Problem 68 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ -(10-6) $$

Step-by-Step Solution

Answer

Answer: The value of the expression \(-(10-6)\) is \(-4\).

Step 1: Solve the inner expression

In line with the order of operations, we will first solve the expression inside the parentheses. We have: $$(10-6)$$ Which is a subtraction. So, we subtract 6 from 10: $$10 - 6 = 4$$

Step 2: Apply the negation

Now, the expression becomes: $$-(4)$$ To find the value, we just need to negate the value inside the parentheses. Negating 4 gives us: $$-(4) = -4$$ Thus, the value of the expression \(-(10-6)\) is \(-4\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental element in algebra that combine numbers, variables, and mathematical operators to describe relationships and values. Much like a sentence in a language, an algebraic expression conveys a mathematical idea. Different parts of an expression are combined by operations like addition, subtraction, multiplication, and division.

For example, in the expression \(10 - 6\), we have two numbers, 10 and 6, combined with a subtraction operation. Understanding expressions is key to mastering algebra because they are the basis upon which more complex algebraic problems are built. Evaluating an algebraic expression often involves following the established order of operations, PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Negation, a concept we'll handle separately, represents the inversion of sign of a number within such expressions.

Negation in Algebra

Negation in algebra is a critical concept that involves changing the sign of a number. It's essentially a flipping operation, where positive values become negative and vice versa. When we speak of negating a number or an expression in algebra, we take its additive inverse. The symbol for negation is typically a minus sign \( - \).

Consider the expression \( -(10 - 6) \). After simplifying the content within the parentheses to 4, the negation symbol outside the parentheses tells us to take the opposite of 4, which is -4. It's important to distinguish between subtraction and negation; subtraction involves taking away value, while negation simply inverts it. This concept becomes even more interesting when negating an already negative value, as this would result in a positive value. Clear understanding of negation is essential not only in basic algebra but also in solving equations and inequalities.

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics. They include addition, subtraction, multiplication, and division, and are used to perform calculations within algebraic expressions and equations. In our example, the subtraction operation simplifies \(10 - 6\) to zero. Once you've completed operations within parentheses, as stipulated by the order of operations, you can proceed with the remaining operations in sequence.

When solving algebraic expressions, it's crucial to perform arithmetic operations in the correct order to arrive at the correct solution. Mistakes in this area can lead to incorrect answers and confusion. Therefore, practice with arithmetic operations not only reinforces your computational skills but also prepares you for the more abstract concepts encountered in algebra.