• **Addition of Integers**: When adding two integers, pay attention to their signs. If numbers have different signs, find the difference and keep the sign of the larger absolute value. For example, in the expression \(-4 + 7\), a positive (+7) and a negative (-4) number are added, resulting in a positive 3.
• **Subtraction of Integers**: Subtraction involves adding the opposite of the number. For 2 - 5, think of it as 2 + (-5). So, the result is -3, as we add a negative five to two.

Understanding these basic operations with integers allows you to solve expressions step-by-step accurately.

Parentheses Manipulation

Parentheses manipulation is essential when evaluating expressions. Parentheses indicate which operations should be performed first. This is part of the order of operations in mathematics, often taught using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)).

When facing the expression from our exercise, we handle the parentheses to simplify the operations:

• First, find the result of \((-4 + 7\), which is within the parentheses: 3.
• Next, solve \((2 - 5)\), within its own set of parentheses: -3.
• Finally, substitute the results back into the expression to complete the evaluation, performing \(3 - (-3)\), which translates to \(3 + 3\) and results in 6.

By correctly handling parentheses, you ensure that all mathematical operations are conducted in the intended sequence, which is crucial for accurate outcomes.