Problem 27 - Elementary and Intermediate Algebra

Question

Find \(-x\) when \(x\) is each of the following. $$ 0.101 $$

Step-by-Step Solution

Answer

-0.101

Step 1: Identify the given value of x

We need to find \(-x\) given that \(x\) is 0.101.

Step 2: Apply the negation

To find \(-x\), we multiply the given value of \(x\) by -1. \(-x = -1 \times 0.101\).

Step 3: Simplify the multiplication

Perform the multiplication: \(-1 \times 0.101 = -0.101\).

Key Concepts

Negation

Negation is a fundamental concept in mathematics, especially when dealing with negative numbers operations. It involves changing the sign of a number. For example, if you have a positive number like 0.101, its negation will be -0.101. Essentially, negating a number means multiplying it by -1.
With negation, you can turn positive numbers into negative ones and vice versa. This concept becomes useful in algebra where we often need to find the opposite value of a given number. Do not confuse negation with subtraction; while they might seem similar, they serve different purposes.

Multiplication

Multiplication is another core concept, vital to understanding operations with negative numbers. Here, to find \(-x\), we multiply a number by -1. For example, when multiplying 0.101 by -1, you get -0.101.
The rules for multiplication with negative numbers are straightforward:

• A positive number times a negative number equals a negative number, e.g. \(0.101 \times -1 = -0.101\).
• A negative number times a negative number equals a positive number, e.g. \(-0.101 \times -1 = 0.101\).

Understanding these rules can help avoid errors when working through algebra problems.

Simplification

Simplification involves making an expression easier to understand or work with by performing arithmetic operations and reducing its components. For instance, when the given problem involves multiplying 0.101 by -1 to find -x, simplifying the multiplication \(-1 \times 0.101\) results in -0.101.
Simplifying expressions is crucial in algebra to make solving equations and other operations more manageable. It often involves combining like terms, reducing fractions, or performing arithmetic to arrive at a final, simplified form. Remember, correctly simplifying steps prevents mistakes and ensures accurate solutions.

Algebra Basics

Understanding algebra basics is essential for tackling problems involving negative numbers and their operations. At its core, algebra consists of manipulating symbols and numbers to solve equations. When working with expressions, you might need to apply operations such as negation, multiplication, and simplification.
For example, in the exercise, finding -x when x is a positive number involves these basic algebra steps: identifying the given number, applying the operation (negation by multiplying by -1), and simplifying the result. Algebra basics form the foundation for more advanced mathematical concepts and problem-solving techniques. Developing a strong grasp of these basics will support you in handling more complex algebraic expressions and equations in the future.