Problem 53 - Elementary Algebra

Question

Determine each of the values, \(|-(2 \cdot 3)|\)

Step-by-Step Solution

Answer

Answer: 6

Step 1: Solve the expression inside the absolute value

First, we have to solve the expression inside the absolute value function, which is \(-(2 \cdot 3)\). To do this, we multiply 2 and 3 and then apply the negative sign on the result. \(-(2 \cdot 3) = -(6) = -6\)

Step 2: Find the absolute value

Now that we have the result of the expression inside the absolute value function, we need to find the absolute value of that result, which is \(|-6|\). The absolute value of a number is its distance from 0 on the number line, regardless of the direction. In this case, the distance of -6 from 0 is 6. The answer is \(|-(2 \cdot 3)| = 6\).

Key Concepts

Solving Algebraic Expressions

When tackling algebraic expressions, it's essential to understand the step-by-step process you must follow to find the solution. Let us take the expression \( |-(2 \cdot 3)| \) as an example.Firstly, you need to address any operations inside parentheses or absolute value symbols. Multiplication, one of the primary operations in algebra, is considered before subtraction or addition due to the order of operations, commonly known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).With our example expression, \( -\(2 \cdot 3\) \) asks us to multiply 2 by 3 first. Then, because multiplication in algebra follows the same rules as in basic arithmetic, we multiply to get 6, and then apply the negative sign, yielding \( -6 \).Understanding each step clearly and in the right sequence is key to solving algebraic expressions effectively.

Multiplication in Algebra

Multiplication in algebra can sometimes seem intimidating, but it follows the same principles as multiplication with ordinary numbers. The crucial point to remember when multiplying in algebra is to perform the multiplication of coefficients (numeric parts) and then to consider the variables (algebraic parts) if present. This is under the assumption that the variables do not have exponents or are not part of a more complex expression.

Applying Multiplication to Negative Numbers

When multiplying a positive number by a negative number, the result is always negative. In our exercise, we multiplied 2 (a positive number) by 3 (also a positive number), and then applied the negative sign afterward, symbolizing a multiplication by -1. Thus, \( 2 \cdot 3 = 6 \) becomes \( -1 \cdot 6 = -6 \).Being thorough with your multiplication, especially with signs, ensures accurate results in algebra.

Properties of Absolute Value

The absolute value function is a fundamental concept in algebra that measures the distance a number is from zero on the number line, irrespective of the direction. This distance is always expressed as a non-negative number. For example, the absolute value of both -6 and 6 is 6, even though they lie on opposite sides of zero on the number line.

Absolute Value of Negative Numbers

One common misunderstanding is when dealing with negative numbers within absolute value symbols. No matter the initial sign of the number, once you apply the absolute value, the outcome is positive because we're talking about distance—and distance cannot be negative. When working with \( |-6| \), we determine the distance of -6 from 0, which is 6.Understanding the properties of absolute value not only aids in solving homework problems but also lays the groundwork for more complex mathematical concepts in functions and calculus.