Problem 69 - Elementary Algebra

Question

Simplify. $$ -2[7-(4+|-7|)] $$

Step-by-Step Solution

Answer

The simplified expression is 8.

Step 1: Solve the Absolute Value

The first step is to solve the absolute value. The expression has a term \(|-7|\) which simplifies to 7 because the absolute value of -7 is 7.

Step 2: Perform the Inner Parentheses Calculation

Substitute the absolute value result into the expression: \ \(-2[7-(4+7)]\). \ Now solve the inside of the parentheses: \ \(4+7 = 11\).

Step 3: Solve the Brackets

Replace the inner expression with its calculated result: \ \(-2[7-11]\). \ Then, calculate the expression inside the brackets: \ \(7-11 = -4\).

Step 4: Multiply by the Coefficient

Now, multiply the result of the bracket by the coefficient outside: \ \(-2 imes -4 = 8\).

Key Concepts

Algebra and Absolute Value

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and operations. It forms the foundation of higher mathematics and helps in solving real-world problems by finding unknown values. One crucial aspect of algebra is understanding absolute value, which is always non-negative and represents the distance of a number from zero on the number line.

• The absolute value is denoted by two vertical bars, for example, \(|x|\).
• It is the same as the number itself if the number is positive.
• If the number is negative, the absolute value is the positive version of that number.

In the exercise, \(|-7|\) simplifies to 7 because the absolute value of \(-7\) is its distance from zero, which is 7, a non-negative result. This initial step ensures the expressions are correctly handled before further simplifications.

Simplifying Expressions

Simplifying expressions is a fundamental process in algebra wherein complex expressions are reduced to their simplest form. This often involves executing basic arithmetic operations and combining like terms.

• Breaking down expressions into simpler parts makes them easier to manage.
• It allows clearer insight into the problem and the final outcome.

For example, in the given exercise, once the absolute value is resolved, the expression becomes easier to understand and manipulate: \(-2[7-(4+7)]\). By performing addition within the parentheses, namely \(4 + 7 = 11\), the expression is simplified further to \(-2[7-11]\), a much clearer form ready for the next operations. This approach reduces the complexity of calculations and aids in systematic progression toward the solution.

Order of Operations

Order of operations is a set of rules that is essential to consistently simplifying mathematical expressions. It provides a guideline for which parts of an expression to solve first, second, and so on, ensuring accuracy.

• These rules are often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Operations within parentheses are performed first.
• The multiplication and division follow, from left to right, and finally, addition and subtraction are solved last, also from left to right.

In our exercise, this order is crucial. After the absolute value was resolved, the expression within the parentheses \(7 - 11\) is handled, leading to \(-4\). Legally, the multiplication step \(-2 \times -4\) wraps up the entire calculation to get the final result, 8. By following this order precisely, we guarantee that our solution is both systematic and correct.