Problem 73 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ -2[(4-8)-(5-11)] $$

Step-by-Step Solution

Answer

Answer: The value of the expression is -4.

Step 1: Simplify the expression inside the two parentheses

First, we simplify the two parentheses separately, starting with (4-8) and (5-11). $$ (4-8) = -4 $$ $$ (5-11) = -6 $$

Step 2: Subtract the results inside the brackets

Now, we will substitute the values of the parentheses back into the expression and subtract the two results. $$ -2[-4-(-6)] $$ $$ -2[-4+6] $$

Step 3: Simplify the expression inside the brackets

Next, we simplify the expression inside the brackets. $$ -2[2] $$

Step 4: Multiply the result by -2

Finally, we multiply the result inside the brackets by -2. $$ -2 * 2 = -4 $$ The value of the given expression is -4.

Key Concepts

Order of Operations

When solving algebraic expressions, it's crucial to follow the **order of operations** to get accurate results. The order of operations is usually remembered by the acronym PEMDAS, which stands for:

• Parentheses (or brackets)
• Exponents (or powers and roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This rule dictates the sequence in which operations should be undertaken within a mathematical expression.
For the original exercise \(-2[(4-8)-(5-11)]\), we first resolved the parentheses (P) before any other operations. By applying PEMDAS, we ensured that subtraction inside the parentheses was dealt with prior to multiplication outside of it. This systematic approach avoids errors and guarantees consistency in mathematical computations.

Negative Numbers

Understanding **negative numbers** is essential when tackling algebraic expressions. A negative number is any number less than zero and is denoted with a minus sign (-). These numbers signify a decrease or a movement in the opposite direction on a number line.
In the context of our exercise, we dealt with negative results from the operations within the parentheses: \((4-8) = -4\) and \((5-11) = -6\). Recognizing when a subtraction yields a negative number is important.
Furthermore, it is crucial to understand operations involving negative numbers:

• Subtracting a negative is the same as adding its positive counterpart.
• Multiplying two negatives results in a positive number.
• Multiplying a positive by a negative results in a negative.

In the given expression, converting \(-(-6)\) to \(+6\) is a practical application of these principles.

Brackets in Expressions

Brackets, or **parentheses**, play a pivotal role in algebraic expressions by indicating which calculations should be performed first. They enclose operations that need prioritization over others outside of them. Solving the contents inside brackets before dealing with the rest of the expression ensures clarity and accuracy.
Consider the expression in the problem \(-2[(4-8)-(5-11)]\):The operations within the brackets \((4-8)\) and \((5-11)\) take precedence over the multiplication outside because brackets are part of the order of operations hierarchy.

• Evaluate the expression in the innermost brackets first.
• Simplify step by step moving outward from the innermost brackets.

Once the bracketed operations are simplified, their result is used for the subsequent operations. This step-by-step simplification is critical, especially in complex expressions, to avoid errors and ensure the correct outcome.