Problem 74 - Elementary Algebra

Question

Find the sums for the the following problems. \(14+[(-3)+5]\)

Step-by-Step Solution

Answer

Answer: The sum of the numbers in the expression is 22.

Step 1: Simplify inner brackets

First, we need to simplify \((-3) + 5\). Since the numbers have different signs, we find their difference and keep the sign of the larger number. In this case, |5-(-3)|=5+3=8 and the sign is positive. So, we simplify the expression to: \(14 + [8]\).

Step 2: Proceed with the final operation

Next, we perform the addition of 14 and 8: \(14+8=22\). So the sum for the given expression is 22.

Key Concepts

Understanding Order of Operations

When solving algebraic expressions, it's essential to follow the order of operations. This ensures that mathematical expressions are evaluated correctly and consistently. You may have heard of the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

This sequence tells us how to approach an expression with multiple operations:

• Parentheses: Always start by simplifying expressions inside parentheses or brackets first.
• Exponents: Next, calculate any exponential expressions.
• Multiplication and Division: These operations are on the same level, so solve them from left to right.
• Addition and Subtraction: Lastly, perform addition and subtraction from left to right.

In this specific problem, the expression inside the brackets was simplified first as per the order of operations, which then allowed us to add the remaining numbers.

Mastering Integer Addition

Integer addition involves combining whole numbers, which include positive numbers, negative numbers, and zero. It is a fundamental skill that is required for solving more complex algebraic expressions.

When adding two numbers, consider their signs:

• Same Sign: If both numbers are positive, add their absolute values and the result is positive. If both are negative, add their absolute values and keep the negative sign.
• Different Signs: Subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

In our exercise, we added (-3) and 5. These numbers have different signs, so we subtracted 3 from 5, resulting in 2, and kept the positive sign, because 5 has a larger absolute value than -3.

This understanding of integer addition is crucial for simplifying expressions correctly.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest or most concise form. This process makes it easier to perform further calculations and understand the expression's value.

Here are the general steps:

• Combine Like Terms: Look for terms that have the same variable raised to the same power, and combine them.
• Simplify the Operations: Use the order of operations to simplify any addition, subtraction, multiplication, or division remaining in the expression.
• Simplify Inside Parentheses: Always solve the expressions inside the parentheses before dealing with the rest of the expression.

In the given example, simplifying involves reducing the contents of the inner expression (-3) + 5 before proceeding to the outer addition with 14.

Simplifying first makes the final steps more straightforward and reduces the risk of errors.