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Chapter 6: Problem 19

\(x-2\) and \(2-x\) are called _____ of each other.

Short Answer

Expert verified
The terms \(x-2\) and \(2-x\) are called 'Negatives' or 'Opposites' of each other.

Step by step solution

01

Identify the given expressions

The given expressions are \(x-2\) and \(2-x\).
02

Recognize relationship between expressions

Notice that the two expressions have the same numeric values but the order is reversed and the sign in between the terms has changed from subtraction to addition, or vice versa.
03

Define the mathematical term

Such expressions are called negations or opposites of each other. Thus the term that completes the sentence is 'Negatives' or 'Opposites'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Algebraic expressions are foundational elements in mathematics that contain variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. Understanding algebraic expressions is key to mastering algebra. They allow us to model real-life problems and solve equations.
To create an algebraic expression, identify the components:
  • **Variables**: Symbols, typically letters like 'x' or 'y', representing unknown values.
  • **Constants**: Numbers that have fixed values.
  • **Operators**: Symbols like '+', '-', '*', or '/' indicating arithmetic operations.
For example, in the expression \( x - 2 \), 'x' is a variable and '2' is a constant. The minus sign suggests subtraction. Algebraic expressions help us describe mathematical situations and solve them.
Opposite Numbers
Opposite numbers, also known as additive inverses, are pairs of numbers that, when added together, equal zero. These numbers are vital in algebra because they help in simplifying expressions and solving equations. The concept is straightforward: for any number 'a', its opposite is '-a'.
Consider the expression \( x - 2 \) and its opposite \( 2 - x \). These expressions are related through the concept of opposite numbers because if you rearrange terms and signs, they negate each other:
  • When \( x - 2 \) is solved, think of it as \( x + (-2) \).
  • The opposite \( 2 - x \) is \( 2 + (-x) \).
Both these expressions have terms that are opposites: 'x' and '-x', '2' and '-2'. Mastery of opposites is beneficial for solving equations, particularly in understanding subtraction.
Subtraction in Algebra
Subtraction in algebra is different from simple arithmetic subtraction. It involves not just numbers but also variables. A clear understanding of subtraction rules is important for simplifying expressions and solving equations. Begin with the structure:
  • Subtraction is essentially adding the additive inverse. Hence, \( a - b \) can be rewritten as \( a + (-b) \).
  • In algebraic expressions, ensure you correctly handle the signs of variables and constants.
For example, in expressions like \( x - 2 \) and \( 2 - x \), the subtraction indicates different ordering of terms and signs:
  • \( x - 2 \) suggests starting with 'x' then subtracting '2'.
  • \( 2 - x \) implies starting with '2' then subtracting 'x'.
Subtraction gives a sense of direction in algebra, showing whether a value is increased or decreased. Mastering this concept aids in forming more complex expressions and understanding their transformation.

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Most popular questions from this chapter

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{x^{3}-1}{x^{2}+x+1}$$

Perform the operations. Assume that no denominators are \(0 .\) $$\frac{x}{3 y}+\frac{2 x}{3 y}-\frac{x}{3 y}$$

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{x^{2}-2 x-15}{x^{2}+2 x-15}$$

Refer to the tax deductions listed in the table. Give each ratio in simplest form. $$\begin{array}{lc}\text { Item } & \text { Amount } \\\\\hline \text { Medical } & \$ 995 \\\\\text { Real estate tax } & \$ 1,245 \\\\\text { Contributions } & \$ 1,680 \\\\\text { Mortgage interest } & \$ 4,580 \\\\\text { Union dues } & \$ 225\end{array}$$ Find the ratio of real estate tax deductions to the total deductions.

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{15 x-3 x^{2}}{25 y-5 x y}$$
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